Datasets:

reichenbach

/

arxiv_ppr_embeds

like 0

the detection of diffuse synchrotron emission ( radio halos ) on mpc scales in an increasing number of galaxy clusters provides good evidence for a distributed magnetic field of @xmath0gauss strength in the hot intracluster medium ( icm ; see e.g. @xcite for a review ) . imaging of faraday rotation of linearly - polarized radio emission from embedded and background sources confirms that there are fields associated with thermal plasma along lines of sight through the clusters ( e.g. @xcite ) . observations of faraday rotation can also be made for radio galaxies in sparser environments , allowing the study of magnetic fields in environments too sparse for radio halos to be detected ( e.g.@xcite ) . the faraday effect @xcite is the rotation suffered by linearly polarized radiation travelling through a magnetized medium , and can be described by the two following relations : @xmath1}= \psi(\lambda)_{[{\rm rad}]}~-~\psi_{0~[{\rm rad}]}=\lambda^2 _ { [ { \rm m}^2]}~{\rm rm}_{[{\rm rad\,m}^{-2}]},\ ] ] with @xmath2}=812\int_{0}^{l_{[{\rm kpc}]}}n_{\rm{e}~[{\rm cm}^{-3}]}b_{z~[\mu{\rm g}]}dz_{[{\rm kpc}]}\ , , \label{equarm}\ ] ] where @xmath3 and @xmath4 are the @xmath5-vector position angle of linearly polarized radiation observed at wavelength @xmath6 and the intrinsic angle , respectively , @xmath7 is the electron gas density , @xmath8 is the magnetic field along the line - of - sight ( @xmath9 ) , and @xmath10 is the integration path . rm is the _ rotation measure_. observations of faraday rotation variations across extended radio galaxies allow us to derive information about the integral of the density - weighted line - of - sight field component . the hot ( @xmath11k ) plasma emits in the x - ray energy band via thermal bremsstrahlung . when high quality x - ray data for a radio - source environment is available , it is possible to infer the gas density distribution and therefore to separate it from that of the magnetic field , subject to some assumptions about the relation of field strength and density . most of the rm images of radio galaxies published so far show patchy structures with no clear preferred direction , consistent with isotropic foreground fluctuations over a range of linear scales ranging from tens of kpc to @xmath12100pc ( e.g. @xcite ) . numerical modelling has demonstrated that this type of complex rm structure can be accurately reproduced if the magnetic field is randomly variable with fluctuations on a wide range of spatial scales , and is spread throughout the whole group or cluster environment ( e.g.@xcite ) . these authors used forward modelling , together with estimators of the spatial statistics of the rm distributions ( structure and autocorrelation functions or a multi - scale statistic ) to estimate the field strength , its relation to the gas density and its power spectrum . the technique of bayesian maximum likelihood has also been used for this purpose @xcite . in order to derive the three - dimensional magnetic field power spectrum , all of these authors had to assume statistical isotropy for the field , since only the component of the magnetic field along the line - of - sight contributes to the observed rm . this assumption is consistent with the absence of a preferred direction in most of the rm images . in contrast , the present paper reports on _ anisotropic _ rm structures , observed in lobed radio galaxies located in different environments , ranging from a small group to one of the richest clusters of galaxies . the rm images of radio galaxies presented in this paper show clearly anisotropic `` banded '' patterns over part or all of their areas . in some sources , these banded patterns coexist with regions of isotropic random variations . the magnetic field responsible for these rm patterns must , therefore , have a preferred direction . one source whose rm structure is dominated by bands is already known : m84 @xcite . in addition , there is some evidence for rm bands in sources which also show strong irregular fluctuations , such as cygnusa @xcite . it is possible , however , that some of the claimed bands could be due to imperfect sampling of an isotropic rm distribution with large - scale power , and we return to this question in section [ many ] . the present paper presents new rm images of three sources which show spectacular banded structures , together with improved data for m84 . the environments of all four sources are well characterized by modern x - ray observations , and we give the first comprehensive description of the banded rm phenomenon . we present an initial attempt to interpret the phenomenon as a consequence of source - environment interactions and to understand the difference between it and the more usual irregular rm structure . the rm images reported in this paper are derived from new or previously unpublished archive very large array ( vla ) data for the nearby radio galaxies 0206 + 35 , m84 @xcite , 3c270 @xcite and 3c353 ( swain , private communication ; see @xcite ) . the paper is organized as follows . in section[sec : data ] the radio and x - ray properties of the sources under investigation are presented and in section[sec : teq ] we briefly summarize the techniques used to analyse depolarization and two - dimensional variations of rm . sections[sec : rm ] and [ dp ] present the rm and depolarization images on which our analysis is based and correlations between the two quantities . in section[sec : sfunc ] , we evaluate the rm structure functions in regions where the fluctuations appear to be isotropic and derive the power spectra . a simple model of the source - environment interaction which characterises the effects of compression of a magnetised igm is described in section [ model ] . this can produce rm bands , but only under implausible special initial conditions . empirical `` draped '' field configurations which are able to reproduce the banded rm distributions are investigated in section[drap ] . in section[discuss ] , we speculate on correlations between radio source morphology , and rm anisotropy , discuss other examples from the literature , consider the effects of an isotropic foreground faraday screen on the detectability of rm bands and briefly discuss possible asymmetries in the amplitude of the rm bands between the approaching and receding lobes . finally , section[concl ] summarizes our conclusions . throughout this paper we assume a @xmath13cdm cosmology with @xmath14 = 71 km s@xmath15mpc@xmath15 , @xmath16 = 0.3 , and @xmath17 = 0.7 . high quality radio and x - ray data are available for all of the sources . in this section we summarize those of their observational properties which are relevant to our rm study . a list of the sources and their general parameters is given in table[propradio ] , while table[propx ] shows the x - ray parameters taken from the literature and equipartition parameters derived from our radio observations . the sources were observed with the vla at several frequencies , in full polarization mode and with multiple configurations so that the radio structure is well sampled . the vla observations , data reduction and detailed descriptions of the radio structures are given for 0206 + 35 and m84 by @xcite , for 3c270 by @xcite , and for 3c353 by @xcite . all of the radio maps show a core , two sided jets and a double - lobed structure with sharp brightness gradients at the leading edges of both lobes . the synchrotron minimum pressures are all significantly lower than the thermal pressures of the external medium ( table[propx ] ) . all of the sources have been observed in the soft x - ray band by more than one satellite , allowing the detection of multiple components on cluster / group and sub - galactic scales . the x - ray morphologies are characterized by a compact source surrounded by extended emission with low surface brightness . the former includes a non - thermal contribution , from the core and the inner regions of the radio jets and , in the case of 0206 + 35 and 3c270 , a thermal component which is well fitted by a small core radius @xmath18 model . the latter component is associated with the diffuse intra - group or intra - cluster medium . parameters for all of the thermal components , derived from x - ray observations , are listed in table[propx ] . because of the irregular morphology of the hot gas surrounding 3c353 and m84 , it has not been possible to fit @xmath18 models to their x - ray radial surface brightness profiles . @c c c c c c c c c c c c source & ra & dec & z & kpc / arcsec & fr class & las & log@xmath19 & @xmath20 & env . & ref . + & [ j2000 ] & [ j2000 ] & & & & [ arcsec ] & [ w hz@xmath15 ] & [ degree ] & & & + 0206 + 35 ( 4c35.03 ) & 02 09 38.6 & + 35 47 50 & 0.0377 & 0.739 & i & 90 & 24.8 & 40 & group & 1 + 3c353 & 17 20 29.1 & -00 58 47 & 0.0304 & 0.601 & ii & 186 & 26.3 & 90 & poor cluster & 2 + 3c270 & 12 19 23.2 & + 05 49 31 & 0.0075 & 0.151 & i & 580 & 24.4 & 90 & group & 3 + m84 & 12 25 03.7 & + 12 53 13 & 0.0036 & 0.072 & i & 150 & 23.2 & 60 & rich cluster & 3 + @c c c c c c c c c c c c c source & band & @xmath22 & @xmath23 & @xmath24 & @xmath25 & @xmath26 & @xmath27 & @xmath28 & @xmath29 & @xmath30 & @xmath31 & ref . + & [ kev ] & [ kev ] & [ kpc ] & [ @xmath32 & & [ kpc ] & [ @xmath32 & & [ dyne @xmath33 & [ dyne @xmath33 & @xmath0 g & + 0206 + 35 & 0.2 - 2.5 & 1.3@xmath34 & 22.2 & 2.4 @xmath35 10@xmath36 & 0.35 & 0.85 & 0.42 & 0.70 & 9.6 @xmath35 10@xmath37 & 4.31@xmath35 10@xmath38 & 5.70 & 1 , 2 + 3c353 & & 4.33@xmath39 & & & & & & & & 1.66@xmath35 10@xmath37 & 11.2 & 3 + 3c270 & 0.3 - 7.0 & 1.45@xmath40 & 36.8 & 7.7 @xmath35 10@xmath36 & 0.30 & 1.1 & 0.34 & 0.64 & 5.75@xmath35 10@xmath37 & 1.64@xmath35 10@xmath38 & 3.71 & 4 + m84 & 0.6 - 7.0 & 0.6@xmath41 & & & & 5.28@xmath420.08 & 0.42 & 1.40@xmath420.03 & 1.70@xmath35 10@xmath43 & 1.07@xmath35 10@xmath37 & 9.00 & 5 + [ propx ] 0206 + 35 is an extended fanaroff - riley class i ( fri ; @xcite ) radio source whose optical counterpart , ugc11651 , is a d - galaxy , a member of a dumb - bell system at the centre of a group of galaxies . at a resolution of 1.2arcsec the radio emission shows a core , with smooth two - sided jets aligned in the nw - se direction and surrounded by a diffuse and symmetric halo . @xcite have estimated that the jets are inclined by @xmath4440with respect to the line of sight , with the main ( approaching ) jet in the nw direction . 0206 + 35 has been observed with both the _ rosat _ pspc and hri instruments @xcite and with _ chandra _ @xcite . the x - ray emission consists of a compact source surrounded by a galactic atmosphere which merges into the much more extended intra - group gas . the radius of the extended halo observed by the _ rosat _ pspc is @xmath442.5arcmin ( fig . the rosat and _ chandra _ x - ray surface brightness profiles are well fit by the combination of @xmath18 models with two different core radii and a power - law component ( hardcastle , private communication ; table[propx ] ) . 3c270 is a radio source classified as fri in most of the literature , although in fact , the two lobes have different fr classifications at low resolution @xcite . the optical counterpart is the giant elliptical galaxy ngc4261 , located at the centre of a nearby group . the radio source has a symmetrical structure with a bright core and twin jets , extending e - w and completely surrounded by lobes . the low jet / counter - jet ratio indicates that the jets are close to the plane of the sky , with the western side approaching @xcite . the xmm - newton image ( fig.[x]c ) shows a disturbed distribution with regions of low surface - brightness ( cavities ) at the positions of both radio lobes . a recent _ chandra _ observation @xcite shows `` wedges '' of low x - ray surface brightness surrounding the inner jets ( see also @xcite , @xcite , @xcite , @xcite ) . the overall surface brightness profile is accurately reproduced by a point source convolved with the _ chandra _ point spread function plus a double @xmath18 model ( * ? ? ? * _ projb model _ ) . @xcite found no evidence for a temperature gradient in the hot gas . the group is characterized by high temperature and low luminosity @xcite , which taken together provide a very high level of entropy . this might be a further sign of a large degree of impact of the agn on the environment . 3c353 is an extended frii radio source identified with a d - galaxy embedded at the periphery of a cluster of galaxies . the best estimate for the inclination of the jets is @xmath4490@xcite . the eastern jet is slightly brighter and ends in a well - defined hot spot . the radio lobes have markedly different morphologies : the eastern lobe is round with sharp edges , while the western lobe is elongated with an irregular shape . the location of the source within the cluster is of particular interest for this work and might account for the different shapes of the lobes . fig.[x](d ) shows the xmm - newton image overlaid on the radio contours . the image shows only the nw part of the cluster , but it is clear that the radio source lies on the edge of the x - ray emitting gas distribution . so that the round eastern lobe is encountering a higher external density and is probably also behind a larger column of faraday - rotating material ( @xcite , @xcite ) . in particular , the image published by @xcite shows that the gas density gradient persists on larger scales . m84 is a giant elliptical galaxy located in the virgo cluster at about 400kpc from the core . optical emission - line imaging shows a disk of ionized gas around the nucleus , with a maximum detected extent of @xmath45arcsec@xmath46 @xcite . the radio emission of m84 ( 3c272.1 ) has an angular extension of about 3arcmin ( @xmath47 11kpc ) and shows an unresolved core in the nucleus of the galaxy , two resolved jets and a pair of wide lobes @xcite . the inclination to the line - of - sight of the inner jet axis is @xmath4860 , with the northern jet approaching , but there is a noticeable bend in the counter - jet very close to the nucleus , which complicates modelling @xcite . after this bend , the jets remain straight for @xmath4440arcsec , then both of them bend eastwards by @xmath4890and fade into the radio emission of the lobes . the morphology of the x - ray emission has a h - shape made up of shells of compressed gas surrounding cavities coincident with both the radio lobes @xcite . this shape , together with the fact that the initial bending of the radio jets has the same direction and is quite symmetrical , suggests a combination of interaction with the radio plasma and motion of the galaxy within the cluster @xcite . the ratio between the x - ray surface brightness of the shells of the compressed gas and their surroundings is @xmath443 and is almost constant around the source . the shells are regions of enhanced pressure and density and low entropy : the amplitude of the density enhancements ( a factor of @xmath443 ) suggests that they are produced by weak shock waves ( mach number @xmath49 ) driven by the expanding lobes @xcite . for a fully - resolved foreground faraday screen , the @xmath50 relation of eq.[equarm ] holds exactly and there is no change of degree of polarization , @xmath51 , with wavelength . even in the presence of a small gradient of rm across the beam , @xmath50 rotation is observed over a wide range of polarization angle . in this case , the emission tends to depolarize with increasing wavelength , following the burn law @xcite : @xmath52 where @xmath53 is the intrinsic value of the degree of polarization and @xmath54=2@xmath55 , with @xmath56 @xcite . + since @xmath57 , eq.[equadp ] clearly illustrates that higher rm gradients across the beam generate higher @xmath54 values and in turn higher depolarization . our observational analysis is based on the following procedure . we first produced rm and burn law @xmath54 images at two different angular resolutions for each source and searched for regions with high @xmath54 or correlated rm and @xmath54 values , which could indicate the presence of internal faraday rotation and/or strong rm gradients across the beam . in regions with low @xmath54 where the variations of rm are plausibly isotropic and random , we then used the structure function ( defined in eq.[sfunction ] ) to derive the power spectrum of the rm fluctuations . finally , to investigate the depolarization in the areas of isotropic rm , and hence the magnetic field power on small scales , we made numerical simulations of the burn law @xmath54 using the model power spectrum with different minimum scales and compared the results with the data . the structure function is defined by @xmath58 ^ 2 > } \label{sfunction}\ ] ] ( @xcite ; @xcite ) where @xmath59 and @xmath60 are vectors in the plane of the sky and @xmath61 is an average over @xmath60 . we assume rm power spectra of the form : @xmath62 where @xmath63 is a scalar spatial frequency and fit the observed structure function ( including the effect of the observing beam ) using the hankel - transform method described by @xcite to derive the amplitude , @xmath64 and the slope , @xmath65 . to constrain the rm structure on scales smaller than the beamwidth , we estimated the minimum scale of the best fitted field power spectrum , @xmath66 , which predicts a mean value of @xmath54 consistent with the observed one . in this paper , we are primarily interested in estimating the rm power spectrum over limited areas , and we made no attempt to determine the outer scale of fluctuations . the use of the structure function together with the burn law @xmath54 represents a powerful technique to investigate the rm power spectrum over a wide range of spatial scales @xcite . the two quantities are complementary , in that the structure function allows us to determine the power spectrum of the fluctuations on scales larger than the beamwidth , while the burn law @xmath54 constrains fluctuations of rm below the resolution limit . @c c c c source & @xmath67 & @xmath68 & beam + & [ mhz ] & [ mhz ] & [ arcsec ] + 0206 + 35 & 1385.1 & 25 & 1.2 + & 1464.9 & 25 & + & 4885.1 & 50 & + 3c353 & 1385.0 & 12.5 & 1.3 + & 1665.0 & 12.5 & + & 4866.3 & 12.5 & + & 8439.9 & 12.5 & + 3c270 & 1365.0 & 25 & 1.65 + & 1412.0 & 12.5 & + & 4860.1 & 100 & + & 1365.0 & 25 & 5.0 + & 1412.0 & 12.5 & + & 1646.0 & 25 & + & 4860.1 & 100 & + m84 & 1413.0 & 25 & 1.65 + & 4885.1 & 50 & + & 1385.1 & 50 & 4.5 + & 1413.0 & 25 & + & 1464.9 & 50 & + & 4885.1 & 50 & + the rm images and associated rms errors were produced by weighted least - squares fitting to the observed polarization angles @xmath3 as a function of @xmath50 ( eq.[pang ] ) at three or four frequencies ( table [ nu ] , see also @xcite and @xcite ) using the rm task in the aips package . each rm map was calculated only at pixels with rms polarization - angle uncertainties @xmath6910at all frequencies . we refer only to the lower - resolution rm and @xmath54 images for 3c270 and m84 ( table [ nu ] ) , as they show more of the faint , extended regions of these sources and are fully consistent with the higher - resolution versions . the rm image of m84 is consistent with that shown by @xcite , but is derived from four - frequency data and has a higher signal - to - noise ratio . in the fainter regions of 0206 + 35 ( for which only three frequencies are available and the signal - to - noise ratio is relatively low ) , the rm task occasionally failed to determine the n@xmath70 ambiguities in position angle correctly . in order to remove these anomalies , we first produced a lower - resolution , but high signal - to - noise rm image by convolving the 1.2arcsec rm map to a beamwidth of 5arcsec fwhm . from this map we derived the polarization - angle rotations at each of the three frequencies and subtracted them from the observed 1.2arcsec polarization angle maps at the same frequency to derive the residuals at high resolution . then , we fit the residuals without allowing any n@xmath70 ambiguities and added the resulting rm s to the values determined at low resolution . this procedure allowed us to obtain an rm map of 0206 + 35 free of significant deviations from @xmath50 rotation and fully consistent with the 1.2-arcsec measurements . we have verified that the polarization angles accurately follow the relation @xmath71 over the full range of position angle essentially everywhere except for small areas around the optically - thick cores : representative plots of @xmath72 against @xmath50 for 0206 + 35 are shown in fig.[fittini ] . the lack of deviations from @xmath50 rotation in all of the radio galaxies is fully consistent with our assumption that the faraday rotating medium is mostly external to the sources . the rm maps are shown in fig.[rm ] . the typical rms error on the fit is @xmath442radm@xmath73 . no correction for the galactic contribution has been applied . all of the rm maps show two - dimensional patterns , _ rm bands _ , across the lobes with characteristic widths ranging from 3 to 12kpc . multiple bands parallel to each other are observed in the western lobe of 0206 + 35 , the eastern lobe of 3c353 and the southern lobe of m84 . in all cases , the iso - rm contours are straight and perpendicular to the major axes of the lobes to a very good approximation : the very straight and well - defined bands in the eastern lobes of both 0206 + 35 ( fig.[rm]a ) and 3c353 ( fig.[rm]d ) are particularly striking . the entire area of m84 appears to be covered by a banded structure , while in the central parts of 0206 + 35 and 3c270 and the western lobe of 3c353 , regions of isotropic and random rm fluctuations are also present . we also derived profiles of along the radio axis of each source , averaging over boxes a few beamwidths long ( parallel to the axes ) , but extended perpendicular to them to cover the entire width of the source . the boxes are all large enough to contain many independent points . the profiles are shown in fig.[rmprof ] . for each radio galaxy , we also plot an estimate of the galactic contribution to the rm derived from a weighted mean of the integrated rm s for non - cluster radio sources within a surrounding area of 10deg@xmath46 @xcite . in all cases , both positive and negative fluctuations with respect to the galactic value are present . in 0206 + 35 ( fig.[rm]a ) , the largest - amplitude bands are in the outer parts of the lobes , with a possible low - level band just to the nw of the core . the most prominent band ( with the most negative rm values ) is in the eastern ( receding ) lobe , about 15 kpc from the core ( fig.[rmprof]a ) . its amplitude with respect to the galactic value is about 40radm@xmath73 . this band must be associated with a strong ordered magnetic field component along the line of sight . if corrected for the galactic contribution , the two adjacent bands in the eastern lobe would have rm with opposite signs and the field component along the line of sight must therefore reverse . m84 ( fig.[rm]b ) displays an ordered rm pattern across the whole source , with two wide bands of opposite sign having the highest absolute rm values . there is also an abrupt change of sign across the radio core ( see also @xcite ) . the negative band in the northern lobe ( associated with the approaching jet ) has a larger amplitude with respect to the galactic value than the corresponding ( positive ) feature in the southern lobe ( fig.[rmprof]c ) . 3c270 ( fig.[rm]c ) shows two large bands : one on the front end of the eastern lobe , the other in the middle of the western lobe . the bands have opposite signs and contain the extreme positive and negative values of the observed rm . the peak positive value is within the eastern band at the extreme end of the lobe ( fig.[rmprof]e ) . the rm structure of 3c353 ( fig.[rm]d ) is highly asymmetric . the eastern lobe shows a strong pattern , made up of four bands , with very straight iso - rm contours which are almost exactly perpendicular to the source axis . as in 0206 + 35 , adjacent bands have rm with opposite signs once corrected for the galactic contribution ( fig.[rmprof]g ) . in contrast , the rm distribution in the western lobe shows no sign of any banded structure , and is consistent with random fluctuations superimposed on an almost linear profile . it seems very likely that the differences in rm morphology and axial ratio are both related to the external density gradient ( fig . [ x]d ) . @l r r r r r r r r source & & & @xmath74 & band & @xmath75 & width & @xmath76 + & & & [ radm@xmath73 ] & [ radm@xmath73 ] & [ kpc ] & [ kpc ] & [ radm@xmath73 ] + 0206 + 35 ( 4c35.03 ) & @xmath7877 & 23 & @xmath79 & & & & + & & & & @xmath78140 & -15 & 10 & 40 + & & & & @xmath7860 & -27 & 4 & + & & & & 34 & 22 & 6 & + & & & & 51 & 8 & 4 & + 3c353 & @xmath7856 & 24 & @xmath80 & & & & + & & & & 122 & -12 & 5 & 50 + & & & & 102 & -19 & 4 & + & & & & -40 & -23 & 4 & + & & & & 100 & -26 & 4 & + 3c270 & 14 & 10 & 12 & & & & + & & & & @xmath788 & 20 & 12 & 10 + & & & & 32 & 37 & 11 & + m84 & @xmath782 & 15 & 2 & & & & + & & & & @xmath7827 & 1 & 3 & 10 + & & & & 22 & -6 & 6 & + in table [ band ] the relevant geometrical features ( size , distance from the radio core , ) for the rm bands are listed . in this section , we use `` depolarization '' in its conventional sense to mean `` decrease of degree of polarization with increasing wavelength '' and define dp = @xmath81 . using the faraday code @xcite , we produced images of burn law @xmath54 by weighted least - squares fitting to @xmath82 as a function of @xmath83 ( eq.[equadp ] ) . only data with signal - to - noise ratio @xmath844 in @xmath85 at each frequency were included in the fits . the burn law @xmath54 images were produced with the same angular resolutions as the rm images . the 1.65arcsec resolution burn law @xmath54 maps for m84 and 3c270 are consistent with the low - resolution ones , but add no additional detail and are quite noisy . this could lead to significantly biased estimates for the mean values of @xmath54 over large areas @xcite . therefore , as for the rm maps , we used only the burn law @xmath54 images at low resolution for these two sources . the burn law @xmath54 maps are shown in fig.[k ] . all of the sources show low average values of @xmath54 ( i.e. slight depolarization ) , suggesting little rm power on small scales . with the possible exception of the narrow filaments of high @xmath54 in the eastern lobe of 3c353 ( which might result from partially resolved rm gradients at the band edges ) , none of the images show any obvious structure related to the rm bands . for each source , we have also compared the rm and burn law @xmath54 values derived by averaging over many small boxes covering the emission , and we find no correlation between them . we also derived profiles of @xmath54 ( fig.[rmprof]b , d , f and h ) with the same sets of boxes as for the rm profiles in the same figure . these confirm that the values of @xmath54 measured in the centres of the rm bands are always low , but that there is little evidence for any detailed correlation . the signal - to - noise ratio for 0206 + 35 is relatively low compared with that of the other three sources , particularly at 4.9ghz ( we need to use a small beam to resolve the bands ) , and this is reflected in the high proportion of blanked pixels on the @xmath54 image . the most obvious feature of this image ( fig . [ k]a ) , an apparent difference in mean @xmath54 between the high - brightness jets ( less depolarized ) and the surrounding emission , is likely to be an artefact caused by our blanking strategy : points where the polarized signal is low at 4.9ghz are blanked preferentially , so the remainder show artificially high polarization at this frequency . for the same reason , the apparent minimum in @xmath54 at the centre of the deep , negative rm band ( fig . [ rmprof]a and b ) is probably not significant . the averaged values of @xmath54 for 0206 + 35 are already very low , however , and are likely to be slightly overestimated , so residual rm fluctuations on scales below the 1.2-arcsec beamwidth must be very small . m84 shows one localised area of very strong depolarization ( @xmath86500rad@xmath46m@xmath87 , corresponding to dp = 0.38 ) at the base of the southern jet ( fig . there is no corresponding feature in the rm image ( fig . the depolarization is likely to be associated with one of the shells of compressed gas visible in the _ chandra _ image ( fig . [ x]b ) , implying significant magnetization with inhomogeneous field and/or density structure on scales much smaller than the beamwidth , apparently independent of the larger - scale field responsible for the rm bands . this picture is supported by the good spatial coincidence of the high @xmath54 region with a shell of compressed gas , as illustrated in the overlay of the 4.5arcsec burn law @xmath54 image on the contours of the _ chandra _ data ( fig.[kx](a ) ) . cooler gas associated with the emission - line disk might also be responsible , but there is no evidence for spatial coincidence between enhanced depolarization and h@xmath88 emission @xcite . despite the complex morphology of the x - ray emission around m84 , its @xmath54 profile is very symmetrical , with the highest values at the centre ( fig.[rmprof](d ) ) . 3c270 also shows areas of very strong depolarization ( @xmath86550rad@xmath46m@xmath87 , corresponding to dp = 0.35 ) close to the core and surrounding the inner and northern parts of both the radio lobes . as for m84 , the areas of high @xmath54 are coincident with ridges in the x - ray emission which form the boundaries of the cavity surrounding the lobes ( fig.[kx](b ) ) . the inner parts of this x - ray structure are described in more detail by @xcite , whose recent high - resolution _ chandra _ image clearly reveals `` wedges '' of low brightness surrounding the radio jets . as in m84 , the most likely explanation is that a shell of denser gas immediately surrounding the radio lobes is magnetized , with significant fluctuations of field strength and density on scales smaller than our 5-arcsec beam , uncorrelated with the rm bands . the @xmath54 profile of 3c270 ( fig.[rmprof](f ) ) is very symmetrical , suggesting that the magnetic - field and density distributions are also symmetrical and consistent with an orientation close to the plane of the sky . the largest values of @xmath54 are observed in the centre , coincident with the features noted earlier and with the bulk of the x - ray emission ( the high @xmath54 values in the two outermost bins have low signal - to - noise and are not significant ) . in the burn law @xmath54 image of 3c353 , there is evidence for a straight and knotty region of high depolarization @xmath4420kpc long and extending westwards from the core . this region does not appear to be related either to the jets or to any other radio feature . as in m84 and 3c270 , the rm appears quite smooth over the area showing high depolarization , again suggesting that there are two scales of structure , one much smaller than the beam , but producing zero mean rm and the other very well resolved . in 3c353 , there is as yet no evidence for hot or cool ionized gas associated with the enhanced depolarization ( contamination from the very bright nuclear x - ray emission affects an area of 1arcmin radius around the core ; @xcite , @xcite ) . the @xmath54 profile of 3c353 ( fig.[rmprof](h ) ) shows a marked asymmetry , with much higher values in the east . this is in the same sense as the difference of rm fluctuation amplitudes ( fig.[rmprof](g ) ) and is also consistent with the eastern lobe being embedded in higher - density gas . the relatively high values of @xmath54 within 20kpc of the nucleus in the western lobe are due primarily to the discrete region identified earlier . we calculated rm structure function for discrete regions of the sources where the rm fluctuations appear to be isotropic and random and for which we expect the spatial variations of foreground thermal gas density , rms magnetic field strength and path length to be reasonably small . these are : the inner 26arcsec of the receding ( eastern ) lobe of 0206 + 35 , the inner 100arcsec of 3c270 and the inner 40arcsec of the western lobe of 3c353 . the selected areas of 0206 + 35 and 3c270 are both within the core radii of the larger - scale beta models that describe the group - scale x - ray emission and the galaxy - scale components are too small to affect the rm statistics significantly ( table [ propx ] ) . in 3c353 , the selected area was chosen to be small compared with the scale of x - ray variations seen in fig . [ x](d ) . in all three cases , the foreground fluctuations should be fairly homogeneous . there are no suitable regions in m84 , which is entirely covered by the banded rm pattern . the structure functions , corrected for uncorrelated random noise by subtracting 2@xmath89 @xcite , are shown in fig.[sfunc ] . all of the observed structure functions correspond to power spectra of approximately power - law form over all or most of the range of spatial frequencies we sample . we initially assumed that the power spectrum was described by eq.[eq - cutoff - pl ] with no high - frequency cut - off ( @xmath90 ) and made least - squares fits to the structure functions , weighted by errors derived from multiple realizations of the power spectrum on the observing grid , as described in detail by @xcite and @xcite . the best - fitting slopes @xmath65 and amplitudes @xmath64 are given in table[spectrum ] . all of the fitted power spectra are quite flat and have low amplitudes , implying that there is little power in the isotropic and random component of rotation measure . indeed , the amplitudes of the largest - scale rm fluctuations sampled in this analysis is a few times less than that of the bands ( see tables[band ] and [ spectrum ] ) . this suggests that the field responsible for the bands is stronger as well as more ordered than that responsible for the isotropic fluctuations . [ cols="^,^,^,^,^,^,^,^ " , ] the structure functions for 0206 + 35 and 3c353 rise monotonically , indicating that the outer scale for the random fluctuations must be larger than the maximum separations we sample . for 3c270 , the structure function levels out at @xmath91 100arcsec ( 15kpc ; fig . [ sfunc]d ) . this could be the outer scale of the field fluctuations , but a better understanding of the geometry and external density distribution would be needed before we could rule out the effects of large - scale variations in path length or field strength ( cf . @xcite ) . in order to constrain rm structure on spatial scales below the beamwidth , we estimated the depolarization as described in section [ sec : teq ] . the fitted @xmath54 values are listed in table[spectrum ] . we stress that these values refer only to areas with isotropic fluctuations , and can not usefully be compared with the integrated depolarizations quoted in in fig.[rm ] . for m84 , using the burn law @xmath54 analysis and assuming that variation of faraday rotation across the 1.65-arcsec beam causes the residual depolarization , we find that @xmath92kpc for any reasonable rm power spectrum . it is clear from the fact that the observed rm bands are perpendicular to the lobe axes that they must be associated with an interaction between an expanding radio source and the gas immediately surrounding it . one inevitable mechanism is enhancement of field and density by the shock or compression wave surrounding the source .. ] the implication of the presence of cavities in the x - ray gas distribution coincident with the radio lobes is that the sources are interacting strongly with the thermal gas , displacing rather than mixing with it ( see @xcite for a review ) . for the sources in the present paper , the x - ray observations of m84 ( * ? ? ? * fig.[x]b ) and 3c270 ( * ? ? ? * fig.[x]c ) show cavities and arcs of enhanced brightness , corresponding to shells of compressed gas bounded by weak shocks . the strength of any pre - existing field in the igm , which will be frozen into the gas , will also be enhanced in the shells . we therefore expect a significant enhancement in rm . a more extreme example of this effect will occur if the expansion of the radio source is highly supersonic , in which case there will be a strong bow - shock ahead of the lobe , behind which both the density and the field become much higher . regardless of the strength of the shock , the field is modified so that only the component in the plane of the shock is amplified and the post - shock field tends become ordered parallel to the shock surface . the evidence so far suggests that shocks around radio sources of both fr classes are generally weak ( e.g. @xcite , @xcite , @xcite ) . there are only two examples in which highly supersonic expansion has been inferred : the southern lobe of centaurusa ( @xmath93 ; @xcite ) and ngc3801 ( @xmath94 ; @xcite ) . there is no evidence that the sources described in the present paper are significantly overpressured compared with the surrounding igm ( indeed , the synchrotron minimum pressure is systematically lower than the thermal pressure of the igm ; table [ propx ] ) . the sideways expansion of the lobes is therefore unlikely to be highly supersonic . the shock mach number estimated for all the sources from ram pressure balance in the forward direction is also @xmath44 1.3 . this estimate is consistent with that for m84 made by @xcite and also with the lack of detection of strong shocks in the x - ray data for the other sources . in this section , we investigate how the rm could be affected by compression . we consider a deliberately oversimplified picture in which the radio source expands into an igm with an initially uniform magnetic field , @xmath95 . this is the most favourable situation for the generation of large - scale , anisotropic rm structures : in reality , the pre - existing field is likely to be highly disordered , or even isotropic , because of turbulence in the thermal gas . _ we stress that we have not tried to generate a self - consistent model for the magnetic field and thermal density , but rather to illustrate the generic effects of compression on the rm structure_. in this model the radio lobe is an ellipsoid with its major axis along the jet and is surrounded by a spherical shell of compressed material . this shell is centred at the mid - point of the lobe ( fig.[prova ] ) and has a stand - off distance equal to 1/3 of the lobe semi - major axis at the leading edge ( the radius of the spherical compression is therefore equal to 4/3 of the lobe semi - major axis ) . in the compressed region , the thermal density and the magnetic field component in the plane of the spherical compression are amplified by the same factor , because of flux - freezing . we use a coordinate system @xmath96 centred at the lobe mid - point , with the @xmath97-axis along the line of sight , so @xmath98 and @xmath99 are in the plane of the sky . the radial unit vector is @xmath100=@xmath101 . @xmath102=@xmath103 and @xmath104=@xmath105 are respectively the pre- and post - shock magnetic - field vectors . then , we consider a coordinate system @xmath106 still centred at the lobe mid - point , but rotated with respect to the @xmath96 system by the angle @xmath107 about the @xmath99 ( @xmath108 ) axis , so that @xmath109 is aligned with the major axis of the lobe . with this choice , @xmath107 is the inclination of the source with respect to the line - of - sight ( fig.[prova ] ) . -axis and the @xmath97-axis represents a generic line - of - sight . , width=226 ] after a spherical compression , the thermal density and field satisfy the equations : @xmath110 \label{freez}\end{aligned}\ ] ] where @xmath111 , @xmath112 and @xmath113 represent the initial thermal density and the components of the field perpendicular and parallel to the compression surface . the same symbols with primes stand for post - shock quantities and @xmath114 is the compression factor . the total compressed magnetic field is : @xmath115 the field strength after compression depends on the the angle between the compression surface and the initial field . maximum amplification occurs for a field which is parallel to the surface , whereas a perpendicular field remains unchanged . the post - shock field component along the line - of - sight becomes : @xmath116 we assumed that the compression factor , @xmath117 , is a function of distance @xmath109 along the source axis from the centre of the radio lobe , decreasing monotonically from a maximum value @xmath118 at the leading edge to a constant value from the centre of the lobe as far as the core . we investigated values of @xmath118 in the range 1.5 4 ( @xmath119 corresponds to the asymptotic value for a strong shock ) . given that there is no evidence for strong shocks in the x - ray data for any of our sources , we have typically assumed that the compression factor is @xmath120 at the front end of the lobe , decreasing to 1.2 at the lobe mid - point and thereafter remaining constant as far as the core . a maximum compression factor of 3 is consistent with the transonic mach numbers @xmath121 estimated from ram - pressure balance for all of the sources and this choice is also motivated by the x - ray data of m84 , from which there is evidence for a compression ratio @xmath122 between the shells and their surroundings ( section[84 ] ) . we produced synthetic rm images for different combinations of source inclination and direction of the pre - existing uniform field , by integrating the expression @xmath123 numerically . we assumed a constant value of @xmath124cm@xmath36 for the density of the pre - shock material ( the central value for the group gas associated with 0206 + 35 ; table [ propx ] ) , a lobe semi - major axis of 21kpc ( also appropriate for 0206 + 35 ) and an initial field strength of 1@xmath0 g . the integration limits were defined by the surface of the radio lobe and the compression surface . this is equivalent to assuming that there is no thermal gas within the radio lobe , consistent with the picture suggested by our inference of foreground faraday rotation and the existence of x - ray cavities and that faraday rotation from uncompressed gas is negligible . as an example , fig.[icm ] shows the effects of compression on the rm for the receding lobe of a source inclined by 40to the line of sight . the initial field is pointing towards us with an inclination of 60with respect to the line - of - sight ; its projection on the plane of the sky makes an angle of 30with the @xmath98-axis . fig.[icm](a ) displays the rm produced without compression ( @xmath126 everywhere ) : the rm structures are due only to differences in path length across the lobe . fig.[icm](b ) shows the consequence of adding a modest compression of @xmath127 : structures similar to bands are generated at the front end of the lobe and the range of the rm values is increased . fig.[icm](c ) illustrates the rm produced in case of the strongest possible compression , @xmath128 : the rm structure is essentially the same as in fig.[icm](b ) , with a much larger range . this very simple example shows that rm bands with amplitudes consistent with those observed can plausibly be produced even by weak shocks in the igm , but the iso - rm contours are neither straight , nor orthogonal to the lobe axis and there are no reversals . these constraints require specific initial conditions , as illustrated in fig.[90 ] , where we show the rm for a lobe in the plane of the sky . we considered three initial field configurations : along the line - of - sight ( fig.[90]a ) , in the plane of the sky and parallel to the lobe axis ( fig.[90]b ) and in the plane of the sky , but inclined by 45to the lobe axis ( fig.[90]c ) . the case closest to reproducing the observations is that displayed in fig.[90](b ) , in which reversals and well defined and straight bands perpendicular to the jet axis are produced for both of the lobes . in fig.[90](a ) , the structures are curved , while in fig.[90](c ) the bands are perpendicular to the initial field direction , and therefore inclined with respect to the lobe axis . for a source inclined by 40to the line of sight , we found structures similar to the observed bands only with an initial field in the plane of the sky and parallel to the axis in projection ( figs[40]a and b ; note that the synthetic rm images in this example have been made for each lobe separately , neglecting superposition ) . we can summarize the results of the spherical pure compression model as follows . 1 . an initial field with a component along the line - of - sight does not generate straight bands the bands are orthogonal to the direction of the initial field projected on the plane of the sky , so bands perpendicular to the lobe axis are only obtained with an initial field aligned with the radio jet in projection . the path length ( determined by the precise shape of the radio lobes ) has a second - order effect on the rm distribution ( compare figs.[icm]a and b ) . thus , a simple compression model can generate bands with amplitudes similar to those observed but reproducing their geometry requires implausibly special initial conditions , as we discuss in the next section . that the pre - existing field is uniform , close to the plane of the sky and aligned with the source axis in projection is implausible for obvious reasons : 1 . the pre - existing field can not know anything about either the radio - source geometry or our line - of - sight and 2 . observations of faraday rotation in other sources and the theoretical inference of turbulence in the igm both require disordered initial fields . this suggests that the magnetic field must be aligned _ by _ rather than _ with _ the expansion of the radio source . indeed , the field configurations which generate straight bands look qualitatively like the `` draping '' model proposed by @xcite , for some angles to the line - of - sight . the analysis of sections[rm ] , [ dp ] and [ sec : sfunc ] suggests that the magnetic fields causing the rm bands are well - ordered , consistent with a stretching of the initial field that has erased much of the small - scale structure while amplifying the large - scale component . we next attempt to constrain the geometry of the resulting `` draped '' field . a proper calculation of the rm from a draped magnetic - field configuration @xcite is outside the scope of this paper , but we can start to understand the field geometry using some simple approximations in which the field lines are stretched along the source axis . we assumed initially that the field around the lobe is axisymmetric , with components along and radially outwards from the source axis , so that the rm pattern is independent of rotation about the axis . _ it is important to stress that such an axisymmetric field is not physical _ , as it requires a monopole and unnatural reversals , it is nevertheless a useful benchmark for features of the field geometry that are needed to account for the observed rm structure . we first considered field lines which are parabolae with a common vertex on the axis ahead of the lobe . for field strength and density both decreasing away from the vertex , we found that rm structures , with iso - contours similar to arcs , rather than bands , were generated only for the approaching lobe of an inclined source . such anisotropic rm structures were not produced in the receding lobes , nor for sources in the plane of the sky . indeed , in order to generate any narrow , transverse rm structures such as arcs or bands , the line - of - sight must pass through a foreground region in which the field lines show significant curvature , which occurs only for an approaching lobe in case of a parabolic field geometry . this suggested that we should consider field lines which are families of ellipses centred on the lobe , again with field strength and density decreasing away from the leading edge . this indeed produced rm structures in both lobes for any inclination , but the iso - rm contours were arcs , not straight lines . because of the non - physical nature of these axisymmetric field models , the resulting rm images are deliberately not shown in this paper . in order to quantify the departures from straightness of the iso - rm contours , we measured the ratio of the predicted rm values at the centre and edge of the lobe at constant @xmath108 , at different distances along the source axis , @xmath129 , for both of the example axisymmetric field models . the ratio , which is 1 for perfectly straight bands , varies from 2 to 3 in both cases , depending on distance from the nucleus . this happens because the variations in line - of - sight field strength and density do not compensate accurately for changes in path length . we believe that this problem is generic to any axisymmetric field configuration . + the results of this section suggest that the field configuration required to generate straight rm bands perpendicular to the projected lobe advance direction has systematic curvature in the field lines ( in order to produce a modulation in rm ) without a significant dependence on azimuthal angle around the source axis ) . we therefore investigated a structure in which elliptical field lines are wrapped around the front of the lobe , but in a two - dimensional rather than a three - dimensional configuration . we considered a field with a two - dimensional geometry , in which the field lines are families of ellipses in planes of constant @xmath108 , as sketched in fig.[famab ] . the field structure is then independent of @xmath108 . the limits of integration are given by the lobe surface and an ellipse whose major axis is 4/3 of that of the lobe . three example rm images are shown in fig . [ synfam ] . the assumed density ( @xmath130cm@xmath36 ) and magnetic field ( 1@xmath0 g ) were the same as the pre - shock values for the compression model of section [ model ] and the lobe semi - major axis was again 21kpc . since the effect of path length is very small ( section [ model ] ) , the rm is also independent of @xmath108 to a good approximation . the combination of elliptical field lines and invariance with @xmath108 allows us to produce straight rm bands perpendicular to the projected lobe axis for any source inclination . furthermore , this model generates more significant reversals of the rm ( e.g. fig.[synfam]c ) than those obtainable with pure compression ( e.g. fig.[40]b ) . we conclude that a field model of this generic type represents the simplest way to produce rm bands with the observed characteristics in a way that does not require improbable initial conditions . the invariance of the field with the @xmath108 coordinate is an essential point of this model , suggesting that the physical process responsible for the draping and stretching of the field lines must act on scales larger than the radio lobes in the @xmath108 direction . the two - dimensional draped field illustrated in the previous section reproduces the geometry of the observed rm bands very well , but can only generate a single reversal , which must be very close to the front end of the approaching lobe , where the elliptical field lines bend most rapidly . we observe a prominent reversal in the receding lobe of 0206 + 35 ( fig . [ rm]a ) and multiple reversals across the eastern lobe of 3c353 and in m84 ( fig . [ rm]b and d ) . the simplest way to reproduce these is to assume that the draped field also has reversals , presumably originating from a more complex initial field in the igm . one realization of such a field configuration would be in the form of multiple toroidal eddies with radii smaller than the lobe size , as sketched in fig.[circle ] . whatever the precise field geometry , we stress that the straightness of the observed multiple bands again requires a two - dimensional structure , with little dependence on @xmath108 . the majority of published rm images of radio galaxies do not show bands or other kind of anisotropic structure , but are characterized by isotropic and random rm distributions ( e.g. @xcite ) . on the other hand , we have presented observations of rm bands in four radio galaxies embedded in different environments and with a range of jet inclinations with respect to the line - of - sight . our sources are not drawn from a complete sample , so any quantitative estimate of the incidence of bands is premature , but we can draw some preliminary conclusions . the simple two - dimensional draped - field model developed in section [ 2d ] only generates rm bands when the line - of - sight intercepts the volume containing elliptical field lines , which happens for a restricted range of rotation around the source axis . at other orientations , the rm from this field configuration will be small and the observed rm may well be dominated by material at larger distances which has not been affected by the radio source . we therefore expect a minority of sources with this type of field structure to show rm bands and the remainder to have weaker , and probably isotropic , rm fluctuations . in contrast , the three - dimensional draped field model proposed by @xcite predicts rm bands _ parallel _ to the source axis for a significant range of viewing directions : these have not ( yet ) been observed . the prominent rm bands described in the present paper occur only in _ lobed _ radio galaxies . in contrast , well - observed radio sources with tails and plumes seem to be free of bands or anisotropic rm structure ( e.g. 3c31 , 3c449 ; @xcite ) . furthermore , the lobes which show bands are all quite round and show evidence for interaction with the surrounding igm . it is particularly striking that the bands in 3c353 occur only in its eastern , rounded , lobe . the implication is that rm bands occur when a lobe is being actively driven by a radio jet into a region of high igm density . plumes and tails , on the other hand , are likely to be rising buoyantly in the group or cluster and we do not expect significant compression , at least at large distances from the nucleus . the fact that rm bands have so far been observed only in a few radio sources may be a selection effect : much rm analysis has been carried out for galaxy clusters , in which most of the sources are tailed ( e.g. @xcite ) . with a few exceptions like cyga ( see below , section [ cyg ] ) , lobed fri and frii sources have not been studied in detail . cygnusa is a source in which we might expect to observed rm bands , by analogy with the sources discussed in the present paper : it has wide and round lobes and _ chandra _ x - ray data have shown the presence of shock - heated gas and cavities @xcite . rm bands , roughly perpendicular to the source axis , are indeed seen in both lobes @xcite , but interpretation is complicated by the larger random rm fluctuations and the strong depolarization in the eastern lobe . a semi - circular rm feature around one of the hot - spots in the western lobe has been attributed to compression by the bow - shock @xcite . @xcite have claimed evidence for rm bands in the northern lobe of hydraa . image @xcite shows a clear cavity with sharp edges coincident with the radio lobes and an absence of shock - heated gas , just as in our sources . despite the classification as a tailed source , it may well be that there is significant compression of the igm . note , however , that the rm image is not well sampled close to the nucleus . the rm image of the tailed source 3c465 published by @xcite shows some evidence for bands , but the colour scale was deliberately chosen to highlight the difference between positive and negative values , thus making it difficult to see the large gradients in rm expected at band edges . the original rm image ( eilek , private communication ) suggests that the band in the western tail of 3c465 is similar to those we have identified . it is plausible that magnetic - field draping happens in wide - angle tail sources like 3c465 as a result of bulk motion of the igm within the cluster potential well , as required to bend the tails . it will be interesting to search for rm bands in other sources of this type and to find out whether there is any relation between the iso - rm contours and the flow direction of the igm . , seen in the plane normal to its axis . the cross represents the radio core position . , width=226 ] the coexistence in the rm images of our sources of anisotropic patterns with areas of isotropic fluctuations suggests that the faraday - rotating medium has at least two components : one local to the source , where its motion significantly affects the surrounding medium , draping the field , and the other from material on group or cluster scales which has not felt the effects of the source . this raises the possibility that turbulence in the foreground faraday rotating medium might `` wash out '' rm bands , thereby making them impossible to detect . the isotropic rm fluctuations observed across our sources are all described by quite flat power spectra with low amplitude ( table[spectrum ] ) . the random small - scale structure of the field along the line - of - sight essentially averages out , and there is very little power on scales comparable with the bands . if , on the other hand , the isotropic field had a steeper power spectrum with significant power on scales similar to the bands , then its contribution might become dominant . we first produced synthetic rm images for 0206 + 35 including a random component derived from our best - fitting power spectrum ( table[spectrum ] ) in order to check that the bands remained visible . we assumed a minimum scale of 2kpc from the depolarization analysis for 0206 + 35 ( sec.[dp ] ) , and a maximum scale of @xmath131kpc , consistent with the continuing rise of the rm structure function at the largest sampled separations , which requires @xmath132arcsec ( @xmath4720kpc ) . the final synthetic rm is given by : @xmath133 where @xmath134 and @xmath135 are the rm due to the draped and isotropic fields , respectively . the terms @xmath136 and @xmath137 are respectively the density and field component along the line - of - sight in the draped region . the integration limits of the term @xmath134 were defined by the surface of the lobe and the draped region , while that of the term @xmath135 starts at the surface of the draped region and extends to 3 times the core radius of the x - ray gas ( table[propx ] ) . for the electron gas density @xmath111 outside the draped region we assumed the beta - model profile of 0206 + 35 ( table[propx ] ) and for the field strength a radial variation of the form ( * ? ? ? * and references therein ) . @xmath138^{~\eta}\ ] ] where @xmath139 is the rms magnetic field strength at the group centre . we took a draped magnetic field strength of 1.8@xmath0 g , in order to match the amplitudes for the rm bands in both lobes of 0206 + 35 , and assumed the same value for @xmath139 . example rm images , shown in figs.[drapicm](a ) and ( b ) , should be compared with those for the draped field alone ( figs.[synfam]b and c , scaled up by a factor of 1.8 to account for the difference in field strength ) and with the observations ( fig.[rm]a ) after correction for galactic foreground ( table[band ] ) . the model is self - consistent : the flat power spectrum found for 0206 + 35 does not give coherent rm structure which could interfere with the rm bands , which are still visible . we then replaced the isotropic component with one having a kolmogorov power spectrum ( @xmath140 ) . we assumed identical minimum and maximum scales ( 2 and 40kpc ) as for the previous power spectrum and took the same central field strength ( @xmath139=1.8@xmath0 g ) and radial variation ( eq.[br ] ) . example realizations are shown in fig.[drapicm](c ) and ( d ) . the bands are essentially invisible in the presence of foreground rm fluctuations with a steep power spectrum out to scales larger than their widths . it may therefore be that the rm bands in our sources are especially prominent because the power spectra for the isotropic rm fluctuations have unusually low amplitudes and flat slopes . we have established these parameters directly for 0206 + 35 , 3c270 and 3c353 ; m84 is only 14kpc in size and is located far from the core of the virgo cluster , so it is plausible that the cluster contribution to its rm is small and constant . we conclude that the detection of rm bands could be influenced by the relative amplitude and scale of the fluctuations of the isotropic and random rm component compared with that from any draped field , and that significant numbers of banded rm structures could be masked by isotropic components with steep power spectra . the well - established correlation between rm variance and/or depolarization and jet sidedness observed in fri and frii radio sources is interpreted as an orientation effect : the lobe containing the brighter jet is on the near side , and is seen through less magnetoionic material ( e.g. @xcite ) . for sources showing rm bands , it is interesting to ask whether the asymmetry is due to the bands or just to the isotropic component . in 0206 + 35 , whose jets are inclined by @xmath44 40to the line - of - sight , the large negative band on the receding side has the highest rm ( fig.[rmprof ] ) . this might suggest that the rm asymmetry is due to the bands , and therefore to the local draped field . unfortunately , 0206 + 35 is the only source displaying this kind of asymmetry . the other `` inclined '' source , m84 , ( @xmath141 60 ) does not show such asymmetry : on the contrary the rm amplitude is quite symmetrical . in this case , however , the relation between the ( well - constrained ) inclination of the inner jets , and that of the lobes could be complicated : both jets bend by @xmath44 90at distances of about 50arcsec from the nucleus @xcite , so that we can not establish the real orientation of the lobes with respect to the plane of the sky . the low values of the jet / counter - jet ratios in 3c270 and 3c353 suggest that their axes are close to the plane of the sky , so that little orientation - dependent rm asymmetry would be expected . indeed , the lobes of 3c270 show similar rm amplitudes , while the large asymmetry of rm profile of 3c353 is almost certainly due to a higher column density of thermal gas in front of the eastern lobe . within our small sample , there is therefore no convincing evidence for higher rm amplitudes in the bands on the receding side , but neither can such an effect be ruled out . in models in which an ordered field is draped around the radio lobes , the magnitude of any rm asymmetry depends on the field geometry as well as the path length ( cf . @xcite for the isotropic case ) . for instance , the case illustrated in figs [ synfam](b ) and ( c ) shows very little asymmetry even for @xmath107= 40 . the presence of a systematic asymmetry in the banded rm component could therefore be used to constrain the geometry . a different mechanism for the generation of rm fluctuations was suggested by @xcite . they argued that large - scale nonlinear surface waves could form on the surface of a radio lobe through the merging of smaller waves generated by kelvin - helmholtz instabilities and showed that rm s of roughly the observed magnitude would be produced if a uniform field inside the lobe was advected into the mixing layer . this mechanism is unlikely to be able to generate large - scale bands , however : the predicted iso - rm contours are only straight over parts of the lobe which are locally flat , even in the unlikely eventuality that a coherent surface wave extends around the entire lobe . the idea that a mixing layer generates high faraday rotation may instead be relevant to the anomalously high depolarizations associated with regions of compressed gas around the inner radio lobes of m84 and 3c270 ( section [ dp ] and fig . we have argued that the fields responsible for the depolarization are tangled on small scales , since they produce depolarization without any obvious effects on the large - scale faraday rotation pattern . it is unclear whether the level of turbulence within the shells of compressed gas is sufficient to amplify and tangle a pre - existing field in the igm to the level that it can produce the observed depolarization ; a plausible alternative is that the field originates within the radio lobe and mixes with the surrounding thermal gas . in this work we have analysed and interpreted the faraday rotation across the lobed radio galaxies 0206 + 35 , 3c270 , 3c353 and m84 , located in environments ranging from a poor group to one of the richest clusters of galaxies ( the virgo cluster ) . the rm images have been produced at resolutions ranging from 1.2 to 5.5arcsec fwhm using very large array data at multiple frequencies . all of the rm images show peculiar banded patterns across the radio lobes , implying that the magnetic fields responsible for the faraday rotation are anisotropic . the rm bands coexist and contrast with areas of patchy and random fluctuations , whose power spectra have been estimated using a structure - function technique . we have also analysed the variation of degree of polarization with wavelength and compared this with the predictions for the best - fitting rm power spectra in order to constrain the minimum scale of magnetic turbulence . we have investigated the origin of the bands by making synthetic rm images using simple models of the interaction between radio galaxies and the surrounding medium and have estimated the geometry and strength of the magnetic field . our results can be summarized as follows . 1 . the lack of deviation from @xmath50 rotation over a wide range of polarization position angle and the lack of associated depolarization together suggest that a foreground faraday screen with no mixing of radio - emitting and thermal electrons is responsible for the observed rm in the bands and elsewhere ( section [ sec : rm ] ) . the dependence of the degree of polarization on wavelength is well fitted by a burn law , which is also consistent with ( mostly resolved ) pure foreground rotation ( section [ dp ] ) . the rm bands are typically 3 10kpc wide and have amplitudes of 10 50radm@xmath73 ( table [ band ] ) . the maximum deviations of rm from the galactic values are observed at the position of the bands . iso - rm contours are orthogonal to the axes of the lobes . in several cases , neighbouring bands have opposite signs compared with the galactic value and the line - of - sight field component must therefore reverse between them . an analysis of the profiles of and depolarization along the source axes suggests that there is very little small - scale rm structure within the bands . 5 . the lobes against which bands are seen have unusually small axial ratios ( i.e.they appear round in projection ; fig . [ rm ] ) . in one source ( 3c353 ) the two lobes differ significantly in axial ratio , and only the rounder one shows rm bands . this lobe is on the side of the source for which the external gas density is higher . structure function and depolarization analyses show that flat power - law power spectra with low amplitudes and high - frequency cut - offs are characteristic of the areas which show isotropic and random rm fluctuations , but no bands ( section [ sec : sfunc ] ) . 7 . there is evidence for source - environment interactions , such as large - scale asymmetry ( 3c353 ) cavities and shells of swept - up and compressed material ( m84 , 3c270 ) in all three sources for which high - resolution x - ray imaging is available . areas of strong depolarization are found around the edges of the radio lobes close to the nuclei of 3c270 and m84 . these are probably associated with shells of compressed hot gas . the absence of large - scale changes in faraday rotation in these features suggests that the field must be tangled on small scales ( section [ dp ] ) . the comparison of the amplitude of with that of the structure functions at the largest sampled separations is consistent with an amplification of the large scale magnetic field component at the position of the bands we produced synthetic rm images from radio lobes expanding into an ambient medium containing thermal material and magnetic field , first considering a pure compression of both thermal density and field , and then including three- and two - dimensional stretching ( `` draping '' ) of the field lines along the direction of the radio jets ( sects . [ model ] and [ drap ] ) . both of the mechanisms are able to generate anisotropic rm structure . 11 . to reproduce the straightness of the iso - rm contours , a two - dimensional field structure is needed . in particular , a two - dimensional draped field , whose lines are geometrically described by a family of ellipses , and associated with compression , reproduces the rm bands routinely for any inclination of the sources to the line - of - sight ( sec . moreover , it might explain the high rm amplitude and low depolarization observed within the bands . the invariance of the magnetic field along the axis perpendicular to the forward expansion of the lobe suggests that the physical process responsible for the draping and stretching of the magnetic field must act on scales larger than the lobe itself in this direction . we can not yet constrain the scale size along the line of sight . 13 . in order to create rm bands with multiple reversals , more complex field geometries such as two - dimensional eddies are needed ( section [ helic ] ) . we have interpreted the observed rm s as due to two magnetic field components : one draped around the radio lobes to produce the rm bands , the other turbulent , spread throughout the surrounding medium , unaffected by the radio source and responsible for the isotropic and random rm fluctuations ( section [ iso ] ) . we tested this model for 0206 + 35 , assuming a typical variation of field strength with radius in the group atmosphere , and found that a magnetic field with central strength of @xmath142 g reproduced the rm range quite well in both lobes . we have suggested two reasons for the low rate of detection of bands in published rm images : our line of sight will only intercept a draped field structure in a minority of cases and rotation by a foreground turbulent field with significant power on large scales may mask any banded rm structure . our results therefore suggest a more complex picture of the magnetoionic environments of radio galaxies than was apparent from earlier work . we find three distinct types of magnetic - field structure : an isotropic component with large - scale fluctuations , plausibly associated with the undisturbed intergalactic medium ; a well - ordered field draped around the leading edges of the radio lobes and a field with small - scale fluctuations in the shells of compressed gas surrounding the inner lobes , perhaps associated with a mixing layer . in addition , we have emphasised that simple compression by the bow shock should lead to enhanced rm s around the leading edges , but that the observed patterns depend on the pre - shock field . mhd simulations should be able to address the formation of anisotropic magnetic - field structures around radio lobes and to constrain the initial conditions . in addition , our work raises a number of observational questions , including the following . 1 . how common are anisotropic rm structures ? do they occur primarily in lobed radio galaxies with small axial ratios , consistent with jet - driven expansion into an unusually dense surrounding medium ? is their frequency qualitatively consistent with the two - dimensional draped - field picture ? 2 . why do we see bands primarily in sources where the isotropic rm component has a flat power spectrum of low amplitude ? is this just because the bands can be obscured by large - scale fluctuations , or is there a causal connection ? 3 . are the rm bands suggested in tailed sources such as 3c465 and hydraa caused by a similar phenomenon ( e.g. bulk flow of the igm around the tails ) ? 4 . is an asymmetry between approaching and receding lobes seen in the banded rm component ? if so , what does that imply about the field structure ? 5 . how common are the regions of enhanced depolarization at the edges of radio lobes ? how strong is the field and what is its structure ? is there evidence for the presence of a mixing layer ? it should be possible to address all of these questions using a combination of observations with the new generation of synthesis arrays ( evla , e - merlin and lofar all have wide - band polarimetric capabilities ) and high - resolution x - ray imaging . we thank m. swain for providing the vla data for 3c353 , j. eilek for a fits image of 3c465 and j. croston , a. finoguenov and j. googder for the x - ray images of 3c270 , m84 and 3c353 , respectively . we are also grateful to a. shukurov , j. stckl and the anonymous referee for the valuable comments . [ de vaucoleurs1991 ] dev91 de vaucouleurs , g. , de vaucouleurs , a. , corwin , jr . , h. g. , et al . , 1991 , third reference catalogue of bright galaxies , adsurl : http://cdsads.u-strasbg.fr/abs/1991rc3..book.....d [ mcnamara et al.2000 ] mcn2000 mcnamara , b.r . , wise , m. , nulsen , p.e.j . , david , l.p . , sarazin , c.l . , bautz , m. , markevitch , m. , vikhlinin , a. , forman , w.r . , jones , c. , harris , d.e . , 2000 , apj , 534 , l135

we present detailed imaging of faraday rotation and depolarization for the radio galaxies 0206 + 35 , 3c270 , 3c353 and m84 , based on very large array observations at multiple frequencies in the range 1365 to 8440mhz . all of the sources show highly anisotropic banded rotation measure ( rm ) structures with contours of constant rm perpendicular to the major axes of their radio lobes . all except m84 also have regions in which the rm fluctuations have lower amplitude and appear isotropic . we give a comprehensive description of the banded rm phenomenon and present an initial attempt to interpret it as a consequence of interactions between the sources and their surroundings . we show that the material responsible for the faraday rotation is in front of the radio emission and that the bands are likely to be caused by magnetized plasma which has been compressed by the expanding radio lobes . we present a simple model for the compression of a uniformly - magnetized external medium and show that rm bands of approximately the right amplitude can be produced , but for only for special initial conditions . a two - dimensional magnetic structure in which the field lines are a family of ellipses draped around the leading edge of the lobe can produce rm bands in the correct orientation for any source orientation . we also report the first detections of rims of high depolarization at the edges of the inner radio lobes of m84 and 3c270 . these are spatially coincident with shells of enhanced x - ray surface brightness , in which both the field strength and the thermal gas density are likely to be increased by compression . the fields must be tangled on small scales . [ firstpage ] galaxies : magnetic fields radio continuum : galaxies ( galaxies : ) intergalactic medium x - rays : galaxies : clusters

the exact string matching is one of the oldest tasks in computer science . the need for it started when computers began processing text . at that time the documents were short and there were not so many of them . now , we are overwhelmed by amount of data of various kind . the string matching is a crucial task in finding information and its speed is extremely important . the exact string matching task is defined as counting or reporting all the locations of given pattern @xmath0 of length @xmath1 in given text @xmath2 of length @xmath3 assuming @xmath4 , where @xmath0 and @xmath2 are strings over a finite alphabet @xmath5 . the first solutions designed were to build and run deterministic finite automaton @xcite ( running in space @xmath6 and time @xmath7 ) , the knuth pratt automaton @xcite ( running in space @xmath8 and time @xmath7 ) , and the boyer moore algorithm @xcite ( running in best case time @xmath9 and worst case time @xmath10 ) . there are numerous variations of the boyer moore algorithm like @xcite . in total more than 120 exact string matching algorithms @xcite have been developed since 1970 . modern processors allow computation on vectors of length 16 bytes in case of sse2 and 32 bytes in case of avx2 . the instructions operate on such vectors stored in special registers xmm0xmm15 ( sse2 ) and ymm0ymm15 ( avx2 ) . as one instruction is performed on all data in these long vectors , it is considered as simd ( single instruction , multiple data ) computation . in the nave approach ( shown as algorithm [ naivesearch2 ] ) the pattern @xmath0 is checked against each position in the text @xmath2 which leads to running time @xmath10 and space @xmath11 . however , it is not bad in practice for large alphabets as it performs only 1.08 comparisons @xcite on average on each character of @xmath2 for english text . the variable _ found _ in algorithm [ naivesearch2 ] is not quite necessary . it is presented in order to have a connection to the simd version to be introduced . like in the testing evironment of hume & sunday @xcite and the smart library @xcite , we consider the counting version of exact string matching . it can be is easily transformed into the reporting version by printing position @xmath12 in line [ naivesearch2-printi ] . @xmath13 @xmath14 @xmath15 and ( @xmath16=p[j]$ ] ) @xmath17[naivesearch2-printi ] [ naivesearch2-out ] and pattern @xmath18 using the simd - nave - search algorithm ( alignment of pattern vector and vector _ found _ to text @xmath2).,scaledwidth=90.0% ] @xmath13 @xmath19 @xmath20 @xmath21[simdnaivesearch2-printi ] [ simdnaivesearch2-out ] using simd instructions ( shown in algorithm [ simdnaivesearch2 ] ) we can compare @xmath22 bytes in parallel , where @xmath23 in case of sse2 or @xmath24 in case of avx2 and ` and ' represents the bit - parallel ` and ' . this allows huge speedup of a run . for a given position @xmath12 in the text @xmath2 , the idea is to compare the pattern @xmath0 with the @xmath22 substrings @xmath25 $ ] , for @xmath26 , in parallel , in @xmath8 time in total . to this end , we use a primitive @xmath27 which , given a position @xmath12 in @xmath2 and @xmath28 in @xmath0 , compares the strings @xmath29 $ ] and @xmath30^\alpha$ ] and returns an @xmath22-bit integer such that the @xmath31-th bit is set iff @xmath32 = s_2[k]$ ] , in @xmath11 time . in other words , the output integer encodes the result of the @xmath28-th symbol comparison for all the @xmath22 substrings . for example , consider the @xmath22 leftmost substrings of length @xmath33 of @xmath2 , corresponding to @xmath34 . for @xmath35 , the function compares @xmath36 $ ] with @xmath37^\alpha$ ] , i.e. , the first symbol of the substrings against @xmath37 $ ] . for @xmath38 , the function compares @xmath39 $ ] with @xmath40^\alpha$ ] , i.e. , the second symbol against @xmath40 $ ] . let _ found _ be the bitwise and of the integers @xmath27 , for @xmath41 . clearly , @xmath25 = p$ ] iff the @xmath31-bit of _ found _ is set . we compute _ found _ iteratively , until we either compare the last symbol of @xmath0 or no substring has a partial match ( i.e. , the vector _ found _ becomes zero ) . then , the text is advanced by @xmath22 positions and the process is repeated starting at position @xmath42 . for a given @xmath12 , the number of occurrences of @xmath43 is equal to the number of bits set in _ found _ and is computed using a popcount instruction . reporting all matches in line [ simdnaivesearch2-printi ] would add an @xmath44 time overhead , as @xmath44 instructions are needed to extract the positions of the bits set in _ found _ , where @xmath45 is the number of occurrences found . the 16-byte version of function simdcompare is implemented with sse2 intrinsic functions as follows : .... simdcompare(x , y , 16 ) x_ptr = _ mm_loadu_si128(x ) y_ptr = _ mm_loadu_si128(s(y,16 ) ) return _ mm_movemask_epi8(_mm_cmpeq_epi8(x_ptr , y_ptr ) ) .... here s(y,16 ) is the starting address of 16 copies of y. the instruction ` _ mm_loadu_si128(x ) ` loads 16 bytes ( = 128 bits ) starting from x to a simd register . the instruction ` _ mm_cmpeq_epi8 ` compares bytewise two registers and the instruction ` _ mm_movemask_epi8 ` extracts the comparison result as a 16-bit integer . for the 32-byte version , the corresponding avx2 intrinsic functions are used . for both versions the sse4 instruction ` _ mm_popcnt_u32 ` is utilized for popcount . in order to identify nonmatching positions in the text as fast as possible , individual characters of the pattern are compared to the corresponding positions in the text in the order given by their frequency in standard text . first , the least frequent symbol is compared , then the second least frequent symbol , etc . therefore the text type should be considered and frequencies of symbols in the text type should be computed in advance from some relevant corpus of texts of the same type . hume and sunday @xcite use this strategy in the context of the boyer moore algorithm . @xmath13 @xmath19 @xmath46 @xmath21[freqssimdnaivesearch2-printi ] [ freqsimdnaivesearch2-out ] algorithm [ freqsimdnaivesearch2 ] shows the nave approach enriched by frequency consideration . a function @xmath47 gives the order in which the symbols of pattern should be compared ( i.e. , @xmath48 , p[\pi(2)],\ldots , p[\pi(m)]$ ] ) to the corresponding symbols in text . an array for the function @xmath47 is computed in @xmath49 time using a standard sorting algorithm on frequencies of symbols in @xmath0 . hume and sunday @xcite call this strategy _ optimal match _ , although it is not necessarily optimal . for example , the pattern ` qui ' is tested in the order ` q'-`u'-`i ' , but the order ` q'-`i'-`u ' is clearly better in practice because ` q ' and ` u ' appear often together . klekci @xcite compares optimal match with more advanced strategies based on frequencies of discontinuous @xmath50-grams$ ] in a position @xmath12 of the pattern @xmath0 matches to the text , compare next the position of @xmath0 that most unlikely matches . ] with conditional probabilities . his experiments show that the frequency is beneficial in case of texts of large alphabets like texts of natural language . computing all possible frequencies of @xmath50-grams is rather complicated and the possible speed - up to optimal match is likely marginal . thus we consider only simple frequencies of individual symbols . guard test @xcite is a widely used technique to speed - up string matching . the idea is to test a certain pattern position before entering a checking loop . instead of a single guard test , two or even three tests have been used @xcite . guard test is a representative of a general optimization technique called loop peeling , where a number of iterations is moved in front of the loop . as a result , the loop becomes faster because of fewer loop tests . moreover , loop peeling makes possible to precompute certain values used in the moved iterations . for example , @xmath48 $ ] is explicitly known . in some cases , loop peeling may even double the speed of a string matching algorithm applying simd computation as observed by chhabra et al . @xcite . in the following , we call the number of the moved iterations the peeling factor @xmath51 . we assume that the first loop test is done after @xmath51 iterations . thus our approach differs from multiple guard test , where checking is stopped after the first mismatch . all @xmath51 iterations are performed in our approach . @xmath13 @xmath52 [ lpfreqsimdnaivesearch2-firstcomparison ] @xmath53 @xmath46 @xmath21[lpfreqssimdnaivesearch2-printi ] [ lpfreqsimdnaivesearch2-out ] loop peeling for @xmath54 is shown in algorithm [ lpfreqsimdnaivesearch2 ] . the first two comparisons of characters are performed regardless the result of the first comparison ( in line [ lpfreqsimdnaivesearch2-firstcomparison ] ) . if we consider string matching in english texts , it is less probable that all the @xmath22 comparisons fail at the same time than the other way round in the case of a pattern picked randomly from the text . therefore it is advantageous to use the value @xmath54 for english . in theory , @xmath55 would be good for dna . namely , every iteration nullifies roughly 3/4 of the remaining set bits of the bitvector _ found_. however , we achieved the best running time in practice with @xmath56 . if the computation of character frequencies is considered inappropriate , there are other possibilities to speed - up checking . in natural languages adjacent characters have positive correlation . to break correlations one can use a fixed order which avoids adjacent characters . we applied the following heuristic order @xmath57 : @xmath37,p[m ] , p[4 ] , p[7],\ldots , p[3 ] , p[6 ] , \ldots , p[2 ] , p[5],\ldots$ ] . in letter - based languages , the space character is the most frequent character . we can transform @xmath57 to a slightly better scheme @xmath58 by moving first all the spaces to the end and then processing the remaining positions as for @xmath57 . we have selected four files of different types and alphabet sizes to run experiments on : ` bible.txt ` ( fig . [ figbible ] , table [ tab@bible ] ) and ` e.coli.txt ` ( fig . [ figecoli ] , table [ tab@ecoli ] ) taken from canterbury corpus @xcite , ` dostoevsky-thedouble.txt ` ( fig . [ figdostoyevsky ] , table [ tab@dostoyevsky ] ) , novel the double by dostoevsky in czech language taken from project gutenberg , and ` protein-hs.txt ` ( fig . [ figprotein ] , table [ tab@protein ] ) taken from protein corpus @xcite . file ` dostoevsky-thedouble.txt ` is a concatenation of five copies of the original file to get file length similar to the other files . ) , scaledwidth=90.0% ] ) , scaledwidth=90.0% ] ) , scaledwidth=90.0% ] ) , scaledwidth=90.0% ] we have compared methods naive16 and naive32 having 16 and 32 bytes processed by one simd instruction respectively . naive16-freq and naive32-freq are their variants where comparison order given by nondecreasing probability of pattern symbols ( section [ sec@frequency_involved ] ) . naive16-fixed and naive32-fixed are the variants where comparison order is fixed ( section [ sec@alternative_checking_orders ] ) . our methods were compared with the fastest exact string matching algorithms @xcite up to now sbndm2 , sbndm4 @xcite and epsm @xcite taken from smart library . the experiments were run on gnu / linux 3.18.12 , with x86_64 intel core i7 - 4770 cpu 3.40ghz with 16 gb ram . the computer was without any other workload and user time was measured using posix function ` getrusage ( ) ` . the average of 100 running times is reported . the accuracy of the results is about @xmath59 . the experiments show for both sse2 and avx2 instructions that for natural text ( ` bible.txt ` ) with the scheme @xmath57 of fixed frequency of comparisons improves the speed of simd - nave - search but it is further improved by considering frequencies of symbols in the text . in case of natural text with larger alphabet ( ` dostoevsky-thedouble.txt ` ) the scheme @xmath57 improves the speed only for avx2 instructions . the comparison based on real frequency of symbols is the bext for both sse2 and avx2 instructions . in case of small alphabets ( ` e.coli.txt ` , ` protein-hs.txt ` ) the order of comparison of symbols does not play any role ( except for ` protein-hs.txt ` and sse2 instructions ) . for files with large alphabet ( ` bible.txt ` , ` dostoevsky-thedouble.txt ` ) the peeling factor @xmath55 gave the best results for all our algorithms except for naive16-freq and naive32-freq where @xmath54 was the best . the smaller the alphabet is , the less selective the bigrams or trigrams are . for file ` protein-hs.txt ` , @xmath55 was still good and but for dna sequences of four symbols , @xmath56 turned to be the best we also tested nave - search . in every run it was naturally considerably slower than simd - nave - search . frequency order and loop peeling can also be applied to nave - search . however , the speed - up was smaller than in case of simd - nave - search in our experiments . in spite of how many algorithms were developed for exact string matching , their running times are in general outperformed by the avx2 technology . the implementation of the nave search algorithm ( freq - simd - nave - search ) which uses avx2 instructions , applies loop peeling , and compares symbols in the order of increasing frequency is the best choice in general . however , previous algorithms epsm and sbndm4 have an advantage for small alphabets and long patterns . short patterns of 20 characters or less are objects of most searches in practice and our algorithm is especially good for such patterns . for texts with expected equiprobable symbols ( like in dna or protein strings ) , our algorithm naturally works well without the frequency order of symbol comparisons . our algorithm is considerably simpler than its simd - based competitor epsm which is a combination of six algorithms . this work was done while jan holub was visiting the aalto university under the asci visitor programme ( dean s decision 12/2016 ) . s. faro and m. o. klekci . fast packed string matching for short patterns . in p. sanders and n. zeh , editors , _ proceedings of the 15th meeting on algorithm engineering and experiments , alenex 2013 _ , pages 113121 . siam , 2013 . s. faro , t. lecroq , s. borz , s. di mauro , and a. maggio . the string matching algorithms research tool . in j. holub and j. rek , editors , _ proceedings of the prague stringology conference 16 _ , pages 99113 , czech technical university in prague , czech republic , 2016 . m. o. klekci . an empirical analysis of pattern scan order in pattern matching . in sio iong ao , leonid gelman , david w. l. hukins , andrew hunter , and a. m. korsunsky , editors , _ world congress on engineering _ , lecture notes in engineering and computer science , pages 337341 . newswood limited , 2007 .

more than 120 algorithms have been developed for exact string matching within the last 40 years . we show by experiments that the nave algorithm exploiting simd instructions of modern cpus ( with symbols compared in a special order ) is the fastest one for patterns of length up to about 50 symbols and extremely good for longer patterns and small alphabets . the algorithm compares 16 or 32 characters in parallel by applying sse2 or avx2 instructions , respectively . moreover , it uses loop peeling to further speed up the searching phase . we tried several orders for comparisons of pattern symbols and the increasing order of their probabilities in the text was the best .

traditional cantor sets are generated by iterations of an operation of down - scaling by fractions which are powers of a fixed positive integer . for each iteration in the process , we leave gaps . for example , the best - known ternary cantor set is formed by scaling down by @xmath1 and leaving a single gap in each step . an associated cantor measure @xmath2 is then obtained by the same sort of iteration of scales , and , at each step , a renormalization . in accordance with classical harmonic analysis , these measures may be seen to be infinite bernoulli convolutions . our present analysis is motivated by earlier work , beginning with @xcite . we consider recursive down - scaling by @xmath3 for @xmath4 and leave a single gap at each iteration - step . it was shown in @xcite that the associated cantor measures @xmath5 have the property that @xmath6 possesses orthogonal fourier bases of complex exponentials ( i.e. fourier onbs ) . more recently , it was shown in @xcite that the scales @xmath3 are the _ only _ values that generate measures with fourier bases . given a fixed cantor measure @xmath2 , a corresponding set of frequencies @xmath7 of exponents in an onb is said to be a _ spectrum _ for @xmath2 . for example , in the case of recursive scaling by powers of @xmath0 , i.e. @xmath8 , a possible spectrum @xmath7 for @xmath9 has the form @xmath7 as shown below in equation . a spectrum for a cantor measure turns out to be a _ lacunary _ ( in the sense of szolem mandelbrojt ) set of integers or half integers . we direct the interested reader to @xcite regarding lacunary series and their riesz products . when @xmath10 and @xmath2 are fixed , we now become concerned with the possible variety of spectra . given @xmath7 some canonical choice of spectrum for @xmath2 , then one possible way to construct a new fourier spectrum for @xmath9 is to scale by an odd positive integer @xmath11 to form a set @xmath12 . while for some values of @xmath11 this scaling produces a spectrum , it is known that other values of @xmath11 do not yield spectra . this particular question is intrinsically multiplicative : since @xmath2 is an infinite bernoulli convolution , the onb questions involve consideration of infinite products of the riesz type . despite this intuition , we show here ( theorem [ thm : main ] ) that there is a connection between this multiplicative construction and a construction of new onbs with an additive operation . we are then able to produce even more examples of these additive - construction spectra . throughout this paper , we consider the hilbert space @xmath13 where @xmath14 is the @xmath0-bernoulli convolution measure . this measure has a rich history , dating back to work of wintner and erds @xcite . more recently , hutchinson @xcite developed a construction of bernoulli measures via iterated function systems ( ifss ) . the measure @xmath14 is supported on a cantor subset @xmath15 of @xmath16 which entails scaling by @xmath0 . in 1998 , jorgensen and pedersen @xcite discovered that the hilbert space @xmath13 contains a fourier basis an orthonormal basis of exponential functions and hence allows for a fourier analysis . for ease of notation , throughout this paper we will write @xmath17 for the function @xmath18 and for a discrete set @xmath7 we will write @xmath19 for the collection of exponentials @xmath20 . there is a self - similarity inherent in the @xmath0-bernoulli convolution @xmath21 which yields an infinite product formulation for @xmath22 : @xmath23 exponential functions @xmath24 and @xmath25 are orthogonal when @xmath26 a collection of exponential functions @xmath19 indexed by the discrete set @xmath7 is an orthonormal basis for @xmath13 exactly when the function @xmath27 is the constant function @xmath28 . we call the function @xmath29 the _ spectral function _ for the set @xmath7 . the fourier basis for @xmath14 constructed in @xcite is the set @xmath30 , where @xmath31 if @xmath19 is an orthonormal basis ( onb ) for @xmath13 , we say that @xmath7 is a _ spectrum _ for @xmath14 . it is straightforward to show that if @xmath7 is a spectrum for @xmath14 and @xmath11 is an odd integer , then @xmath32 is an orthogonal collection of exponential functions . in many cases , we find that @xmath32 is actually another onb @xcite . this is rather surprising , or at least very different behavior from the usual fourier analysis on an interval with respect to lebesgue measure . we often refer to the spectrum in equation as the _ canonical spectrum _ for @xmath13 , while other spectra for the same measure space can be called _ alternate spectra_. in this section , we describe two naturally occurring isometries on @xmath13 which are defined via their action on the canonical fourier basis @xmath19 . observe from equation that @xmath7 satisfies the invariance equation @xmath33 where @xmath34 denotes the disjoint union . we then define @xmath35 since @xmath36 and @xmath37 map the onb elements into a proper subset of the onb , they are proper isometries . therefore , for @xmath38 we have @xmath39 and @xmath40 is a projection onto the range of the respective operator . the adjoints of @xmath41 are readily computed ( see @xcite for details ) : @xmath42 and @xmath43 it is shown in ( * ? ? ? * section 2 ) that the definitions of @xmath36 and @xmath37 extend to all @xmath44 for @xmath45 , i.e. @xmath46 for every integer @xmath47 , there is a @xmath48-algebra with @xmath49 generators called the cuntz algebra , which we denote by @xmath50 @xcite . we will describe representations of @xmath51 which are generated by two isometries on @xmath13 satisfying the conditions below . [ defn : cuntzrel ] we say that isometry operators @xmath52 on @xmath13 satisfy _ cuntz relations _ if 1 . @xmath53 , 2 . @xmath54 for @xmath55 . when these relations hold , @xmath56 generate a representation of the cuntz algebra @xmath57 . from @xcite , we know that @xmath36 and @xmath37 defined in equation satisfy the cuntz relations for @xmath58 , hence yield a representation of the cuntz algebra @xmath51 ( in fact , an irreducible representation ) within the algebra of bounded operators @xmath59 . as we mentioned above , given a spectrum @xmath7 , the frequencies @xmath12 , for @xmath11 an odd integer , generate an orthonormal collection of exponential functions in @xmath9 . given @xmath7 from equation , one question of interest is the characterization of the odd integers @xmath11 for which the scaled spectrum @xmath12 generates an onb . as a means of exploring this question , we let @xmath60 be the operator @xmath61 since @xmath60 maps an onb to an orthonormal collection , @xmath60 is an isometry and is unitary if and only if @xmath32 is an onb . the following lemmas provide useful relationships between the isometries @xmath36 , @xmath37 , and @xmath60 . [ lem : vp ] let @xmath36 and @xmath37 be the isometry operators from equation . if @xmath62 is a @xmath63-automorphism on @xmath59 , then the operator @xmath64 is unitary . assume @xmath62 is a @xmath63-automorphism . the cuntz relations on @xmath36 and @xmath37 give @xmath65 a similar computation proves that @xmath66 , hence @xmath67 is unitary . [ lem : us1 ] let @xmath68 be the multiplication operator @xmath69 . given @xmath70 such that @xmath60 is unitary , we define the map @xmath71 on @xmath59 . then @xmath72 and @xmath73 . it was proved in @xcite that @xmath60 commutes with @xmath36 for all odd @xmath11 , so @xmath72 . since @xmath60 is unitary , we have @xmath74 . we prove that @xmath75 , which is thus equivalent to the statement of the lemma . @xmath76 therefore , @xmath77 we now discover a connection between the scaled spectrum @xmath12 and what we call an _ additive spectrum _ @xmath78 . it will turn out that this connection tells us more about the additive spectra than the scaled spectra . [ thm : main ] given any odd natural number @xmath11 , if @xmath32 is an onb then @xmath78 is also an onb . since @xmath32 is an onb , we have that the operator @xmath60 from equation is a unitary operator . we define the map on @xmath79 @xmath80 since @xmath60 is unitary , it is straightforward to verify that @xmath81 is a @xmath63-automorphism on @xmath59 . if we apply @xmath81 to our operators @xmath36 and @xmath37 , we have by lemma [ lem : us1 ] , @xmath82 define the operator @xmath83 then @xmath84 is unitary by lemma [ lem : vp ] . we see that if @xmath85 , i.e. @xmath86 for some @xmath87 , that @xmath88 since @xmath89 and @xmath90 by the cuntz relations . similarly , if @xmath91 , hence @xmath92 for some @xmath87 , then @xmath93 . in fact , @xmath84 maps @xmath94 bijectively onto @xmath95 . therefore , since @xmath84 is unitary , we can conclude that @xmath78 is an onb for @xmath13 . we now address the spectral functions recall equation for our additive sets . we can use the splitting @xmath96 to divide the spectral function for @xmath7 into the corresponding terms @xmath97 denote the sums on the right - hand side of the equation above by @xmath98 and @xmath99 respectively . more generally , denote @xmath100 [ prop : per ] the function @xmath101 is @xmath102-periodic . by theorem [ thm : main ] the sets @xmath103 and @xmath104 are both spectra for @xmath2 this follows because it is known ( see , for example , @xcite ) that the scaled sets @xmath105 and @xmath106 are spectra . we therefore have @xmath107 using the fact that the set @xmath7 itself is also a spectrum , we have @xmath108 for all @xmath109 . hence @xmath110 but we also observe that @xmath111 and @xmath112 , so the function @xmath101 is both @xmath113-periodic and @xmath114-periodic , hence is @xmath102-periodic . the function @xmath115 is @xmath102-periodic . we next observe that theorem [ thm : main ] is a stepping stone to the following result . [ thm : dhs ] given any odd integer @xmath11 , the set @xmath116 $ ] is an onb for @xmath9 . this is a direct result of proposition [ prop : per ] . the spectral function for@xmath116 $ ] can be written in the two parts @xmath117 when @xmath118 , we have the canonical onb in the @xmath0 case . otherwise , using the 2-periodicity of @xmath115 , we have @xmath119 since the spectral function is identically @xmath28 , the set @xmath116 $ ] is an onb for @xmath9 . the authors would like to thank allan donsig for helpful conversations while writing an earlier version of this work . we mention here that the existence of the spectra that we call the _ additive spectra _ for @xmath14 is not new . they are among the examples described , from a different perspective , in section 5 of @xcite .

in this paper , we add to the characterization of the fourier spectra for bernoulli convolution measures . these measures are supported on cantor subsets of the line . we prove that performing an odd additive translation to half the canonical spectrum for the @xmath0 cantor measure always yields an alternate spectrum . we call this set an additive spectrum . the proof works by connecting the additive set to a spectrum formed by odd multiplicative scaling .

lacoo@xmath0 is a perovskite , which has the rare earth la on the a - site and co on the b - site , which corresponds to the center of the oxygen octahedron . however , like most of the perovskites , lacoo@xmath0 has oxygen octahedral rotations ( fig . [ sfig : structure ] ) which involve the oxygen octahedra rotating out - of - phase around the [ 111 ] axes of the undistorted cubic highsymmetry structure . the rotation pattern in glazer notation is @xmath14 , which corresponds to the space group r@xmath37c ( # 167 ) . as noted by thornton et . @xcite , lacoo@xmath0 has large thermal expansion and the octahedral rotation angle also changes with temperature . in our study , we used four different crystal structures to isolate and study the effect of different lattice parameters on the spin state transition . we used the two different experimental structures observed at 1143k and 4k , which we denote by @xmath15 and @xmath16 . comparing the electronic structure for these two crystal structures provides a means to study the temperature evolution of the electronic structure . in addition to these two , we also built two cubic perovskite crystal structures with the same inter - lanthanum spacing as them , but with no octahedral rotations . these structures , denoted by @xmath17 and @xmath18 , enabled us to isolate the effect of oxygen octahedral rotations on the electronic structure of lacoo@xmath0 . . each of the oxygen octadra rotates in the opposite direction to all nearest neighbour octahedra by the same amount relative to all three coordinate axes ( @xmath14 structure).,height=288 ] we performed fully charge self - consistent embedded dft+dmft calculations@xcite@xcite@xcite on @xmath12 . our implementation is based on the wien2k all - electron dft package.@xcite for the dft functional we use the gga - pbe functional . we employed a 10 atom unit cell for the 2 structures with rotations and 5 atom unit cells for the two structures without rotations and used 512 k points in the first brillouin zone . our dmft implementation uses state - of - the - art ctqmc@xcite impurity solver based on the hybridization expansion ( ctqmc - hyb ) and all simulations are iterated to self - consistency . it is to be noted that our implementation makes use of projectors to embed / project out the impurity self - energy onto the lattice degrees of freedom . this prevents errors associated with downfolding using wannier orbitals and allows us to achieve highly accurate charge self - consistency.@xcite we also account for octahedral rotations , wherever present , by applying local rotations so as to align our correlated orbitals with the local crystal field set up by the neighboring atoms . in our simulations , we use a hubbard u of 6.0 ev and hund s coupling ( j ) of 0.7ev . we have also investigated the effects of varying the value of j , and found that this does not change the temperature at which the transitions occur , but merely changes the value of the observed @xmath24 by a small amount , with higher j values resulting in slightly larger values . we ignored the spin - orbit interaction , as it is expected to be small in this 3d transition metal oxide . the typical self - consistent calculation requires 30 self consistency cycles each with around 10 dft iterations and one ctqmc iteration per cycle . our lda+dmft calculations gave us the green s function ( @xmath38 ) on the imaginary ( matsubara ) axis which we analytically continued using the maximum entropy method to get the density of states on the real axis . dmft is based on mapping the original lattice problem to an auxilliary impurity problem . we minimize the action of this impurity problem defined by : @xmath39 where @xmath40 is imaginary time , @xmath41 and @xmath42 denote angular momentum and spin labels,@xmath43 and @xmath44 are the annihilation and creation operators for impurity electrons , @xmath45 is the hybridization function between our impurity and the bath , which encodes most of the lattice information . the on - site coulomb repulsion between the co d - electrons is given by @xmath46 where @xmath47 where @xmath48 denotes spherical harmonics and @xmath49 denote slater integrals . the ctqmc impurity solver gives us the impurity self - energy @xmath50 . since correlations are very local in real space , we embed this self - energy by expanding it in terms of quasi - localized atomic orbitals,@xmath51 ; @xmath52 where @xmath53 denote the sites of co atoms in the most general cluster - dmft implementation and @xmath54 denote the atomic degrees of freedom for each site . since , we perform single site dmft , @xmath55 becomes @xmath56 . we then solve the dyson equation in real space ( or an equivalent complete basis such as the kohn - sham basis ) according to the equation : @xmath57 this procedure is what we define as embedded dmft , because the self energy is embedded into a large hilbert space instead of constructing a hubbard - like model by downfolding to a few bands using wannier orbitals . this method has the advantage that the correlations are much more localized in real space compared to the wannier representation , which makes dmft a much better approximation . in addition , this formulation of dft+dmft can be shown to be derivable from the luttinger ward functional which makes the formulation stationary and conserving @xcite . while calculating the density of states(dos ) at @xmath58 for the different structures at different temperatures , we ensured that their fermi energies were adjusted such that the energy levels for the oxygen densities of states lay at the same energy values . this was required because there was an ambiguity in the value of the chemical potential at temperatures where the structure gave rise to an insulating band - gap and we believe an accurate comparison can only be made if some features of the dos are held fixed . this procedure required a shift in the chemical potential of some of the simulations of the order of 0.1 ev . the results we plot in fig 2b are obtained after these shifts are put in . figure 1 on the other hand plots the densities of states before any such post - processing has been done . this leads to small differences between the two figures . instead of fixing the oxygen levels , we also tried fixing the lanthanum f levels and this gave rise to very similar results . we firmly believe that our results displayed in figure 2b are robust and it is merely the relevant magnitudes of the y - axis values at high temperatures that fluctuate by a small amount ( depending on which features are held fixed ) and not the actual temperature at which the charge - gap closure takes place . we also do not plot the dos at @xmath58 but the average of the dos at five points around @xmath58 as this takes care of some of the numerical noise that creeps into our calculation due to both monte carlo noise and the errors in analytic continuation . we also tested our results by averaging over different number of points and no significant changes take place that would affect our claims .

the spin state transition in lacoo@xmath0 has eluded description for decades despite concerted theoretical and experimental effort . in this study , we approach this problem using charge consistent density functional theory + embedded dynamical mean field theory ( dft+dmft ) . we show , from first principles , that lacoo@xmath0 can not be described by a single , pure spin state at any temperature , but instead shows a gradual change in the population of higher spin multiples as temperature is increased . we explicitly elucidate the critical role of the lattice expansion and oxygen octahedral rotations in the spin state transition . we also show that the spin state transition and the metal - insulator transition in lacoo@xmath0 occur at different temperature scales . in addition , our results shed light on the importance of electronic entropy , which has so far been ignored in all first principles studies of this material . the spin state transition in lacoo@xmath0 has been the subject of intensive investigation for decades.@xcite this compound is established to be a narrow bandgap insulator at low temperature with pauli - like magnetic susceptibility . however between 90 - 150 k , it transitions to a local moment phase with a curie - weiss like susceptibility which reaches its peak around 150k . it also undergoes a gradual closing of the insulating gap and is known to be metallic above 600k @xcite . there is considerable debate regarding the mechanism of this transition , mainly due to the uncertainty regarding the multiplet of the @xmath1 ion which characterizes the excited state of the compound . the cobalt ion in lacoo@xmath0 is commonly considered to have a formal valence of @xmath2 and to be in the @xmath3 state . the scale of the crystal field splitting is comparable to hunds coupling energy scale . as a result , one would expect that as temperature is increased , there would be an entropy - driven transition from the low spin ( ls ) @xmath4 state with a fully filled @xmath5 shell ( @xmath6 ) to an @xmath7 high spin ( hs ) state ( @xmath8)@xcite . indeed there is experimental evidence that does support such a scenerio . electron spin resonance@xcite , neutron scattering @xcite , x - ray absorbtion spectroscopy , and magnetic circular dichroism experiments@xcite point towards a transition to an hs state . in addition , no inequivalent co - o bond is found in exafs experiements , which also supports the formation of an hs state due to the hs state not being strongly jahn - teller active @xcite . however , it has been noted that in order to explain the xas experimental data , one would have to assume that the crystal field grows with temperature , which is counter - intuitive.@xcite this led to some authors suggesting that there is an ls - hs alternating structure caused by breathing distortions@xcite @xcite and interatomic repulsion between the hs atoms.@xcite a competing explanation , whereby the excited state is the @xmath9 intermediate spin ( is ) state ( @xmath10 ) , has also become popular@xcite , mainly because of lda+u results which show that the is state is lower in energy compared to the hs state.@xcite the stability of the is state has been justified by the large hybridization of the co 3d electrons with neighboring o 2p electrons . this causes charge transfer between the ions resulting in the co ion having a @xmath11 structure according to the zaanen - sawatzky - allen scheme,@xcite which in turn would cause stabilization of the is state . the intermediate spin state hypothesis also seems to explain experimental findings such as raman spectroscopy , x - ray photoemission , other xas and eels spectroscopies , as well as susceptibility and thermal expansion measurements . @xcite . to summarize , there has been a significant amount of debate regarding the true nature of the spin state transition in lacoo@xmath0 . interest in this compound has also been enhanced in light of recent discoveries of ferromagnetism induced by sr ( hole ) doping @xcite , as well as experiments reporting strain induced magnetism in epitaxially grown thin films.@xcite . in addition , there have been reports of the emergence of a striped phase in thin films with alternating ls and hs / is regions.@xcite low temperature ferromagnetism has also been reported in experiments on @xmath12 nanoparticles@xcite . hence , there is great interest in understanding the true behavior of this material . in this letter , we use density functional theory + embedded dynamical mean field theory ( dft+dmft ) to analyze spin state transition of bulk @xmath12 . our implementation extremizes the dft+dmft functional in real space , thereby avoiding the downfolding approximation and uses the numerically exact ctqmc impurity solver@xcite . even though there are multiple recent studies that use dft+dmft on this compound @xcite , to our knowledge none of them provide a comprehensive analysis of all of the factors governing the transition such as octahedral rotations and electronic entropy . we show that i ) lacoo@xmath0 has large charge fluctuations and it is not possible to explain the spin state with a single multiplet at any temperature , ii ) the crystal field splitting very sensitively depends on the details of the crystal structure , and taking into account not only the thermal expansion but also the oxygen octahedral rotations is very important , and iii ) it is possible to stabilize an insulating phase ( without orbital order ) at intermediate temperatures where local moments are present , thereby showing that the metal - insulator transition is distinct from the spin state transition in this compound . we also show that iv ) electronic entropy difference between the high and low temperature states is necessary for the stabilization of the different spin states , which is a fact overlooked in various first principle studies so far . _ _ crystal structure of lacoo@xmath0__@xmath12 is a perovskite with space group @xmath13 with significant oxygen octahedral rotations ( @xmath14 ) and large thermal expansion . in order to investigate the effect of both of these structural parameters on the spin state transition , we not only use two different experimental structures for 4k and 1143k @xcite , but also create two cubic perovskite structures with the same inter - lanthanum separation as the two experimental structures but with no oxygen octahedral rotations . we refer to these 4 structures as @xmath15,@xmath16,@xmath17 and @xmath18 with ht(high temperature ) and lt(low temperature ) denoting the temperature for the structural data , and @xmath19 and @xmath20 denoting the presence and absence of rotations respectively . in fig 1 we show the density of states for all 4 structures , calculated at both low temperature and high temperature ( 116k and 1160k ) using density functional theory + dynamical mean field theory ( dft+dmft ) . unlike dft , which always predicts a metallic state , our calculations correctly reproduce an insulating ground state at low temperature for all the structures . the @xmath5 orbitals are below the fermi level whereas the @xmath21 orbitals are above the fermi level . the charge gap closes continuously with increasing temperature , and as a result , there is a large overlap in energy between the @xmath5 and @xmath21 orbitals at high temperatures . this overlap , however , is much smaller if the structures without rotations are simulated . ( see fig . 2b ) . the @xmath17 structure shows some overlap at high temperatures , and the @xmath18 structure almost remains an insulator for the entire range of temperature studied , with only a small overlap developing above 900k . this shows clearly that octahedral rotations play a large role in decreasing the strength of the crystal field splitting . this can be explained by a combination of factors . the rotation of the oxygen octahedra causes the misalignment of the crystal field of the o atoms with that of the la atoms , which normally reinforce each other in a perovskite with no octahedral rotations . this leads to an overall reduction of the effective crystal field which reduces the charge gap between the @xmath5 and the @xmath21 orbitals . in addition , this trigonal distortion also leads to a splitting of the @xmath5 orbitals into 2 + 1 orbitals , thereby again reducing the gap with the @xmath21 orbitals . the combination of these two effects seems to overcome the expected decrease in the bandwith of the @xmath21 orbitals.finally , note that there is a considerable overlap in energy of the o 2p orbitals with the co 3d orbitals , which is very important in producing charge fluctuations on the co ion , making it highly mixed - valent . with temperature for all four structures . ( b ) evolution of density of states at fermi level with temperature for all four structures . ] _ the spin state transition _ in order to focus on the spin state of the co ion , we calculate the expectation value of the magnitude of z - component of the spin @xmath22 . note that all our calculations are in the paramagnetic state and hence the value of @xmath23 . the results are presented in fig . 2a as a function of temperature . the largest value of @xmath24 at 1160k is seen for the @xmath15 structure , followed by the @xmath16 structure . this is in line with the stronger crystal field in the @xmath16 structure due to the smaller lattice constant . we also observe that the spin state transition starts at a higher temperature for the @xmath16 structure ( @xmath25580k ) compared to the the @xmath15 ( @xmath25380k ) . this is also consistent with the the low temperature structure having a higher stability for the ls state . the structures without rotations consistently show a lower buildup of higher spin states than the ones with rotations . the @xmath17 structure displays a spin state transition , but with an eventual high temperature value of @xmath24 that is lower . on the other hand , the @xmath18 structure shows almost no transition . this shows that the role that the octahedral rotations play in the reduction of the crystal field is essential for the spin state transition . figures 2a and 2b also show that the spin state transition and the charge gap closing occur at different temperatures , which is a trend that has been observed in experiment but has not been captured in earlier dmft simulations . for example , fig 2b shows that both the @xmath15 and the @xmath16 structures show a complete closure of the charge gap at @xmath25 600k whereas fig 2a shows that the spin state transition in the two structures occur at very different temperatures . _ nature of the excited spin state _ because of the large hybridization between co and o , the d orbitals of co have large charge fluctuations and all the four structures have an effective d - shell occupation of @xmath26 . therefore any analysis of the spin states in terms of the ls , is and hs states of the @xmath3 configuration of the co ion is necessarily inadequate . in fact , our calculations show that the @xmath11 configuration has a higher occupation probability than @xmath3 , and there are also significant probabilities for @xmath27 and @xmath28 fig . 3 shows the evolution of the occupation probabilities for the different values of @xmath24 with temperature . even at high temperatures , @xmath29 and @xmath30 ( the ls states for the even and odd occupancy sectors of the d orbital ) remain the states with the highest probability . however , with the increase of temperature , the weight of the higher spin states increases . at the onset of the transition , the initial change in the value of the spin state is predominantly caused by the excitation of the @xmath31 and the @xmath32 multiplets . the @xmath33 multiplet sees an increase in probability at higher temperatures ( above 500k ) and also follows a similar trend for all the structures except the @xmath18 structure , where all changes are very small . therefore , the initial signature of the transition is best seen in the behavior of the @xmath34 and @xmath32 multiplets , which can be said to be the hs multiplets for the @xmath3 and @xmath11 occupancies respectively . in fig . 4 , we show the occupancy histograms below and above the transition ( at 116k and 1160k).(ctqmc gives us access to the state space probability for each of the 1024 states of the d orbital . however , in order to aid visualization , we only show states which have an occupation probability above 0.001 in any of the structures at any temperature . ) this figure displays clearly how the transition is marked by the excitation of states in the higher spin multiplets . we see that the low temperature state for all of the structures is marked by the presence of a few states with large probability ( mainly corresponding to the @xmath35 and @xmath30 states ) . as the spin state transition sets in , a large number of higher spin states get excited and the ls spin states lose weight . note that the high spin states are highly degenerate so there is no one large peak for the high spin states , but a multitude of lower peaks . this supports the idea that the transition is primarily an entropy driven transition . we can also get a good idea of the relative strengths of the transition for the different structures : the largest change occurs in the @xmath15 structure , and the smallest one happens in the @xmath18 . _ contribution of electronic entropy _ according to the entropy driven transition scenerio , which is supported by calorimetric measurements@xcite , lacoo@xmath0 favors higher spin multiplets at elevated temperatures because of the associated gain in electronic entropy as a result of the high degeneracy of these high spin states - a point missed by first principles calculations at the level of dft . access to higher spin states is also made easier by a larger lattice constant due to the reduced crystal field , so the gain in electronic entropy could also be a driving factor for the large thermal expansion seen in this material . we calculated the contribution of the electronic entropy to the free energy using our state of the art dft+dmft implementation@xcite . in particular , we evaluated the free energy and the electronic entropy for both the 4k and 1143k structures ( @xmath16 and @xmath15 ) at 1160k to predict if the structural changes make a considerable difference . the @xmath15 structure is indeed much higher in electronic entropy compared to the @xmath16 structure at 1160k ; the difference in @xmath36 between these two structures is @xmath25 110 mev per formula unit . this unusually large difference emphasises the importance of electronic entropy to the transition.we also calculate the energy difference between the @xmath15 and @xmath16 structures to be @xmath25 70mev at 1160k with the @xmath16 being lower in energy . thus we see that when the entropy is taken into account and the free energy ( f = e - ts ) is calculated , the high temperature structure @xmath15 becomes more stable purely due to the contribution of electronic entropy . this result therefore confirms the structural phase transition that is observed as a function of temperature . so , we can conclude that the electronic entropy , which has been ignored in many first - principles studies of this material , is a leading factor in creating an anomalous thermal expansion and driving the material to a high spin state . _ summary _ we studied the spin state transition of lacoo@xmath0 using state of the art fully charge self consistent dft+dmft . by using different experimental and hypothetical crystal structures , we disentangled the effect of different components of the crystal structure and showed that both the thermal expansion and the presence of oxygen octahedral rotations have tremendous effect on the spin state transition of lacoo@xmath0 . our single site dmft approach reproduced not only the spin state transition but also the intermediate phase which has nonzero magnetic moment but is insulating . this shows that the spin state and the metal - insulator transitions occur at different temperature scales and that the magnetic - insulating phase can be reproduced without necessarily involving cell doubling via mechanisms such as breathing distortions of spatially inhomogenous mixed spin states . our results emphasize the importance of charge fluctuations on the co ion due to hybridization with the o anions , and thus point to the inadequacy of a simple spin state picture with only one formal valence . while the spin state transition is concurrent with a sudden change in occupation in the high spin multiplets , low and intermediate spin states also have significant occupation in the whole temperature range . finally , our work is the first calculation of the electronic entropy of lacoo@xmath0 and it points to the fact that the difference of the contribution of entropy to the free energy is significant and is large enough to drive the spin state transition in this material . _ acknowledgements _ tb was supported by the rutgers center for materials theory.bc and kh were supported by the nsf - dmr 1405303 . 46 natexlab#1#1bibnamefont # 1#1bibfnamefont # 1#1citenamefont # 1#1url # 1`#1`urlprefix[2]#2 [ 2][]#2 , , , * * , ( ) , issn , http://www.sciencedirect.com/science/article/pii/003189146490182x . , , , , , * * ( ) . , * * , ( ) , http://link.aps.org/doi/10.1103/physrev.155.932 . , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.6.1021 . , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.65.220407 . , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.55.4257 . , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.67.172401 . , , , , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevlett.97.247208 . , , , , , , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevlett.97.176405 . , , , , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevlett.102.026401 . , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.81.035101 . , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.5.4466 . , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.71.024418 . , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.66.094408 . , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.54.5309 . , , , , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.77.045123 . , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.68.235113 . , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevlett.55.418 . , , , , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.47.16124 . , , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevlett.71.4214 . , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevlett.99.047203 . , , , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.66.020402 . , , , , , , , , , * * ( ) . , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.67.224423 . , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.67.140401 . , , , , , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.69.094417 . , , , , * * , ( ) , , http://dx.doi.org/10.1143/jpsj.63.1486 . , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevlett.110.267204 . , , , , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.88.035125 . , , , , , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.75.144402 . , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.77.014434 . , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.79.054409 . , , , * * , ( ) , http://scitation.aip.org/content/aip/journal/apl/94/12/10.1063/1.3097551 . , , , , , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.79.024424 . , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.79.092409 . , , , , , , , , , , , * * , ( ) , , , http://dx.doi.org/10.1021/nl302562 . , , , , , , , , , * * , ( ) , http://stacks.iop.org/0953-8984/27/i=17/a=176003 . , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.77.045130 . , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.86.184413 . , , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.86.195104 . , , ( ) . . , , , , , * * , ( ) , http://link.aps.org/doi/10.1103/physrevb.55.14103 . , * * , ( ) , http://link.aps.org/doi/10.1103/physrevlett.115.256402 . * supplementary material for + `` role of entropy and structural parameters in the spin state transition of @xmath12 '' *

discovering planets outside the solar system is one of the key goals of modern astronomy . since the first detection ( mayor & queloz 1995 ) using the radial velocity technique , we have come to know of the existence of @xmath1 extra - solar planets . while radial velocity monitoring of nearby stars remains the most successful technique in this venture , a promising alternative is slowly gaining ground . this so - call ` transit ' method focuses on detecting planets that transit their host stars . it requires continuous observing of a large number of stars , but can provide independent information concerning planet characteristics otherwise unobtainable by the radial velocity technique . a couple dozens planet transit searches are currently underway ( see horne 2003 for a review ) . among these , the optical gravitational lensing experiment ( ogle ) has announced a large number of planetary transit candidates ( udalski et al . 2002a ; and follow - ups ) , and a number of these candidates have been confirmed spectroscopically ( e.g. , konacki et al . 2003a , 2004a , 2004b ; bouchy et al . 2004 ; pont et al . 2004 ; torres et al . these are selected to resemble close - in gaseous planets ( period @xmath2 days ) transiting main - sequence stars in the galactic disk . many of the ogle candidates ( e.g. ogle - tr-56 , @xmath3 days ) have orbital periods a factor of @xmath4 shorter than the closest planets discovered by the radial velocity technique . the questions arise whether this is a new population of planets and why they are not seen by the radial velocity method . if confirmed to be genuine planets , they pose intriguing challenges for understanding planet formation , migration and survival . what types of objects can masquerade as planet transits ? the success of ogle in detecting planet transits relies partly on the extreme crowdiness and hence large base numbers in its fields . however , this advantage also brings on the masqueraders a faint eclipsing binary system can project coincidentally ( or in some cases , associate physically ) near a brighter disk star . the deep eclipses and the ellipsoidal variations from the binary are then diluted by the light from the brighter star into shallow eclipses and little variations out of transit , mimicking the signatures of a transiting planet . sirko & paczynski ( 2003 ) carefully studied the light - curves of these candidates and concluded that on average @xmath5 of these are contaminations by eclipsing binaries , with the shorter - period ones more likely to be so . spectroscopic follow - up of a large number of these candidates ( konacki et al . 2003a ; dreizler et al . 2003 ) also reached a similar conclusion , though at a much greater observational expense . moreover , spectroscopic observations are not always able to separate the blends from genuine planetary objects as the blended main star may show little or no velocity variations ( see , e.g. torres 2004 ) . high - quality photometric light - curves can be used to rule out the blends ( seager & mallen - ornelas 2003 ) , but such data are difficult to obtain for the crowded ogle fields . it is also possible to exclude some blending configurations by comparing the observed light - curves against synthetic light - curves constructed using model isochrones ( torres 2004 , 2005 ) . this latter technique is more powerful if the blend and the main star are physical triples and therefore are likely coeval . our aim in this work is to provide an independent new method to recognize blends . our method is efficient , assembly - line in style , and robust . it uses original imaging data and does not require any follow - up work . as such , this method may be broadly adopted in light of the fact that ogle and other transiting searches are likely to produce an increasing number of planet candidates in the future . moreover , our method is more suitable for detecting blends that are not physically associated ( coincidental alignment ) and thereby complements the light - curve method of torres ( 2004,2005 ) . we propose to use the fact that a blended system , albeit unresolved in the images , always leaves a tell - tale sign : the shapes of their images are not round . the magnitude of the ellipticity depends on the angular separation and the relative brightness between the primary star and the seconary blend . as we show below , we can measure the shape of a typical blend in the ogle fields with great precision . comparison of the shape in and out of transit allows us to identify blends with eclipsing binaries . for instance , a star blended with an eclipsing binary with an undiluted eclipsing depth of @xmath6 is expected to exhibit a factor of 2 change in its ellipticity between the two phases . the actual change in shape may be smaller , though still detectable , as a typical star in the ogle fields is multiply blended . the success of this technique depends critically on how well we measure the shape of a star , in relation to other stars in the same image . this is where the only major obstacle in this method arises : the point - spread - function ( psf ) varies across the image due to a multitude of distortions in the photon pathway . it also varies with time as the pathway changes and the seeing fluctuates . psf anisotropy and seeing change the shapes of the objects and renders raw measurements of the ellipticity unreliable . a similar problem exists in weak gravitational lensing , where one has to disentangle the lensing induced distortions in the shapes of faint galaxies from these observational effects . fortunately , the weak lensing community has studied this problem in great detail and has come up with solutions which we adapt to the case in hand . we note that the method we develop here have aspects unique to the stellar problem . among the hundreds of transiting systems published by the ogle - iii team ( udalski et al . 2002a , 2002b , 2002c ) , we choose to focus our initial efforts on two candidates , ogle - tr-3 and ogle - tr-56 . on the basis of spectroscopic follow - up observations with 8 m class telescopes , these two candidates were identified as likely planetary candidates since their host stars show little or no velocity variations ( konacki et al . 2003a ; dreizler et al . 2003 ) . tr-56 undergoes genuine flat - bottom transit and has detectable radial velocity variations , both consistent with a planet explanation . for this candidate , the blending scenario was examined in detail by torres et al . ( 2005 ) who were able to confirm the planetary nature of this object using a combined analysis of the light curve and radial velocity measurements . the interpretation for tr-3 , however , is more open to debate . it shows no significant velocity variations , its light curve contains hints of a secondary eclipse as well as out - of - eclipse fluctuations ( sirko & paczynski 2003 ; konacki 2003a ) . the method presented here provides a completely independent assessment of the identities of these two objects . we briefly describe the data in 2 . the shape measurement technique is described in detail in 3 . in 4 we provide an extensive test of our analysis and present the results for the two planet transit candidates in 5 . the data we analyze were obtained during the third phase of ogle ( ogle iii , udalski et al . these were collected using the 1.3 m warsaw telescope at the las campanas observatory , equipped with the 8k mosaic camera . the field of view of the camera is about 35 by 35 arcminutes , with a pixel scale of @xmath7/pixel . the observations were done in the @xmath8-band , and have exposure times of 120s . our analysis does not require the full field , so instead we use small cuts of 600 by 600 pixels , not necessarily centered on the target candidate . for tr-3 we have 109 images in - transit and 308 images out - of - transit , whereas we have 65 and 259 images , respectively , for tr-56 . we retrieved all in - transit images , which results in a broad range in seeing . to minimize the systematic errors caused by the seeing correction ( see 3.2 ) , we have selected out - of - transit images such that their seeing distribution resembles that of the in - transit data . this is illustrated in figure [ seeing_dist ] . in this section we discuss the shape measurements , focussing on how to deal with the variable psf . the methodology is based on the techniques developed for weak gravitational lensing applications ( e.g. , see kaiser , squires & broadhurst 1995 ; hoekstra et al . 1998 ) , and we adopt their notation . the correction for the psf can be split into two separate steps . the first one is the correction for the anisotropic part of the psf , which induces an ellipticity in addition to the intrinsic ellipticity of the object under investigation . the second step is the correction for the circularization by the psf ( i.e. , seeing ) , which typically lowers the ellipticity . for both steps , we require a set of comparison stars which can be presumed to be intrinsically round . to quantify the shapes , we use the central second moments @xmath9 of the image fluxes and form the two - component polarisation @xmath10 because of photon noise , unweighted second moments can not be used . instead we use a circular gaussian weight function , with a dispersion @xmath11 : @xmath12 where @xmath13 is the pixel number in the direction of the @xmath14-axis , pointing away from the centroid of the object . for the weight function @xmath15 we adopt a gaussian with a dispersion @xmath11 . for the analysis presented here , the weighted moments are measured from the images within an aperture with a radius of 6 pixels , and we take @xmath16 pixels , which is the optimal width for a seeing of 09 . these choices suppress the contributions from nearby stars . in practice , the psf will not be isotropic . instead , the images are typically concolved with an anisotropic psf , which induces coherent ellipticities in the images . in order to recover the true `` shape '' of the blend , we need to undo the effect of the psf anisotropy . the correction scheme we use is based on that developed by kaiser , squires & broadhurst ( 1995 ) , with modifications described in hoekstra et al . ( 1998 ) . the effect of an anisotropic psf on the polarisation @xmath17 of an object is quantified by the `` smear polarisability '' @xmath18 , which measures the response of the polarisation to a convolution with an anisotropic psf , and can be estimated for each object from the data ( see hoekstra et al . 1998 for the correct expressions ) . having measured the polarisations and smear polarisabilities , the corrected polarisations are given by @xmath19 where @xmath20 is a measure of the psf anisotropy . it is measured using a true point source by @xmath21 where @xmath22 are the measured ellipticity of the point source and @xmath23 the diagonal components of its smear polarisability tensor . formally , the correction requires the use of the full two by two tensor , but the off - diagonal terms are typically small . examination of the measured values indicates that they are consistent with noise . we therefore only use the diagonal terms in the correction for psf anisotropy . this correction has been tested extensively in the case of galaxies convolved with an anisotropic psf ( e.g. , hoekstra et al . 1998 ; erben et al . 2001 ) . for this application , the correction works well , because galaxies are centrally concentrated , and their shapes are well characterized by the quadrupole moments . in the case of two or more nearby point sources the situation is somewhat different : the shape is not well described by a simple quadrupole , and higher order moments are expected to contribute to the polarisation . to explore this in more detail , we created images that were convolved with a moffat ( 1969 ) profile and then convolved with a line ( which simulates the psf anisotropy ) . a detailed discussion of this study can be found in the appendix . here we summarize the main conclusion . the simulations indicate that the correction given by eqn . 3 is incomplete and that an additional term proportional to @xmath24 ( the total size of the anisotropy ) is needed . this leads to an improved correction for psf anisotropy , albeit empirical , given by @xmath25 where the value of @xmath26 depends on the configuration of the point sources ( separation and flux ratio ) and the seeing . we found that the size of @xmath26 is proportional to the polarisation of the object . this is supported by an examination of the residuals in the shapes of the objects in the ogle data . the fact that @xmath26 is proportional to the polarisation is not surprising : when the polarisation is larger , the higher order moments become more important . however , in the case of ogle , blends with more than one source are likely . consequently , it is difficult to compute the expected value of @xmath26 . instead , we determine the value empirically by fitting a term proportional to @xmath24 to the shape measurements . the psf anisotropy depends on the position of the object on the chip and it typically varies with time . fortunately , it is possible to characterize the spatial variation of the psf anisotropy with a low order polynomial model fitted to a subsample of the objects identified as suitable stars ( i.e. , the stars should be bright but not saturated ) . this works particularly well for the data used here , as we use relatively small regions around the ogle transit candidates . for the analysis here we model the spatial variation by a second order polynomial . such a model is derived for each exposure and used to undo the effect of the psf anisotropy . the derivation of the psf anisotropy model implicitely assumes that the set of comparison stars are intrinsically round : i.e. , the observed polarisation is solely caused by psf anisotropy . it is possible to reject wide separation binaries ( or blends ) from this set on the basis of their large ellipticities , but it is more difficult to reject stars that have a small intrinsic ellipticity because of a companion . however , so long as the number of comparison stars is sufficiently large , because their position angles are uncorrelated with each other and with the psf anisotropy , we still can obtain an unbiased model for the psf anisotropy . in the case of ogle , severe crowding means most bright stars are blended with fainter stars . as a result , the noise introduced by the blends can be substantial . nevertheless , we can find a set of brighter stars which are comparatively less affected by blending and provide good estimates for the psf anisotropy . the result of this procedure carried over one frame is presented in figure [ psf_an ] : we detected significant psf anisotropy , and found that a second - order polynomial is sufficient to remove the psf anisotropy across the whole image , leaving residual @xmath27 and @xmath28 scattering randomly around the zero - level . in the absence of blending and shot noise , the corrected ellipticities should all be zero ( assuming the model used to correct for the psf anisotropy is perfect ) . however , blending gives rise to non - zero ellipticities for the stars and is partly responsible for the residual anisotropy in figure [ psf_an ] . we measure this `` intrinsic '' ellipticity of the stars used in the psf anisotropy correction using repeated observations taken by the ogle team ( as this procedure reduces the shot noise ) . we then subtract the `` intrinsic '' ellipticities from the observed ones and obtain an improved fit . we found that this iteration had little effect on the results , because of the random orientations of the blends . large values of psf anisotropy are typically a nuisance , as they imply larger corrections . however , the large range of psf anisotropy ( fig . [ psfan_dist ] ) exhibited by the ogle observations is helpful for the purpose of our paper : it allows us to examine the accuracy of the correction for psf anisotropy in more detail , and to understand the limitation of our algorithm . the second step in our correction procedure is to account for the effect of seeing , i.e. , the isotropic part of the psf . typically , an object will appear rounder with increasing seeing . an example is presented in the left panels of figure [ seeing_var ] , which shows the ellipticities for one of the stars in tr-3 field as the seeing varies . the dependence on seeing can be rather complicated , with some configurations appearing more eccentric with increasing seeing . in the simple case of a single blend , one could use simulations to attempt to determine the seeing dependence . this is not feasible here , because the target stars are on average blended with 1.5 objects . instead , we make use of the fact that the observations span a large range in seeing ( see fig . [ seeing_dist ] and fig . [ seeing_var ] ) to remove the seeing dependence empirically . using the following model , individually tuned for each star , @xmath29 this second - order fitting is sufficient to remove any visible seeing dependence for , e.g. , the object shown in figure [ seeing_var ] . from now on , we report the shape measurement for a fiducial seeing , taken to be 1 arcsecond . we selected a sample of a total of 171 stars around the transit candidate in the two fields . these stars have some range in brightness and ellipticity . we demonstrate below our capabilities in removing the effects of psf anisotropy and seeing . to examine the accuracy of the psf anisotropy correction , we split the in - transit data into two subsets of similar sizes : one with large psf anisotropy ( @xmath24 ) and one with small @xmath24 . the two subsets have a similar range in seeing . in figure [ psfcor ] , we present the differences in ellipticity measurements for these comparison stars , without correcting for psf anisotropy ( upper panels ) , after using equation ( 3 ) to correct for the anisotropy ( middle panels ) , and after using equation ( 5 ) to correct for the anisotropy ( lower panels ) . this experiment convinced us that we can remove psf anisotropy successfully from our data . the reduced @xmath30 for the results in the lower panels of figure [ psfcor ] is close to unity , indicating that the estimated errors are a fair estimate of the statistical uncertainty in the measurements . in producing figure [ psfcor ] , we have applied a seeing correction that is based on the combined in - transit data , minimizing the systematics caused by the latter correction . as mentioned above , the seeing ranges are similar for both samples , so psf anisotropy is the only systematic relevant for comparison . upper panels in figure [ shape_zoom ] expand the view from the lower panels of figure [ psfcor ] for the small ellipticity objects , while lower panels in figure [ shape_zoom ] shows results from the same procedure using out - of - transit data . for some of the brighter objects , the achieved error bars are as small as @xmath31 . this capability to measure shapes accurately brings about another potential application for the algorithm described here : finding binary stars that are too close , or are too different in fluxes , to be resolved ( also see 6 ) . we now examine the reliability of the correction for seeing . if this is successful , we will be able to accurately measure the in and out - of - transit ellipticity changes in planet candidates , and constrain the blending scenario . as shown in figure [ seeing_dist ] , the seeing distributions for the in and out - of - transit data were chosen to be similar , i.e. we selected out - of - transit images such that the two distributions match . this approach minimizes the sensitivity of our results to systematic errors caused by the adopted seeing correction . we fit equation ( 6 ) to the in and out - of - transit measurements separately . the resulting differences between the in and out - of - transit data are presented in figure [ difference ] . panels a and b show the differences in @xmath32 and and @xmath33 respectively . the lower panels in figure [ difference ] show histograms of the differences in units of the estimated measurement uncertainty . for a normal distribution , this should be a gaussian with a dispersion of 2 , which is indicated by the solid smooth line . this gaussian provides a fair match to the observed scatter , but the data show more outliers than what would be expected from a normal distribution . for @xmath34 we obtain a reduced @xmath35=1.31 and for @xmath36 we find a similar value of @xmath35=1.37 , larger than the expected value around unity . these values reduce to 1.14 and 1.16 respectively when we reject objects that are more than @xmath37 away from zero . the bootstrap analysis provides an estimate of the random error but not of the systematic error . the results presented in figure [ difference ] suggest that the estimated errors are typically correct , but in a few cases , residual systematic errors are still present in the data leading to the excess of outlyers . nevertheless , the results presented in figure [ difference ] do suggest that for most objects we can measure the difference in shapes between the in and out - of - transit data accurately . we have attempted to identify what is causing some of the outlyers , but have not been able to find an obvious way to improve the measurements . we suspect that it might be due to imperfections in the correction for psf anisotropy . we also note that some of the objects are not present in all exposures ( as they lie too close to the edge ) , which might lead to differences in the actual seeing distributions , which in turn can lead to systematic errors in the shape differences . the size of the final error bar as used in figure [ difference ] depends on the number of frames used as well as on the apparent magnitude of the object : the shape measurements in a single frame will be noisier for fainter stars . this is demonstrated in figure [ error ] , which shows the error in @xmath32 and @xmath33 as a function of apparent magnitude for the out - of - transit shape measurements . as expected , the errors increase with magnitude . this is more clearly seen for `` rounder '' objects , which are affected less by the correction for seeing and the last term in equation 5 for the psf anisotropy correction . we also computed the smallest possible error bar as a function of apparent magnitude using simulated images . in these images , which have the same noise properties as the ogle data , we measured the scatter in the shape of a point source . in this case , the error is solely due to poisson noise . the result is given by the dashed curve in figure [ error ] . the actual error bars are larger , because of the uncertainties introduced by the empirical corrections for psf anisotropy and seeing . finally , figure [ error ] also demonstrates that the accuracy with which one can measure shapes is excellent : the typical uncertainty for a star with @xmath38 is @xmath39 . the results presented in the previous section demonstrates our ability to accurately measure the shapes of objects in the ogle fields . in this section we present results for the two ogle planet transit candidates . table [ tab_shapes]a lists the final polarisations for tr-3 and tr-56 at a fiducial seeing of 1 arcsecond measured from the out - of - transit images . we detect a significant polarisation for both transit candidates , thus implying that they are both blended with other sources . in fact , most stars studied in the crowded ogle fields show evidence of blending ( or even multiple blending ) . within a circle of @xmath40 radius , an average star is surrounded by @xmath41 companions , with a mean flux ratio of @xmath42 and a mean sepration of @xmath43 . the resulting average ellipticity of the blend depends mainly on the brightness of the primary star : the brighter the star , the smaller the ellipticity . figure [ edist ] shows the distribution of @xmath32 and @xmath33 for the analysed stars in the fields of tr-3 and tr-56 ( indicated by the crosses ) . the two transit candidates are indicated by the open circles , with tr-3 being the point on the left . although the distribution is peaked towards round objects , the observed ellipticities for the transit candidates are by no means anomalous . to test whether the planet - like transit is caused by a blended eclipsing binary , we also list in table [ tab_shapes]b differences in shape between the out and in transit data for these two stars : a significant change in shape would confirm that the blend is an eclipsing binary . we detect no change in shape in tr-56 , suggesting that the observed transit is a genuine planetary transit , in line with evidences from radial velocity , line - curve analysis and isochrone fitting ( e.g. , torres et al . 2005 ) . in tr-3 , however , we do observe a change in shape : the ellipticity in - transit is larger . the errors inferred from the bootstrap analysis suggest a significance of @xmath44 . however , the results presented in section 4.2 and figure [ difference ] indicate that the distribution of errors is not exactly gaussian , but has tails . we therefore need to account for the possibility that the change in shape is caused by residual systematics . to this end , a more conservative estimate of the significance of the change in shape for tr3 can be obtained by considering the fraction of studied objects that show a difference at least as large as tr-3 . of the objects in the fields of tr-3 and tr-56 , accurate shapes could be determined for 171 of them . none of these objects show an ellipticity change as large as tr-3 , and we can only derive a lower limit to the probability for the observed shape change in tr-3 to be caused by systematic effects : the probability is less than @xmath45 . this is larger than the probability of a @xmath46 event ( @xmath47 ) but still sufficiently small for us to conclude that it is very likely that tr-3 is indeed a blend with an eclipsing binary system . if tr-3 is only singly blended , its polarisation should decrease by a factor of 2 during eclipse , an effect that should be emminently detectable . however , we find that its ellipticity increases by @xmath48 during eclipse . this can be explained if tr-3 is multiply blended . in fact , we have also measured tr-3 s polarization ( out - of - transit ) using different weight functions ( @xmath11 in eq . [ 2 ] ) . it varies with @xmath11 differently than a singly - blended object would , suggesting that it is indeed multiply blended . in the case of multiple blending , provided that the primary star is much brighter than the blending stars , the resulting ellipticity is given by @xmath49 where @xmath50 is the number of blends , and @xmath51 is the contribution from each blend . consequently , if two blends have opposite signs for @xmath52 , the polarisation can actually increase during an eclipse . if we assume that the observed eclipse is caused by a blend with an eclipsing binary , with a eclipse depth @xmath6 ( i.e. , a full eclipse of an equal mass system ) , we can place limits on the configuration of the blend . to do so , we note that the depth of the observed transit is @xmath53 , which implies that the flux of the presumed binary contributes @xmath42 to the total flux . under these assumptions , the change in ellipticity indicates that the binary is located @xmath54 arcseconds from the brighter star . if the eclipse depth is reduced to 25% , the presumed binary contributes 8% of the flux instead , and the separation decreases to @xmath55 arseconds . these numbers are below the resolution limit of the photometry ( by analysing centroid shift in and out of eclipse , @xmath56 ) , and the blend is likely a background source ( as opposed to a physical triple with the main star ) . interestingly , by examining the light - curve in detail , konacki ( 2003b ) have also come to a similar conclusion that tr-3 is likely a blend of a background eclipsing binary with a forground bright star . our result here confirms their suggestion and predicts the position of the blend . the sepration from the main star is small but should be detectable by hst observations . we have presented an algorithm that can detect blends of bright stars with fainter eclipsing binaries . such systems contaminate searches for transiting planets , in particular in crowded fields where blends are common . this technique provides a cheap way to find such blends , thus minimizing the amount of time required on large aperture telescope for spectroscopic follow - up of planet candidates . we have demonstrated the accuracy with which shapes can be measured using imaging data from the optical gravitational lensing experiment ( ogle ) . our method requires a careful correction of the point spread function which varies both with time and across the field . to this end we have adopted a method developed in weak gravitational lensing with modifications necessary for this particular application . we have tested the correction for psf anisotropy in great detail , using a sample of 171 stars surrounding the two planet transit candidates studied here . comparison of samples with large and small psf anisotropy indicates that this correction can be applied with great accuracy . for a star with an apparent magnitude @xmath38 , we obtain a @xmath57 uncertainty of @xmath58 in the polarisation . applied to ogle - tr-3 and ogle - tr-56 , two of the planetary candidates , we show that both systems are indeed blended with fainter stars , as are most other stars in the ogle fields . in the case of tr-56 we do not detect a change in shape in and out of transit , consistent with it indeed being a genuine planetary object . for tr-3 we observe a significant change in shape . if we adopt the error bars from the bootstrap analysis , the significance is @xmath44 . however , the distribution of errors is not precisely gaussian , but has tails . a more conservative estimate of the significance , estimated from the observed distribution of shape differences , provides an upper limit of @xmath59 to the probability that the observed change is caused by residual systematics . our results favour the scenario where tr-3 is caused by a blend with a background eclipsing binary , in line with evidences from other studies . a number of studies have appeared since the ogle announcement of transit candidates , mostly aiming at distinguishing blends from genuine planets . in contrast to some of these studies which carry out follow - up spectroscopy using large telescopes , our approach uses original imaging data and is a value - added application . moreover , unlike studies which perform detailed light - curve fitting or isochrone stellar model fitting , our method is assembly - line in style and can be applied to a large number of transit candidates without too much human interaction . lastly , our technique is especially suited to finding blends that are not physically associated with the bright star , and is therefore complementary to the isochrone fitting technique which is more powerful for the physical triple case . given the efficiency in dealing with a large number of objects without requiring additional data , the shape method may also be useful for other planetary transit searches , in particular the nasa kepler mission . this transit mission aims to detect @xmath60 giant inner planets and @xmath61 terrestrial planets . recently the target survey area has been moved to a higher galactic latittude to reduce the confusion by blends with eclipsing binaries . a quick examination of the usno - b catalogue ( monet 2003 ) in this new field suggests that the stellar density is @xmath62 times less dense than that in the ogle field , with a similar number distribution in stellar magnitudes . however , stars in kepler have a psf of @xmath63 radius , we therefore expect each bright star ( @xmath64 ) to have @xmath65 companions within the psf envelope , compared to @xmath41 ( @xmath66 ) in the ogle case . the probability of blending with an eclipsing binary is likely enhanced by a similar ratio . more study is necessary to determine the false - positive rate due to blending in kepler , armed with the experience from ogle . nevertheless , we expect that our shape technique can be readily applied to this mission . the achieved accuracy in measuring the shape of stars also bodes well for another potential application of our algorithm : finding binary stars that are too close to be resolved , yet too far apart for radial velocity studies . by detecting small deviations from circularity , we should be able to discover intermediate separation binaries ( @xmath67 ) with flux ratio as low as @xmath68 , within a large volumn of our galaxy . this will not only complement existing binary searches , but its high efficiency may also disclose binary population with an unprecedented rate such as to enable new and meaningful statistical studies . in a subsequent paper we will investigate this application in more detail , and apply it to wide field imaging data from the explore project ( malln - ornelas et al . 20003 ; yee et al . 2003 ) , which were obtained with the aim of finding transiting planets . au acknowledges support from the polish kbn grant 2p03d02124 and the grant `` subsydium profesorskie '' of the foundation for polish science . as indicated by figure [ psfcor ] the correction for psf anisotropy using equation 3 leaves a systematic residual , roughly proportional to the polarisation . this correction scheme has been used extensively in weak lensing applications , and has been tested in great detail . the difference between the analysis of galaxies and the blends considered here , is that the shapes of galaxies are well characterized by their quadrupole moments . in the case of two point sources , higher order moments contribute to the moments . in this section we examine how to improve the correction for psf anisotropy , in particular we justify the use of equation 5 . unlike the case for galaxies ( kaiser et al . 1995 ) , this problem is too complicated to solve analytically . instead we study the effect of psf anisotropy on simulated images of two point sources . we also note that new methods have been developed in which the images of the objects are decomposed into a series of localized basis functions . for instance , bernstein & jarvis ( 2002 ) use laguerre expansions , whereas refregier ( 2003 ) adopted weighted hermite polynomials . the advantage of these methods might be that they can quantify higher order moments of the images . nevertheless , as we will show below ( and in 4 ) , the empirical extension of the kaiser et al . ( 1995 ) method is adequate for the results presented here . we create well oversampled images of two point sources , and convolve these with a moffat function , with a width given by the required seeing . these images are then convolved with a `` line '' , which simulates the effect of psf anisotropy . examples for two configurations are indicated by the thin solid lines in figure [ cormod ] . the results presented in this figure are for a case where the psf anisotropy is given by @xmath69 alone , with the other component set to zero . a single point source would show a linear trend with @xmath69 , but because of the second point source the slope changes when @xmath69 changes sign . the next step is to correct these polarisations for psf anisotropy . if we use equation 3 , we obtain the dashed lines in figure [ cormod ] . in both cases we see a clear residual @xmath70 , and not just @xmath71 . these results have led us to consider an additional term in the correction @xmath72 , leading to equation 5 . based on a large set of simulations , we found that the slope of the trend is proportional to @xmath17 . this is not completely surprising , because the amplitude of the polarisation is a measure of the importance of the second point source , and consequently a measure of the relevance of higher order moments . hence the additional term in the correction for psf anisotropy is given by @xmath73 . we examined what value for @xmath74 yields the best correction . the results indicate that @xmath74 depends on the configuration , in particular on the flux ratio . for instance , in figure [ cormod]a we used @xmath75 and in fig . [ cormod]b we obtained the best result for @xmath76 to obtain the improved corrections , indicated by the dotted lines . overall , the range in @xmath74 appears to be fairly small , although its value is difficult to determine if the polarisations are small ( i.e. , when the additional correction is small ) . we note that the conclusions listed above are based on our empirical study of the simulated images . using equation 5 , the correction for psf anisotropy works well for most of the situations ( as is the case for the ones presented in fig . [ cormod ] ) . in some extreme cases , however , with large values for the polarisation and psf anisotropy , the correction leaves significant residuals ( relative errors as large as 10% ) . yee , h.k.c . , malln - ornelas , g , seager , s. , gladders , m.d . , brown , t. , minniti , d. , ellison , s. , & malln - fulleron , g. 2003 , to appear in the proceedings of the spie conference : astronomical telescopes and instrumentation , astro - ph/0208355

we present an algorithm that can detect blends of bright stars with fainter , un - associated eclipsing binaries . such systems contaminate searches for transiting planets , in particular in crowded fields where blends are common . spectroscopic follow - up observations on large aperture telescopes have been used to reject these blends , but the results are not always conclusive . our approach exploits the fact that a blend with a eclipsing binary changes its shape during eclipse . we analyze original imaging data from the optical gravitational lensing experiment ( ogle ) , which were used to discover planet transit candidates . adopting a technique developed in weak gravitational lensing to carefully correct for the point spread function which varies both with time and across the field , we demonstrate that ellipticities can be measured with great accuracy using an ensemble of images . applied to ogle - tr-3 and ogle - tr-56 , two of the planetary transit candidates , we show that both systems are blended with fainter stars , as are most other stars in the ogle fields . moreover , while we do not detect shape change when tr-56 undergoes transits , tr-3 exhibits a significant shape change during eclipses . we therefore conclude that tr-3 is indeed a blend with an eclipsing binary , as has been suggested from other lines of evidence . the probability that its shape change is caused by residual systematics is found to be less than @xmath0 . our technique incurs no follow - up cost and requires little human interaction . as such it could become part of the data pipeline for any planetary transit search to minimize contamination by blends . we briefly discuss its relevance for the kepler mission and for binary star detection . = -15 mm

in recent years there has been a renewed interest in the study of doped transition metal oxides like @xmath1 . these materials exhibit interesting phenenoma like the correlation induced metal insulator transition . although there are several experimental data available right now @xcite it is still quite difficult to tackle these substances theoretically . realistic models have to take into account several bands and are to be explored at finite doping . the most promising way towards a theoretical description , perhaps , is the limit of large spatial dimensions @xcite , which defines a dynamical mean field theory for the problem . this limit can be mapped onto an impurity model together with a selfconsistency condition which is characteristic for the specific model under consideration @xcite . the mapping allows to apply several numerical and analytical techniques which have been developed to analyse impurity models over the years . there are different approaches which have been used for this purpose : qualitative analysis of the mean field equations @xcite , quantum montecarlo methods @xcite , iterative perturbation theory ( ipt ) @xcite , exact diagonalization methods @xcite , and the projective self - consistent method , a renormalization techique @xcite . however , each of these methods has its shortfalls . while quantum montecarlo calculations are not applicable in the zero temperature limit , the exact diagonalization methods and the the projective self consistent method yield only a discrete number of pols for the density of states . moreover , the computational requirements of the exact diagonalization and the quantum montecarlo methods are such that they can only be implemented for the simplest hamiltonians . to carry out realistic calculations it is necessary to have an accurate but fast algorithm for solving the impurity model . in this context iterative perturbation theory has turned out to be a useful and reliable tool for the case of half filling @xcite . however , for finite doping the naive extension of the ipt scheme is known to give unphysical results . there is still no method which can be applied away from half filling and which at the same time is powerful enough to treat more complicated models envolving several bands . the aim of this paper is to close this gap by introducing a new iterative perturbation scheme which is applicable at arbitrary filling . for simplicity , we treat the single band case here . but we believe that the ideas can be generalized to more complicated models involving several bands . the ( asymmetric ) anderson impurity model @xmath2 describes an impurity @xmath3 coupled to a bath of conduction electrons @xmath4 . the hybridization function is given by @xmath5 . once a solution is known for arbitrary parameters a large number of lattice models can be solved by iteration . an example is the hubbard hamiltonian : @xmath6 which can actually serve as an effective hamiltonian for the description of doped transition metal oxides @xcite . on a bethe lattice with infinite coordination number @xmath7 the hubbard model is connected to the impurity model by the following selfconsistency condition : @xmath8 and @xmath9 . the mapping requires that the propagator of the lattice problem is given by the impurity green function ( @xmath10 ) . below we set @xmath11 . in the next section we will derive the perturbation scheme for the impurity model . afterwards the scheme is applied to the hubbard model ( section [ su3 ] ) . some results for the doped system are presented and the accuracy of our scheme is discussed . we conclude with a summary and an outlook on further extensions ( section [ su4 ] ) . in this section , we derive the approximation scheme which , given the hybridization function @xmath12 and the impurity level @xmath13 , provides a solution of model ( [ eu1 ] ) . for simplicity , we assume that there is no magnetic symmetry breaking ( @xmath14 ) . we also restrict us to zero temperature . the procedure is an extension of the ordinary ipt scheme to finite doping . the success of ipt at half filling can be explained by that it becomes exact not only in the weak but also in the strong coupling limit @xcite . moreover , this approach captures the right low and high frequency behavior so that we are dealing with an interpolation scheme between correct limits . the idea of our approach is to construct a self energy expression which retains these features at arbitrary doping and reduces at half filling to the ordinary ipt result . ordinary ipt approximates the self energy by its second order contribution : @xmath15 where @xmath16 with @xmath17 . here , the ( advanced ) green function @xmath18 is defined by @xmath19 the parameter @xmath20 is given by @xmath21 . in particular it vanishes at half filling . the full green function follows from @xmath22 to ensure the correctness of this approximation scheme in different limits , we modify the self energy functional as well as the definition of the parameter @xmath20 . we start with an ansatz for the self energy : @xmath23 here , @xmath24 is the normal second order contribution defined in equation ( [ eu3 ] ) . we determine the parameter @xmath25 from the condition that the self energy has the exact behavior at high frequencies . afterwards , @xmath26 is determined from the atomic limit . the leading behavior for large @xmath27 can be obtained by expanding the green function into a continuous fraction @xcite : @xmath28 . here , @xmath29 marks the @xmath30th order moment of the density of states . one can compute these quantities by evaluating a commutator ( see @xcite ) . we obtain for our model @xmath31 . the leading term of the self energy is therefore given by @xmath32 here , @xmath33 is the physical particle number given by @xmath34 . ( [ eu7 ] ) has to be compared with the large frequency limit of ( [ eu3 ] ) : @xmath35 where @xmath36 is a fictitious particle number determined from @xmath37 ( i. e. @xmath38 ) . from ( [ eu6 ] ) , ( [ eu7 ] ) , and ( [ eu8 ] ) we conclude @xmath39 chosing @xmath25 in this way guaranties that our self energy is correct to order @xmath40 . it should be noted from the continuous fraction considered above that consequently the moments of the density of states up to second order are reproduced exactly . next , we have to fix @xmath26 . the exact impurity green function for @xmath41 is given by @xcite @xmath42 this can be written as @xmath43 where @xmath44 this expression is to be compared with the atomic limit of our ansatz ( [ eu6 ] ) . since @xmath45 , we obtain @xmath46 thus , the final result for our interpolating self energy is @xmath47 yet , @xmath20 is still a free parameter . we fix it imposing the friedel sum rule @xcite : @xmath48 this statement , which is equivalent to the luttinger theorem @xcite @xmath49 , should be viewed as a condition on the zero frequency value of the self energy to obtain the correct low energy behavior . the use of the friedel sum rule is the main difference to an earlier approximation scheme @xcite and is essential to obtain a good agreement with the exact diagonalization method . so far , we considered three different limits : strong coupling , zero frequency and large frequency . it remains to check the weak coupling limit . taking into account that @xmath50 and @xmath51 for @xmath52 , it follows that ( [ eu12 ] ) is indeed exact to order @xmath53 . the actual solution of the impurity model is determined by a pair ( @xmath20 , @xmath33 ) which satisfies equations ( [ eu3 ] ) , ( [ eu4 ] ) , ( [ eu5 ] ) , ( [ eu12 ] ) , and ( [ eu13 ] ) . for the numerical implementation broyden s method @xcite , a generalization of newton s method , has turned out to be very powerful . defining two functions @xmath54(\omega)$ ] and @xmath55 $ ] the impurity problem can be solved by searching for the zeros of @xmath56 and @xmath57 ( @xmath58 is the particle number determined from the friedel sum rule ) . the algorithm is very efficient as in most cases a solution is found within 4 to 10 iterations . after treating the anderson impurity model , we now apply the perturbation scheme to the solution of the hubbard model . starting with a guess for @xmath12 one can solve the impurity model using the scheme described above . this yields a propagator @xmath59 , which can be used to determine a new hybridization function @xmath12 according to ( [ ue20 ] ) . the iteration is continued until convergence is attained . it is most accurate to perform the calculation first on the imaginary axis . once the constants @xmath25 and @xmath26 in the interpolating self energy are determined in this way , they can be used to perform the iteration on the real axis . in the case of the hubbard model , the luttinger theorem takes the simple form @xcite @xmath60 this can be used to simplify the selfconsistency procedure if @xmath20 rather than @xmath61 is fixed . starting with a guess for @xmath62 and @xmath61 , one can compute @xmath37 , @xmath33 , and @xmath36 . afterwards ( [ eu12 ] ) yields @xmath63 and a new @xmath61 is obtained from ( [ ue21 ] ) . to illustrate the accuracy of our method we compare it with results obtained using the exact diagonalization algorithm of caffarel and krauth @xcite . both methods are in close agreement when used on the imaginary axis ( see figure [ fig_compi ] ) . the real advantage of our perturbation scheme compared to the exact diagonalization is disclosed when we display the spectral functions obtained by these two methods on the real axis ( figure [ fig_comp ] ) . = 2.8truein = 2.8truein it is clear that the exact diagonalization is doing its best in producing the correct spectral distribution . but it is unable to give a smooth density of states . instead several sharp structures occur as a consequence of treating only a finite number of orbitals in the anderson model . figure [ fig3 ] shows the evolution of the spectral density of the doped mott insulator ( @xmath64 ) with increasing hole doping @xmath65 . the qualitative features are those expected from the spectra of the single impurity @xcite and are in agreement with the quantum montecarlo calculations @xcite . for small doping there is a clear resonance peak at the fermi level . as @xmath65 is increased , the peak broadens and is shifted through the lower hubbard band . at the same time the weight of the upper band decreases . the most striking feature of the evolution of the spectral density as a function of doping is the finite shift of the kondo resonance from the insulating band edge as the doping goes to zero . it was demonstrated analytically that this is a genuine property of the exact solution of the hubbard model in infinite dimensions using the projective self - consistent method @xcite and is one of the most striking properties of the hubbard model in large dimensions . this feature did not appear in the earlier studies of hubbard model in large dimensions using montecarlo techniques @xcite at higher temperatures , and is also not easily seen in exact diagonalization algorithms @xcite . in this paper we introduced a new perturbation scheme for the solution of lattice models away from half filling . the basic idea is to construct an expression for the self energy which interpolates between correct limits . in the weak coupling limit our approximate self energy is exact to order @xmath53 , and it is also exact in the atomic limit . the proper low frequency behavior is ensured by the friedel sum rule ( or , equivalently , the luttinger theorem ) . this is important to obtain the right low energy features in the spectral density . the overall distribution of the density of states on the other hand is determined by the spectral moments , which are reproduced exactly up to second order by satisfying the proper large frequency behavior . in the light of these features it might not be too astonishing that we obtain a good agreement with the exact diagonalization method . since the algorithm decribed here is accurate and very fast ( a typical run to solve the hubbard model takes 60 seconds on a dec alpha station 200 4/233 ) it has a wide range of applications . two examples that come to mind are the effects of disorder on the hubbard model away from half filling and the study of realistic models with orbital degeneracy . the latter is very important to make contact with realistic three dimensional transition metal oxides . 99 y. tokura , y. taguchi , y. okada , y. fujishima , and t. arima , k. kumagai and y. iye , _ phys . lett . _ * 70 * , 2126 ( 1993 ) . y. okimoto , t. katsufuji , and y. tokura , _ phys . * 51 * , 9581 ( 1995 ) . t. katsufuji , y. okimoto , and y. tokura , _ preprint _ ( 1995 ) w. metzner and d. vollhardt , _ phys . lett . _ * 62 * , 324 ( 1989 ) . a. georges , g. kotliar , _ phys . b _ * 45 * 6479 ( 1992 ) . m. jarell , _ phys . lett . _ * 69 * , 168 ( 1992 ) . m. rozenberg , x. y. zhang and g. kotliar , _ phys . lett . _ * 69 * , 1236 ( 1992 ) . a. georges and w. krauth , _ phys . _ * 69 * , 1240 ( 1992 ) . x. y. zhang , m. j. rozenberg and g. kotliar , _ phys . lett . _ * 70*,1666 ( 1993 ) . m. caffarel and w. krauth _ phys . lett . _ * 72 * , 1545 ( 1994 ) , q. si , m. rozenberg g. kotliar and a. ruckenstein , _ phys . lett . _ * 72 * , 2761 ( 1994 ) . g. moeller , q. si , g. kotliar , m. rozenberg and d. s. fisher , _ phys . lett . _ * 74 * , 2082 ( 1995 ) . h. kajueter , g. kotliar and g. moeller , submitted to _ phys . b _ ( 1995 ) . m. j. rozenberg , g. kotliar and x. y. zhang , _ phys . b _ * 49 * 10181 ( 1994 ) . m. j. rozenberg , g. kotliar , h. kajueter , g. a. thomas , d. h. rapkine , j. m. honig , and p. metcalf , _ phys . lett . _ * 75 * , 105 ( 1995 ) . r. g. gordon , _ j. math . phys . _ * 9 * , 655 ( 1968 ) . w. nolting and w. borgie , _ phys . b. _ * 39 * , 6962 ( 1989 ) . w. brenig , k. schnhammer , _ z. phys . _ * 257 * , 201 ( 1974 ) a. martin - rodero , f. flores , m. baldo and r. pucci , _ sol . state comm . _ * 44 * , 911 ( 1982 ) . d. c. langreth , _ phys . _ * 150 * , 516 ( 1966 ) . j. m. luttinger , j. c. ward , _ phys . _ * 118 * , 1417 ( 1960 ) . e. mller - hartmann , _ z. phys . b. _ * 76 * , 211 ( 1989 ) . m. jarrell and t. pruschke _ z. phys . * 90 * , 187 ( 1993 ) . d. fisher , g. kotliar and g. moeller , to appear in _ phys . b_. w. h. press et al . , _ numerical recipes in fortran _ , 2nd edition , p. 382 , cambridge university press .

we derive a new perturbation scheme for treating the large d limit of lattice models at arbitrary filling . the results are compared with exact diagonalization data for the hubbard model and found to be in good agreement . # 1@xmath0#1

probabilistic models of application domains are central to pattern recognition , machine learning , and scientific modeling in various fields . consequently , unifying frameworks are likely to be fruitful for one or more of these fields . there are also more technical motivations for pursuing the unification of diverse model types . in multiscale modeling , models of the same system at different scales can have fundamentally different characteristics ( e.g. deterministic vs. stochastic ) and yet must be placed in a single modeling framework . in machine learning , automated search over a wide variety of model types may be of great advantage . in this paper we propose stochastic parameterized grammars ( spg s ) and their generalization to dynamical grammars ( dg s ) as such a unifying framework . to this end we define mathematically both the syntax and the semantics of this formal modeling language . the essential idea is that there is a `` pool '' of fully specified parameter - bearing terms such as \{@xmath0 , @xmath1 , @xmath2 } where @xmath3 and @xmath4 might be position vectors . a grammar can include rules such as @xmath5 which specify the probability per unit time , @xmath6 , that the macrophage ingests and destroys the bacterium as a function of the distance @xmath7 between their centers . sets of such rules are a natural way to specify many processes . we will map such grammars to stochastic processes in both continuous time ( section [ xref - section-92874322 ] ) and discrete time ( section [ xref - section-9287443 ] ) , and relate the two definitions ( section [ xref - section-92874423 ] ) . a key feature of the semantics maps is that they are naturally defined in terms of an algebraic _ ring _ of time evolution operators : they map operator addition and multiplication into independent or strongly dependent compositions of stochastic processes , respectively . the stochastic process semantics defined here is a mathematical , algebraic object . it is independent of any particular simulation algorithm , though we will discuss ( section [ xref - section-92871922 ] ) a powerful technique for generating simulation algorithms , and we will demonstrate ( section [ xref - section-92872143 ] ) the interpretation of certain subclasses of spg s as a logic programming language . other applications that will be demonstrated are to data clustering ( @xcite ) , chemical reaction kinetics ( section [ xref - section-92872456 ] ) , graph grammars and string grammars ( section [ xref - section-92872523 ] ) , systems of ordinary differential equations and systems of stochastic differential equations ( section [ xref - section-926213036 ] ) . other frameworks that describe model classes that may overlap with those described here are numerous and include : branching or birth - and - death processes , marked point processes , mgs modeling language using topological cell complexes , interacting particle systems , the blog probabilistic object model , adaptive mesh refinement with rewrite rules , stochastic pi - calculus , and colored petri nets . the mapping @xmath8 to an operator algebra of stochastic processes , however , appears to be novel . the present paper is an abbreviated summary of @xcite . consider the rewrite rule @xmath9 where the @xmath10 and @xmath11 denote symbols @xmath12 chosen from an arbitrary alphabet set @xmath13 of `` types '' . in addition these type symbols carry expressions for parameters @xmath14 or @xmath15 chosen from a base language @xmath16 defined below . the @xmath17 s can appear in any order , as can the @xmath18 s . different @xmath17 s and @xmath18 s appearing in the rule can denote the same alphabet symbol @xmath12 , with equal or unequal parameter values @xmath14 or @xmath15 . @xmath19 is a nonnegative function , assumed to be denoted by an expression in a base language @xmath20 defined below , and also assumed to be an element of a vector space @xmath21 of real - valued functions . informally , @xmath19 is interpreted as a nonnegative probability rate : the independent probability per unit time that any possible instantiation of the rule will `` fire '' if its left hand side precondition remains continuously satisfied for a small time . this interpretation will be formalized in the semantics . we now define @xmath22 . each term @xmath23 or @xmath24 is of type @xmath12 and its parameters @xmath14 take values in an associated ( ordered ) cartesian product set @xmath25 of @xmath26 factor spaces chosen ( possibly with repetition ) from a set of base spaces @xmath27 . each @xmath28 is a measure space with measure @xmath29 . particular @xmath28 may for example be isomorphic to the integers @xmath30 with counting measure , or the real numbers @xmath31 with lebesgue measure . the ordered choice of spaces @xmath28 in @xmath32 constitutes the type signature @xmath33 of type @xmath12 . ( as an aside , polymorphic argument type signatures are supported by defining a derived type signature @xmath34 . for example we can regard @xmath30 as a subset of @xmath31 . ) correspondingly , parameter expressions @xmath14 are tuples of length @xmath26 , such that each component @xmath35 is either a constant in the space @xmath36 , or a variable @xmath37 that is restricted to taking values in that same space @xmath38 . the variables that appear in a rule this way may be repeated any number of times in parameter expressions @xmath14 or @xmath15 within a rule , providing only that all components @xmath35 take values in the same space @xmath36 . a _ substitution _ @xmath39 of values for variables @xmath40 assigns the same value to all appearances of each variable @xmath40 within a rule . hence each parameter expression @xmath14 takes values in a fixed tuple space @xmath25 under any substitution @xmath41 . this defines the language @xmath22 . we now constrain the language @xmath20 . each nonnegative function @xmath42 is a probability rate : the independent probability per unit time that any particular instantiation of the rule will fire , assuming its precondition remains continuously satisfied for a small interval of time . it is a function only of the parameter values denoted by @xmath43 and @xmath44 , and not of time . each @xmath19 is denoted by an expression in a base language @xmath20 that is closed under addition and multiplication and contains a countable field of constants , dense in @xmath31 , such as the rationals or the algebraic numbers . @xmath6 is assumed to be a nonnegative - valued function in a banach space @xmath45 of real - valued functions defined on the cartesian product space @xmath46 of all the value spaces @xmath47 of the terms appearing in the rule , taken in a standardized order such as nondeccreasing order of type index @xmath48 on the left hand side followed by nondecreasing order of type index @xmath48 on the right hand side of the rule . provided @xmath20 is expressive enough , it is possible to factor @xmath49 within @xmath20 as a product @xmath50=@xmath51@xmath52 of a conditional distribution on output parameters given input parameters @xmath53 and a total probability rate @xmath51 as a function of input parameters only . with these definitions we can use a more compact notation by eliminating the @xmath17 s and @xmath18 s , which denote types , in favor of the types themselves . ( the expression @xmath54 is called a parameterized _ term , _ which can match to a parameter - bearing _ object _ or _ term instance _ in a `` pool '' of such objects . ) the caveat is that a particular type @xmath55 may appear any finite number of times , and indeed a particular parameterized term @xmath54 may appear any finite number of times . so we use multisets @xmath56 ( in which the same object @xmath57 may appear as the value of several different indices @xmath58 ) for both the lhs and rhs ( left hand side and right hand side ) of a rule : @xmath59 here the same object @xmath60 may appear as the value of several different indices @xmath58 under the mappings @xmath61 and/or @xmath62 . finally we introduce the shorthand notation @xmath63 and @xmath64 , and revert to the standard notation @xmath65 for multisets ; then we may write @xmath66 @xmath67 . in addition to the * with * clause of a rule following the lhs@xmath68rhs header , several other alternative clauses can be used and have translations into * with * clauses . for example , `` * subject to * @xmath69 '' is translated into `` * with * @xmath70 '' where @xmath71 is an appropriate dirac or kronecker delta function that enforces a contraint @xmath72 . other examples are given in @xcite . the translation of `` * solving * @xmath73 '' or `` * solve * @xmath73 '' will be defined in terms of * with * clauses in section [ xref - section-926213036 ] . as a matter of definition , stochastic parameterized grammars do not contain * solving*/*solve * clauses , but dynamical grammars may include them . there exists a preliminary implementation of an interpreter for most of this syntax in the form of a _ mathematica _ notebook , which draws samples according to the semantics of section [ xref - section-92795243 ] below . a stochastic parameterized grammar ( spg ) @xmath74 consists of ( minimally ) a collection of such rules with common type set @xmath75 , base space set @xmath76 , type signature specification @xmath77 , and probability rate language @xmath20 . after defining the semantics of such grammars , it will be possible to define semantically equivalent classes of spg s that are untyped or that have richer argument languages @xmath22 . we provide a semantics function @xmath78 in terms of an operator algebra that results in a _ stochastic process _ , if it exists , or a special `` undefined '' element if the stochastic process does nt exist . the stochastic process is defined by a very high - dimensional differential equation ( the master equation ) for the evolution of a probability distribution in continuous time . on the other hand we will also provide a semantics function @xmath79 that results in a discrete - time stochastic process for the same grammar , in the form of an operator that evolves the probability distribution forward by one discrete rule - firing event . in each case the stochastic process specifies the time evolution of a probability distribution over the contents of a `` pool '' of grounded parameterized terms @xmath80 that can each be present in the pool with any allowed multiplicity from zero to @xmath81 . we will relate these two alternative `` meanings '' of an spg , @xmath82 in continuous time and @xmath83 in discrete time . a state of the `` pool of term instances '' is defined as an integer - valued function @xmath84 : the `` copy number '' @xmath85 of parameterized terms @xmath80 that are grounded ( have no variable symbols @xmath40 ) , for any combination @xmath86 of type index @xmath87 and parameter value @xmath88 . we denote this state by the `` indexed set '' notation for such functions , @xmath89 . each type @xmath12 may be assigned a maximum value @xmath90 for all @xmath91 , commonly @xmath92 ( no constraint on copy numbers ) or 1 ( so @xmath93 which means each term - value combination is simply present or absent ) . the state of the full system at time @xmath94 is defined as a probability distribution on all possible values of this ( already large ) pool state : @xmath95 . the probability distribution that puts all probability density on a particular pool state @xmath96 is denoted @xmath97 . for continuous - time we define the semantics @xmath98 of our grammar as the solution , if it exists , of the master equation @xmath99 , which can be written out as : @xmath100 and which has the formal solution @xmath101 . for discrete - time semantics @xmath83 there is an linear map @xmath102 which evolves unnormalized probabilities forward by one rule - firing time step . the probabilities must of course be normalized , so that after @xmath103 discrete time steps the probability is : @xmath104 which , taken over all @xmath105 and @xmath106 , defines @xmath83 . in both cases the long - time evolution of the system may converge to a limiting distribution @xmath107 which is a key feature of the semantics , but we do not define the semantics @xmath108 as being only this limit even if it exists . thus semantics - preserving transformations of grammars are fixedpoint - preserving transformations of grammars but the converse may not be true . the master equation is completely determined by the _ generators _ @xmath109 and @xmath102 which in turn are simply composed from elementary operators acting on the space of such probability distributions . they are elements of the operator polynomial ring @xmath110 $ ] defined over a set of basis operators @xmath111 in terms of operator addition , scalar multiplication , and noncommutative operator multiplication . these basis operators @xmath111 provide elementary manipulations of the copy numbers @xmath112 . the simplest basis operators @xmath111 are elementary creation operators @xmath113 and annihilation operators @xmath114 that increase or decrease each copy number @xmath112 in a particular way ( reviewed in @xcite ) : @xmath115 where @xmath116is the kronecker delta function . these two operator types then generate @xmath117 : @xmath118 we can write these operators @xmath119 as finite or infinite dimensional matrices depending on the maximum copy number @xmath90 for type @xmath12 . if @xmath90=1 ( for a fermionic term ) , and we omit the type which are all assumed equal below , then @xmath120 likewise if @xmath90=@xmath121 ( for a bosonic term ) , @xmath122 . by truncating this matrix to finite size @xmath123 we may compute that for some polynomial @xmath124 of degree @xmath125 - 1 in @xmath126 with rational coefficients , @xmath127 = \delta ( x - y ) [ i+n q ( n| n^{\left ( \max \right ) } ) ] \ ] ] where @xmath71 is the dirac delta ( generalized ) function appropriate to the ( product ) measure @xmath128 on the relevant value space @xmath46 . eg . if @xmath129=1 then @xmath130 ; if @xmath129=@xmath121 then @xmath131 . for a grammar rule number `` @xmath132 '' of the form of ( equation [ xref - equation-924145912 ] ) we define the operator that first ( instantaneously ) destroys all parameterized terms on the lhs and then ( immediately and instantaneously ) creates all parameterized terms on the rhs . this happens independently of time or other terms in the pool . assuming that the parameter expressions @xmath3 contain no variables @xmath40 , the effect of this event is : @xmath133 \ \ \ \left[\prod \limits_{j\in \operatorname{lhs } ( r ) } a_{b ( j ) } ( y_{j } ) \right ] % \label{xref - equation-922211956}\ ] ] if there are variables @xmath134 , we must sum or integrate over all their possible values in @xmath135 : @xmath136 \ \ \ \left[\prod \limits_{j\in \operatorname{lhs } ( r ) } a_{b ( j ) } ( y_{j } ( \left\ { x_{c}\right\ } ) ) \right ] % \label{xref - equation-922212022}\end{gathered}\ ] ] thus , syntactic variable - binding has the semantics of multiple integration . a `` monotonic rule '' has all its lhs terms appear also on the rhs , so that nothing is destroyed . unfortunately @xmath137 does nt conserve probability because probability inflow to new states ( described by @xmath137 ) must be balanced by outflow from current state ( diagonal matrix elements ) . the following operator conserves probability : @xmath138 . for the entire grammar the time evolution operator is simply a sum of the generators for each rule : @xmath139 this superposition implements the basic principle that every possible rule firing is an exponential process , all happening in parallel until a firing occurs . note that ( equation [ xref - equation-922211956 ] ) , ( equation [ xref - equation-922212022 ] ) and @xmath140 are encompassed by the polynomial ring @xmath110 $ ] where the basis operators include all creation and annihilation operators . ring addition ( as in equation [ xref - equation-922211931 ] or equation [ xref - equation-922212022 ] ) corresponds to independently firing processes ; ring operator multiplication ( as in equation [ xref - equation-922211956 ] ) corresponds to obligatory event co - ocurrence of the constituent events that define a process , in immediate succession , and nonnegative scalar multiplication corresponds to speeding up or slowing down a process . commutation relations between operators describe the exact extent to which the order of event occurrence matters . the operator @xmath102 describes the flow of probability per unit time , over an infinitesimal time interval , into new states resulting from a single rule - firing of any type . if we condition the probability distribution on a single rule having fired , setting aside the probability weight for all other possibilities , the normalized distribution is @xmath141 . iterating , the state of the discrete - time grammar after @xmath103 rule firing steps is @xmath142 as given by ( equation [ xref - equation-102111134 ] ) , where @xmath140 as before . the normalization can be state - dependent and hence dependent on @xmath103 , so @xmath143 . this is a critical distinction between stochastic grammar and markov chain models , for which @xmath144 . an execution algorithm is directly expressed by ( equation [ xref - equation-102111134 ] ) . an indispensible tool for studying such stochastic processes in physics is the time - ordered product expansion @xcite . we use the following form : @xmath145 \cdot p_{0}% \label{xref - equation-92363221}\end{gathered}\ ] ] where @xmath146 is a solvable or easily computable part of @xmath109 , so the exponentials @xmath147 can be computed or sampled more easily than @xmath148 . this expression can be used to generate feynman diagram expansions , in which @xmath84 denotes the number of interaction vertices in a graph representing a multi - object history . if we apply ( equation [ xref - equation-92363221 ] ) with @xmath149 and @xmath150 , we derive the well - known gillespie algorithm for simulating chemical reaction networks @xcite , which can now be applied to spg s . however many other decompositions of @xmath109 are possible , one of which is used in section [ xref - section-926213036 ] below . because the operators @xmath109 can be decomposed in many ways , there are many valid simulation algorithms for each stochastic process . the particular formulation of the time - ordered product expansion used in ( equation [ xref - equation-92363221 ] ) has the advantage of being recursively self - applicable . thus , ( equation [ xref - equation-92363221 ] ) entails a systematic approach to the creation of novel simulation algorithms . _ proposition . _ given the stochastic parameterized grammar ( spg ) rule syntax of equation [ xref - equation-924145912 ] , ( a ) there is a semantic function @xmath151 mapping from any continuous - time , context sensitive , stochastic parameterized grammar @xmath74 via a time evolution operator @xmath152 to a joint probability density function on the parameter values and birth / death times of grammar terms , conditioned on the total elapsed time , @xmath94 . ( b ) there is a semantic function @xmath142 mapping any discrete - time , sequential - firing , context sensitive , stochastic parameterized grammar @xmath74 via a time evolution operator @xmath153 to a joint probability density function on the parameter values and birth / death times of grammar terms , conditioned on the total discrete time defined as number of rule firings , @xmath103 . ( c ) the short - time limit of the density @xmath78 conditioned on @xmath154 and conditioned on @xmath103 is equal to @xmath83 . proof : ( a ) : section [ xref - section-92874322 ] . ( b ) : section [ xref - section-9287443 ] . ( c ) equation [ xref - equation-92363221 ] ( details in @xcite , @xcite ) . given a new kind of mathematical object ( here , spg s or dg s ) it is generally productive in mathematics to consider the transformations of such objects ( mappings from one object to another or to itself ) that preserve key properties . examples include transformational geometry ( groups acting on lines and points ) and functors acting on categories . in the case of spg s , two possibilities for the preserved property are immediately salient . first , an spg syntactic transformation @xmath155 could preserve the semantics @xmath156 either fully or just in fixed point form : @xmath157 . preserving the full semantics would be required of a simulation algorithm . alternatively , an inference algorithm could preserve a joint probability distribution on unobserved and observed random variables , in the form of bayes rule , @xmath158 where @xmath159 are collections of parameterized terms that are inpuuts to , internal to , and outputs from the grammar @xmath74 respectively .. a number of other frameworks and formalisms can be expressed or reduced to spgs as just defined . for example , data clustering models are easily and flexibly described @xcite . we give a sampling here . given the chemical reaction network syntax @xmath160 define an index mapping @xmath161 and likewise for @xmath162 as a function of @xmath163 . then ( equation [ xref - equation-926214051 ] ) can be translated to the following equivalent grammar syntax for the multisets of parameterless terms @xmath164 whose semantics is the time - evolution generator @xmath165 \ \ \ \left[\prod \limits_{j\in \operatorname{lhs } ( r ) } a_{b ( j ) } \right ] \ \ .\ ] ] this generator is equivalent to the stochastic process model of mass - action kinetics for the chemical reaction network ( equation [ xref - equation-926214051 ] ) . consider a logic program ( e.g. in pure prolog ) consisting of horn clauses of positive literals @xmath166 axioms have @xmath167 . we can _ translate _ each such clause into a monotonic spg rule @xmath168 where each different literal @xmath169 denotes an unparameterized type @xmath12 with @xmath170 . since there is no * with * clause , the fule firing rates default to @xmath171 . the corresponding time - evolution operator is @xmath172 \ \ \ \left[\prod \limits_{j\in \operatorname{lhs } ( r ) } n_{b ( j ) } \right]\ ] ] the semantics of the logic program is its least model or minimal interpretation . it can be computed ( knaster - tarski theorem ) by starting with no literals in the `` pool '' and repeatedly drawing all their consequences according to the logic program . this is equivalent to converging to a fixed point @xmath173 ) . more general clauses include negative literals @xmath174 on the lhs , as @xmath175 , or even more general cardinality constraint atoms @xmath176@xmath177 @xcite . these constraints can be expressed in operator algebra by expanding the basis operator set @xmath178 beyond the basic creation and annihilation operators @xcite . finally , atoms with function symbols may be admitted using parameterized terms @xmath179 . graph grammars are composed of local rewrite rules for graphs ( see for example @xcite ) . we now express a class of graph grammars in terms of spg s . the following syntax introduces object identifier ( oid ) labels @xmath180 for each parameterized term , and allows labelled terms to point to one another through a graph of such labels . the graph is related to two subgraphs of neighborhood indices @xmath181 and @xmath182 specific to the input and output sides of a rule . like types or variables , the label symbols appearing in a rule are chosen from an alphabet @xmath183 . unlike types but like variables @xmath40 , the label symbols @xmath184actually denote nonnegative integer values - unique addresses or object identifiers . a graph grammar rule is of the form , for some nonnegative - integer - valued functions @xmath185 , @xmath186 , @xmath187 , @xmath188 for which @xmath189 , @xmath190 : @xmath191 ( compare to ( equation [ xref - equation-924145912 ] ) ) . note that the fanout of the graph is limited by @xmath192 . let @xmath193 be mutually exclusive and exhaustive , and the same for @xmath194 . define @xmath195 , @xmath196 , and @xmath197 . then the graph syntax may be translated to the following ordinary non - graph grammar rule ( where nextoid is a variable , and oidgen and null are types reserved for the translation ) : @xmath198 which already has a defined semantics @xmath8 . note that all set membership tests can be done at translation time because they do not use information that is only available dynamically during the grammar evolution . optionally we may also add a rule schema ( one rule per type , @xmath12 ) to eliminate any dangling pointers @xcite . strings may be encoded as one - dimensional graphs using either a singly or doubly linked list data structure . string rewrite rules are emulated as graph rewrite rules , whose semantics are defined above . this form is capable of handling many l - system grammars @xcite . there are spg rule forms corresponding to stochastic differential equations governing diffusion and transport . given the sde or equivalent langevin equation ( which specializes to a system of ordinary differential equations when @xmath199 ): @xmath200 under some conditions on the noise term @xmath201 the dynamics can be expressed @xcite as a fokker - planck equation for the probability distribution @xmath202 : @xmath203 let @xmath204 be the solution of this equation given initial condition @xmath205 ( with dirac delta function appropriate to the particular measure @xmath128 used for each component ) . then at @xmath206 , @xmath207 thus the probability rate @xmath208 is given by a differential operator acting on a dirac delta function . by ( equation [ xref - equation-922212022 ] ) we construct the evolution generator operators @xmath209 , where @xmath210 the second order derivative terms give diffusion dynamics and also regularize and promote continuity of probability in parameter space both along and transverse to any local drift direction . calculations with such expressions are shown in @xcite . diffusion / drift rules can be combined with chemical reaction rules to describe reaction - diffusion systems @xcite . the foregoing approach can be generalized to encompass partial differential equations and stochastic partial differential equations@xcite . these operator expressions all correspond to natural extended - time processes given by the evolution of continuous differential equations . the operator semantics of the differential equations is given in terms of derivatives of delta functions . a special `` * solve * '' or `` * solving * '' keyword may be used to introduce such ode / sde rule clauses in the spg syntax . this syntax can be eliminated in favor of a `` * with * '' clause by using derivatives of delta functions in the rate expression @xmath211 , provided that such generalized functions are in the banach space @xmath212 as a limit of functions . if a grammar includes such de rules along with non - de rules , a solver can be used to compute @xmath213 in the time - ordered product for @xmath148 as a hybrid simulation algorithm for discontinuous ( jump ) stochastic processes combined with stochastic differential equations . the relevance of the modeling language defined here to _ artificial intelligence _ includes the following points . first , pattern recognition and machine learning both benefit foundationally from better , more descriptively adequate probabilistic domain models . as an example , @xcite exhibits hierarchical clustering data models expressed very simply in terms of spg s and relates them to recent work . graphical models are probabilistic domain models with a fixed structure of variables and their relationships , by contrast with the inherently flexible variable sets and dependency structures resulting from the execution of stochastic parameterized grammars . thus spg s , unlike graphical models , are variable - structure systems ( defined in @xcite ) , and consequently they can support compositional description of complex situations such as multiple object tracking in the presence of cell division in biological imagery @xcite . second , the reduction of many divergent styles of model to a common spg syntax and operator algebra semantics enables new possibilities for hybrid model forms . for example one could combine logic programming with probability distribution models , or discrete - event stochastic and differential equation models as discussed in section [ xref - section-926213036 ] in possibly new ways . as a third point of ai relevance , from spg probabilistic domain models it is possible to derive _ algorithms _ for simulation ( as in section [ xref - section-92871922 ] ) and inference either by hand or automatically . of course , inference algorithms are not as well worked out yet for spg s as for graphical models . spg s have the advantage that simulation or inference algorithms could be expressed again in the form of spg s , a possibility demonstrated in part by the encoding of logic programs as spg s . since both model and algorithm are expressed as spg s , it is possible to use spg transformations that preserve relevant quantities ( section [ xref - section - emj001 ] ) as a technique for deriving such novel algorithms or generating them automatically . for example we have taken this approach to rederive by hand the gillespie simulation algorithm for chemical kinetics . this derivation is different from the one in section [ xref - section-92871922 ] . because spg s encompass graph grammars it is even possible in principle to express families of valid spg transformations as meta - spg s . all of these points apply _ a fortiori _ to dynamical grammars as well . the relevance of the modeling language defined here to _ computational science _ includes the following points . first , as argued previously , multiscale models must encompass and unify heterogeneous model types such as discrete / continuous or stochastic / deterministic dynamical models ; this unification is provided by spg s and dg s . second , a representationally adequate computerized modeling language can be of great assistance in constructing mathematical models in science , as demonstrated for biological regulatory network models by cellerator @xcite and other cell modeling languages . dg s extend this promise to more complex , spatiotemporally dynamic , variable - structure system models such as occur in biological development . third , machine learning techniques could in principle be applied to find simplified approximate or reduced models of emergent phenomena within complex domain models . in that case the forgoing ai arguments apply to computational science applications of machine learning as well . both for artificial intelligence and computational science , future work will be required to determine whether the prospects outlined above are both realizable and compelling . the present work is intended to provide a mathematical foundation for achieving that goal . we have established a syntax and semantics for a probabilistic modeling language based on independent processes leading to events linked by a shared set of objects . the semantics is based on a polynomial ring of time - evolution operators . the syntax is in the form of a set of rewrite rules . stochastic parameterized grammars expressed in this language can compactly encode disparate models : generative cluster data models , biochemical networks , logic programs , graph grammars , string rewrite grammars , and stochastic differential equations among other others . the time - ordered product expansion connects this framework to powerful methods from quantum field theory and operator algebra . useful discussions with guy yosiphon , pierre baldi , ashish bhan , michael duff , sergei nikolaev , bruce shapiro , padhraic smyth , michael turmon , and max welling are gratefully acknowledged . the work was supported in part by a biomedical information science and technology initiative ( bisti ) grant ( number r33 gm069013 ) from the national institue of general medical sciences , by the national science foundation s frontiers in biological research ( fibr ) program award number ef-0330786 , and by the center for cell mimetic space exploration ( cmise ) , a nasa university research , engineering and technology institute ( ureti ) , under award number # ncc 2 - 1364 . 000 mjolsness , e. ( 2005 ) . _ stochastic process semantics for dynamical grammar syntax_. uc irvine , irvine . uci ics tr # 05 - 14 , http://computableplant.ics.uci.edu/papers/#frameworks . [ stochsem05 ] mattis , d. c. , & glasser , m. l. ( 1998 ) . _ the uses of quantum field theory in diffusion - limited reactions_. reviews of modern physics , * 70 * , 9791001 . [ mattisglasser98 ] risken , h. ( 1984 ) . _ the fokker - planck equation_. berlin : springer.[riskenfp ] gillespie , d. j. , ( 1976 ) . 22 , 403 - 434 . [ gillespie76 ] cenzer , d. , marek , v. w. , & remmel , j. b. ( 2005 ) . _ logic programming with infinite sets_. annals of mathematics and artificial intelligence , volume 44 , issue 4 , aug 2005 , pages 309 - 339 . [ remmel04 ] cuny , j. , ehrig , h. , engels , g. , & rozenberg , g. ( 1994 ) . _ graph grammars and their applications to computer science_. springer.[graphgram94 ] prusinkiewicz , p. , & lindenmeyer , a. ( 1990 ) . _ the algorithmic beauty of plants_. new york : springer - verlag.[prusinkiewiczalgb ] e. mjolsness ( 2005 ) . _ variable - structure systems from graphs and grammars_. uc irvine school of information and computer sciences , irvine . uci ics tr # 05 - 09 , http://computableplant.ics.uci.edu/papers/vbl-struct_gg_tr.pdf . [ vsstr05 ] victoria gor , tigran bacarian , michael elowitz , eric mjolsness ( 2005 ) . _ tracking cell signals in fluorescent images_. computer vision methods for bioinformatics ( cvmb ) workshop at computer vision and pattern recognition ( cvpr ) , san diego . [ cvprfluor05 ] bruce e. shapiro , andre levchenko , elliot m. meyerowitz , barbara j. wold , and eric d. mjolsness ( 2003 ) . _ cellerator : extending a computer algebra system to include biochemical arrows for signal transduction simulations_. bioinformatics 19 : 677 - 678 . [ cellerator ]

we define a class of probabilistic models in terms of an operator algebra of stochastic processes , and a representation for this class in terms of stochastic parameterized grammars . a syntactic specification of a grammar is mapped to semantics given in terms of a ring of operators , so that grammatical composition corresponds to operator addition or multiplication . the operators are generators for the time - evolution of stochastic processes . within this modeling framework one can express data clustering models , logic programs , ordinary and stochastic differential equations , graph grammars , and stochastic chemical reaction kinetics . this mathematical formulation connects these apparently distant fields to one another and to mathematical methods from quantum field theory and operator algebra . accepted for : ninth international symposium on artificial intelligence and mathematics , january 2006

qcd analysis of deep - inelastic scattering ( dis ) data provides one with new knowledge of hadron physics and serves as a test of reliability of our theoretical understanding of the hard scattering of leptons and hadrons . at large momentum transfer @xmath3 gev@xmath4 we have the reliable description of dis that is based on the twist expansion and `` factorization '' theorems . at small ( moderate ) transfer @xmath5 ( a few ) gev@xmath4 this qcd description faces two main problems : ( i ) the high twist corrections to the leading twist contribution become important but remains poorly known ; ( ii ) perturbative qcd ( pqcd ) becomes unreliable due to the fact that the qcd running coupling @xmath6 grows and `` feels '' infra - red landau singularity appearing at the scale @xmath7 of a few tenth of gev . we discuss in this paper a solution of the last problem by applying to dis analysis a _ nonpower perturbative theory _ whose couplings have no singularity at @xmath8 and whose corresponding series possess a better convergence at low @xmath9 . a widely used approach to resolve the aforementioned problem is to apply the analytic perturbation theory ( apt ) developed by shirkov , solovtsov _ et al . _ @xcite . there , the running qcd coupling @xmath10 of pqcd is transformed into an analytic ( holomorphic ) function of @xmath9 , @xmath11 , apt coupling . this was achieved by keeping in the dispersion relation the spectral density @xmath12 unchanged on the entire negative axis in the complex @xmath9-plane ( i.e. , for @xmath13 ) , and setting it equal to zero along the unphysical cut @xmath14 . in the framework of apt the images @xmath15 of integer powers of the originals @xmath16 , @xmath17 following the same dispersion relations were also constructed . at low @xmath9 the couplings @xmath18 change slowly with @xmath9 in contrast with the original @xmath16 behaviour while at high @xmath9 @xmath19 . later , the correspondence @xmath20 was extended to noninteger powers / indices @xmath21 in @xcite and was called fractional apt ( fapt ) , which provides the basis for application to dis . in this respect let us mention a recent papers @xcite where the processing of the dis data has been performed in fapt in the one - loop approximation and the reasonable results for hadron characteristics has been obtained . various analytic qcd models can be constructed , and have been proposed in the literature , among them in refs . these models fulfill certain additional constraints at low and/or at high @xmath9 . for further literature on various analytic qcd models , we refer to review articles @xcite . some newer constructions of analytic models in qcd of @xmath22 include those based on specific classes of @xmath23 functions with nonperturbative contributions @xcite or without such contributions @xcite and those based on modifications of the the spectral density @xmath24 @xmath25 \equiv { \rm i m } \ ; { { \mathcal{a}}}_{1}(q^2=-\sigma - i \epsilon)/\pi\right]$ ] at low ( positive ) @xmath26 where @xmath27 is parameterized in a specific manner by adding two positive delta functions to @xmath28 , cf . @xcite . the possibility to extend the dis analysis formally in the whole @xmath9 range together with the effect of slowing - down of the fapt evolution of the parton distribution functions ( pdf ) in the low @xmath9 region are attractive phenomenological features of fapt . a number of works deal with this task in a naive form @xcite , where the authors show that at very low @xmath9 and bjorken variable @xmath29 apt agrees with experimental data . besides , the applicability of the apt approach was analyzed in the bjorken polarized sum rule @xcite confirming that the range of validity of apt is down to @xmath30 mev , as compared to experimental data . the common feature of these works was taking into consideration some nonperturbative effects against the background of apt , i.e. , higher twists in @xcite or an effective constant gluon mass in @xcite . the basis for applying fapt to low energies in this approach is the factorization theorem that allows one to shift the frontier between the perturbative and nonperturbative effects via the variation of the factorization scale . therefore , we shift the range where perturbation series is applicable in fapt , as it was demonstrated in @xcite ( see reviews of this issue in @xcite , where this phenomenon was also discussed for pion form factors ) . our goal here is to elaborate a general scheme of dis data processing in the framework of fapt taking as a pattern the dis analysis at nlo . in this respect the discussion here can be considered as an extension of the partonic results of the article @xcite on the higher - loop level . we shall focus on the specifics of the dokshitzer - gribov - lipatov - altarelli - parisi ( dglap ) evolution of the pdf @xmath31 in fapt . we involve into consideration the coefficient function @xmath32 of the process and compare the final result with a similar one in pqcd . an important problem of higher twist contribution remains untouched here , but higher twist effects can be taken as an unknown function @xmath33 , i.e. @xmath34 @xcite , or as a constant @xcite @xmath35 , or as an effective sum of all twists contributions in @xcite . we stress that ht effects are only indirectly affected by the analytization procedure . the behavior of ht will be given by the fit of experimental data together with the corresponding parton distribution functions @xcite . besides , in @xcite the authors included more terms in the ht expansion and demonstrated that they are essentially smaller and quickly decreasing . because of this ( theoretically ) unknown behavior we avoid this problem since we pretend to provide perturbation tools how to deal with fapt , while the pure phenomenological analysis is transferred to future investigations . let us recall that the dis analysis can be performed in a few different ways : one of them is provided by the mellin moments defined via inelastic structure functions ( sfs ) @xmath36 , @xmath37 the second approach is based on the direct application of the dglap integro - differential evolution equations @xcite to pdf @xmath38 , while the observable sf is the mellin convolution of the coefficient function and pdf , @xmath39 . the third approach makes use of the jacobi polynomial expansion method @xcite . just this method will be used in this work . the paper is organized as follows : in sec . [ tb ] , we present a theoretical background where we describe the jacobi polynomial ( jp ) method and how to calculate free parameters in order to obtain the nonsinglet structure function . in sec . [ fapt ] , we briefly describe the fapt approach and derive the dglap evolution for the moments @xmath40 in fapt . we present in sec . [ nr ] the free parameters obtained in the analysis of the so called mstw parameterization , see @xcite for details , and the nonsinglet sfs at lo . section [ nr ] contains the results of the analysis of numerical realization of the fapt evolution and the comparison with the results of analogous calculations in pqcd . finally , in sec . [ sec : summ ] we summarize our conclusions . important technical details including new findings are collected in four appendices . we shall focus here on nonsinglet ( ns ) structure functions , @xmath41 , with their corresponding mellin moments @xmath42 ( via eq . ( [ moments ] ) ) to avoid technical complications of the coupled system solution in the singlet case . the pdfs @xmath43 are universal process - independent densities explaining how the whole hadron momentum @xmath44 is partitioned in @xmath45 , i.e. , the momentum carried by the struck parton ( see , for instance @xcite ) . the @xmath29-dependence of pdf is formed at a hadron scale of an order of @xmath46 by nonperturbative forces , while its dependence on factorization / renormalization scale @xmath47 can be obtained within perturbation theory . a brief description of the evolution of the mellin moments in pqcd , up to nlo , is outlined in appendix [ app1 ] as well as the theoretical background with our notation and conventions . we consider the scale @xmath48 as a reference scale for the solution of the evolution equation ( [ mns ] ) where the pdfs are regarded as functions of @xmath29 and the parameters are fixed by comparison with dis data . in particular , we use here the data - based mstw pdfs ( see @xcite , where @xmath49 ) . namely , x u_v(x , q_0 ^ 2)&=&a_u x^_1(1-x)^_2(1+_u + _ u x ) , + x d_v(x , q_0 ^ 2)&=&a_d x^_3(1-x)^_4(1+_d + _ d x ) , [ mstw ] where the values of @xmath50 ( @xmath51 ) , @xmath52 and @xmath53 can be found in @xcite . we use only the valence quark pdfs because the ns pdf @xmath54 can be expressed as @xmath55 ( see appendix [ app1 ] for details ) . the ns sf @xmath56 is represented as the mellin convolution of coefficient function @xmath57 of the process and the corresponding pdf @xmath58 . the @xmath59 can be expanded in the jacobi polynomials @xmath60 , which was developed in refs . @xcite , in truncating the expansion at @xmath61 , where the method converges ( see , for review @xcite ) : @xmath62 here @xmath42 are the mellin moments of nonsinglet sf calculated explicitly in eq . ( [ mns ] ) ; @xmath63 is the weight function and the parameters @xmath64 will be obtained by fitting to the data . the jacobi polynomials @xmath65 are defined as an expansion series by means of @xmath66 they satisfy the orthogonality relation @xmath67 another way to obtain sf @xmath0 is to take the inverse mellin transform @xmath68 under the moments @xmath42 ( i.e. , the inverse of eq . ( [ moments ] ) ) . choosing a convenient path of integration one obtains for @xmath69 @xmath70 here we take the path along a vertical line @xmath71 . we perform the `` exact '' numerical inverse mellin transform , further comparing the results with the jacobi polynomial method , only at the one - loop level due to technical limitations . in this way , we estimate the accuracy of the applied polynomial method , the results of this numerical verification are outlined in appendix [ app3 ] . let us finally mention that one can take sf @xmath72 instead of the ns @xmath73 to consider , , the neutrino dis results of the ccfr collaboration , like it was started in @xcite . this replacement will lead to only minor changes of technical details in the procedure elaborated below . it is known that the perturbative qcd coupling suffers from unphysical ( landau ) singularities at @xmath74 . this prevents the application of perturbative qcd in the low - momentum spacelike regime and , in part , impedes the investigation of high twists in dis . our goal here is not to discuss the motivation and complete construction of fapt , which couplings @xmath75 are free of the aforementioned problems , but present to reader illustrations of the properties of this nonpower perturbation theory that are important for further dis analysis . application of the cauchy theorem to the running coupling @xmath76 , established in @xcite and developed in @xcite , gives us the following dispersion relation ( or klln - lehmann spectral representation ) for the images @xmath77 in the spacelike domain @xmath78}{\sigma+q^2}d\sigma = \int_{-\infty}^{\infty } \frac{\rho_\nu^{(l)}(l_\sigma)}{1+{\rm exp}(l - l_\sigma)}dl_\sigma\ , , \label{anusd}\ ] ] ( where @xmath79 ) that has no unphysical ( landau ) singularities . for the timelike regime analogous coupling reads @xmath80}{\sigma}d\sigma = \int_{l_s}^{\infty } \rho_\nu^{(l)}(l_\sigma ) dl_\sigma . \label{unusd}\ ] ] here , @xmath81 is the fapt image of the qcd coupling @xmath82 in the euclidean ( spacelike ) domain with @xmath83 and the label @xmath84 denotes running in the @xmath84-loop approximation , whereas in the minkowski ( timelike ) domain , we used in ( [ unusd ] ) @xmath85 . it is convenient to use the following representation for the spectral densities @xmath86 : & & _ ^(l)(l _ ) ( a_s(l)(l - i))^= , [ eq : rho ] + & & r_(l)(l)= |a_s(l)(l - i ) | ; _ ( l)(l)=arg(a_s(l)(l - i ) ) . from the definition ( [ anusd ] ) and eq.([eq : rho ] ) it follow that there is no standard algebra for the images @xmath75 , @xmath87 that justifies the name _ nonpower perturbative theory_. in the one - loop approximation , the @xmath88 has the simplest form , i.e. , @xmath89 substituting eq.([eq : rho-1 ] ) in eq.([eq : rho ] ) for @xmath90 and then the result @xmath91 in eq.([anusd ] ) , one reproduces at @xmath92 the well - known expression for maximum value of @xmath93 , @xmath94 @xcite , @xmath95 * * at the two - loop level , they have a more complicated form . to be precise , one gets a_s(2)=- , and r_(2)(l ) & = & c_1(n_f)| 1+w_-1(z_w ( l+i))| , + _ ( 2)(l)&= & arccos , with @xmath96 being the appropriate branch of the lambert function , @xmath97 , @xmath98 , where @xmath99 are the qcd @xmath23-function coefficients and @xmath100 is the number of active quarks , see the expressions in appendix [ app0 ] . for our purpose we use here only the two - loop couplings like @xmath101 . extensions up to four - loops can be found in @xcite . now we implement this formalism with the help of numerical calculation with the main module mathematica package fapt.m of @xcite ( confirmed by a recent program in @xcite ) . according to this and using the corresponding notation from @xcite in the rhs of eqs.([anufaptprog]-[alphafaptprog ] ) , we have _ ^(l)(l)&= & , ( l=14 ; n_f=3 6 ) [ anufaptprog ] + _ ^(l)(l_s)&= & , ( l=14 ; n_f=3 6 ) [ unufaptprog ] for the coupling in pqcd we obtain @xmath102}{4\pi } , \quad ( l=1\div 4).\label{alphafaptprog}\ ] ] the correspondence between the pqcd expansion and fapt one is based on the linearity of the transforms in eqs.([anusd ] ) and ( [ unusd ] ) , see @xcite . this can be illustrated for the simple case of a single scale quantity @xmath103 , calculated within minimal subtraction renormalization schemes and taken at the renormalization scale @xmath104 . the expansions for @xmath105 and for its image @xmath106 are written as d(q^2 ) & = & d_0a_s^(q^2 ) + _ n d_n a_s^n+(q^2 ) + d(q^2)&=&d_0_(q^2)+ _ n d_n _ ( n+)(q^2 ) [ eq : dtod ] at the _ same coefficients @xmath107 _ that are numbers at @xmath104 . we start with the well - known solution of dglap equation for the nonsinglet pdf @xmath108 in nlo approximation . this solution is combined with the corresponding coefficient function @xmath32 the parton cross - section taken at the parton momentum @xmath109 . this is presented in appendix [ app1 ] in the form of eq.([mns ] ) for the moments @xmath110 of the ns sf @xmath69 . rewriting eq . ( [ mns ] ) in the approximate form , i.e. , neglecting the @xmath111 terms in the two - loop evolution factor , one arrives at the commonly used relation @xmath112 the use of fapt will change in this scheme the sense of expansion parameters @xmath113 in accordance with ( [ eq : dtod ] ) . an analogous evolution relation for the analytic images of the moments @xmath110 , @xmath114 , can be obtained from eq.([appmpqcd ] ) by replacing the powers @xmath115 with the fapt couplings @xmath75 ( with @xmath21 being here an index rather than a power ) @xcite and reads @xmath116 the implementation of the proposed calculation in the form of ( [ anufaptprog],[alphafaptprog ] ) is quite direct . the fapt evolution relation ( [ mnsapt ] ) for the moments is the main result of the section . further , we shall use code ( [ anufaptprog ] ) from @xcite to obtain @xmath117 numerically . the last approximation was taken up to @xmath118 since the contribution of the next term in the fapt expansion in eq.([mnsapt ] ) is negligible in comparison with the previous one ( as we demonstrate in appendix [ app2 ] ) . this analytic version of the moment evolution does not face any problems at low energies due to the boundedness of couplings and rapid convergence of the fapt series . in the absence of a fit of experimental data for the fapt model we propose a relation for the initial moments at @xmath119 : f_(n , q^2_0)&= & + & = & , [ inmom ] where the moment of pdf ( see eq.([eq : nsmpdf ] ) ) in pqcd stands in the lhs , while the moment for pdf in fapt stands in the rhs of the second equation . in other words , we take the same initial pdf as in pqcd from the mstw data for these both cases ( in @xcite the parameters were taken the same since the difference between them was negligible ) . we can use either the jacobi polynomial expansion or directly the inverse mellin transform ( appendix [ app3 ] ) . the accuracy of the sf approximation by a finite number of jacobi polynomials ( truncated at @xmath120 ) depends on the choice of the weight - function parameters . therefore , we test the nonsinglet sf , given by the mstw data , by searching for the minimum of ( @xmath121 ) : @xmath122 where we have used eqs . ( [ moments ] ) and ( [ mns ] ) at @xmath121 . thus , we have @xmath123 and from eq . ( [ jpsf ] ) @xmath124 . then , we determine the values of @xmath125 and @xmath23 that provide the best fit to the data for different values of @xmath120 . at the one loop level we find : @xmath126 , @xmath127 , and @xmath128 for @xmath129 , whereas for two loops we get ( for even pdfs only ) : @xmath126 , @xmath130 , and @xmath131 for @xmath129 . to evolve nonsinglet moments , we need to fix the values of the qcd scale @xmath132 in the leading and next - to - leading order , taken in @xcite from the comparison with data , where @xmath133 and @xmath134 . in the case of fapt , the scales @xmath135 must be taken into account very carefully . the authors of @xcite fixed the @xmath136 value directly from the comparison with the data in the leading order ( where @xmath137 ) and obtained @xmath138 that corresponds to @xmath139 . we can see that the perturbative and the analytic values of @xmath136 are close to each other at least inside the margin of errors . for this reason , we will take @xmath140 for simplicity ( recalling that an appropriate value should be taken from the analysis of the experimental data but this goes beyond the scope of this work ) . the couplings in pqcd and in fapt were calculated with the mathematica package developed by bakulev and khandramai in @xcite where the heavy flavour thresholds were taken into account . taking into account the above estimates of the initial parameters , we substitute eqs . ( [ appmpqcd ] ) and ( [ mnsapt ] ) into eq . ( [ jpsf ] ) , and obtain the evolution of sfs up to nlo in pqcd or fapt , respectively . we show the final results of the evolution in figs . [ figsf1 ] , [ figsf2 ] using for dis the character interval @xmath141 . vs @xmath9 at ( a ) lo and ( b ) nlo . the bjorken @xmath142 for ( a.1 ) and ( b.1 ) , and @xmath143 for ( a.2 ) or ( b.2 ) . the solid line represents the fapt results and the dashed line the pqcd ones . ] vs @xmath29 at ( a ) lo and ( b ) nlo . the energy scale is @xmath144 in ( a.1 ) , ( b.1 ) , @xmath145 in ( a.2 ) , ( b.2 ) and @xmath146 in ( a.3 ) , ( b.3 ) . the solid line represents the fapt results and the dashed line the pqcd ones . ] in fig . [ figsf1 ] we fix @xmath29 at two different values : @xmath142 in ( a.1 ) , ( b.1 ) , and @xmath143 in ( a.2 ) , ( b.2 ) , where ( a ) and ( b ) represent the lo and nlo results , respectively . in fig . [ figsf2 ] , we fix @xmath9 at three different values : @xmath144 in ( a.1 ) , ( b.1 ) , with the initial point @xmath145 in ( a.2 ) , ( b.2 ) , and @xmath146 in ( a.3 ) , ( b.3 ) , where again ( a ) and ( b ) represent the lo and nlo , respectively . the main goal of this work is to propose a new theoretical tool for the dis analysis , based on fractional analytic perturbation theory , to the dis community . this approach allows one to analyze formally the leading - twist structure function in the whole @xmath9 range . this conclusion is explicitly shown in figs . [ figsf1 ] and [ figsf2 ] . the scheme of the approach is formulated in sec . [ fapt ] and applied for data processing in sec . our consideration is restricted to the leading twist . the higher twist contributions ( ht ) can be taken into account by a fit of experimental data together with pdfs . moreover , the role of the stability of apt for this fit was pointed out in @xcite ( and in introduction here ) . our investigation reveals the following main features of applying fapt : * structure function @xmath0 at fixed @xmath29 changes very slowly in the entire range of @xmath9 . * at high @xmath9 evolution ( @xmath147 ) the pqcd and fapt distributions become practically equal . * the evolution in fapt is more gradual ( i.e. , it evolves slower ) and smoother than in pqcd . * the new analytic ( fapt ) series converge faster than the pqcd series . from inspection of figs . [ figsf1 ] , [ figsf2 ] it is obvious that the one- and the two - loop fapt approximations do not differ significantly from each other ( the difference is less than @xmath148 ) . + in this work , we have analyzed only the nonsinglet part , the consideration of the singlet part can be performed along the same line but requires more complicated formulas and cumbersome numerical calculations . this is the task for forthcoming investigation . other important issues to complete this fapt approach as the reliable tool for dis is to add the target mass corrections ( tmc ) and the aforementioned ht contributions in our scheme of calculation . these improvements will help one to clarify in future the behavior at very low energies ( @xmath149 ) in more detail . it would be important to emphasize , that the fapt approach admits investigation of the ht contributions in the most sensitive regime of moderate / small @xmath9 due to the high stability of the radiative corrections . this investigation was started by the late a. p. bakulev to whom we dedicate this work . we are grateful to g. cveti for useful comments and to n. g. stefanis for careful reading of the paper and many valuable critical remarks . we thank a. v. sidorov and o. p. solovtsova for the useful remarks . this work was supported by the scientific program of the the russian foundation for basic research grant no . 14 - 01 - 00647 , belrffr jinr grant f14d-007 ( s.v.m ) , and by conicyt fellowship `` becas chile '' grant no.74150052 ( c.a ) . the renormalization group equation for @xmath150 at the expansion of the @xmath23-function up to the nlo approximation is given by [ eq : betaf ] a_s(l ) = -(a_s(l))= - _ 0 a_s^2(l ) - _ 1 a_s^3(l)+ , + where the first two beta coefficients are _ 0 = c_a - t_r n_f , _ 1 = c_a^2 - ( 4c_f + c_a)t_r n_f . [ eq : beta0&1 ] the anomalous dimensions of composite operators in lo , @xmath151 , nlo @xmath152 and the coefficient function @xmath153 are expressed by means of transcendental sums @xmath154 , see , e.g. , @xcite , [ eq : ad - lo - nlo ] _ ^(0)(n)&=&2c_f , _ ^(1)(n)&= & ( c_f^2-c_f c_a ) \{16s_1(n)+16 . + 64 ^(n)+24s_2(n)-3 - 8s_3^ ( ) . + & & .-816 } + & & + c_fc_a \ { s_1(n)-16s_1(n)s_2(n ) . + s_2(n ) --4 } + & & + c_f n_f t_r \{-s_1(n)+s_2(n)++16 } , [ ad ] c_^(1)(n)&=&c_f ( 2s_1 ^ 2(n)+3s_1(n)-2s_2(n)-+++ -9 ) . on the other hand , the series @xmath155 can be expressed via the generalized riemann @xmath156 functions , see @xcite , that are analytic functions in _ both variables _ @xmath157 : [ eq : s ] s_1 ( n)&=&(n+1)-(1 ) , + s_2 ( n)&=&(2)-(n+1)=(2)-(2,n+1 ) , + s_(n)&=&()-(,n+1 ) . ^(n)&=&s_-2,1= -(3 ) _ k=1^ ( ( k+n+1)- ( 1 ) ) . for the @xmath158 and @xmath159 series we use the notation given in @xcite . performing the analytic continuation from even @xmath160 , @xmath161 , and from odd @xmath160 , @xmath162 ( see for details @xcite ) one obtains [ eq : san ] [ eq : s+ ] & & s^+_(n/2)2 ^ -1= 2 ^ -1- ( ) , + & & s^-_(n/2)2 ^ -1= 2 ^ -1- ( ) , [ eq : s- ] where @xmath163 is the lerch transcendent function @xcite . the expressions on the r.h.s . of eqs.([eq : san ] ) are now _ analytic functions in both variables @xmath157 _ this is a new result . the pdfs are the nonsinglet @xmath164 and singlet @xmath165 parton distribution functions , f_ns(x , q^2)&=&u_v(x , q^2)-d_v(x , q^2 ) , + f_s(x , q^2 ) & = & u_v(x , q^2)+d_v(x , q^2 ) + s(x , q^2 ) v(x , q^2)+s(x , q^2 ) , [ nss ] whereas @xmath166 is the distribution of valence quarks and @xmath167 is the sea quark distribution . more generally , the ns pdf is a combination of the forms @xmath168 and @xmath169 but for our consideration we focus on the nucleon scattering provided by combination ( b1 ) . the moments representation for pdfs is defined as f_ns(n , q^2)&= & _ 0 ^ 1dx x^n-1f_ns(x , q^2 ) , [ eq : nsmpdf ] + f_s(n , q^2)&=&_0 ^ 1dx x^n-1f_s(x , q^2 ) . [ eq : mpdf]the moments @xmath170 for the structure function @xmath171 follow from the mellin convolution @xmath172 , [ eq : func - mom ] f_(z,^2)&=&(c _ f_)(z,^2)_0 ^ 1c_(y , a_s ) f_(x,^2 ) ( z - xy ) dydx , + m_(n,^2)&= & c_(n , a_s(^2 ) ) f_(n,^2 ) . here @xmath173 is the nonsinglet coefficient function of the process that can be presented as the perturbation series @xmath174 ; @xmath153 in appendix [ app0 ] is the moment of the @xmath175 . the qcd evolution of the moments @xmath110 up to nlo of is given by ( see ref @xcite ) m_(n , q^2 ) = ( ) ^p(n ) ^d_(n ) + m_(n , q_0 ^ 2 ) , [ mns ] where @xmath176 and : d_(n)=_^(0)(n)/2_0 , p(n ) = ( - ) . the coefficients of anomalous dimension in lo and nlo and the coefficient function in nlo are given in eqs.([eq : ad - lo - nlo ] ) , appendix [ app0 ] . in the case of the nucleon structure function @xmath0 , one needs to take into account only even values of @xmath160 in the nlo anomalous dimension . the accuracy of the evaluation of the structure functions depends on the method we use ; therefore , it is indispensable to verify it in our approach . the jacobi polynomial method promises us a good enough accuracy for the evolution , as was shown in previous works ( see @xcite ) . this method is applied directly to the terms of the bjorken variable @xmath29 , but it affects the @xmath9-dependence indirectly . therefore , it is necessary to confirm the @xmath29-range applicability of the jp method . to this end , we compare the results of the jp approach with the `` exact '' numerical calculations of inverse mellin moments following eq . ( [ exinvm ] ) but only in the one - loop approximation due to technical limitations . vs @xmath29 in lo ( a ) , at the energy scale @xmath177 gev@xmath4 and ( b ) @xmath178 gev@xmath4 . the solid ( red ) line represents the fapt result and the dashed ( blue ) line the pqcd one in the jp method . the thick squares ( red ) and spheres ( blue ) represent the result of the `` exact '' numerical inverse mellin transform.,width=566 ] the comparison of these two results in fig . [ figsf3 ] demonstrates a very good accuracy . so , in order to clarify it , we perform a zoom in @xmath29 , going to a lower @xmath29-region ( @xmath179 ) . we see in fig . [ figsf4 ] that the jp method gradually loses precision starting at @xmath180 . also , we can see that in this range , the difference between these two methods reaches @xmath181% . vs @xmath29 at lo , at energy scale ( a ) @xmath177 gev@xmath4 , and ( b ) @xmath178 gev@xmath4 . the solid ( red ) line represents the fapt outcome and the dashed ( blue ) line the pqcd one in the jp method . the thick squares ( red ) and spheres ( blue ) represent the result of the `` exact '' numerical inverse mellin transform.,width=566 ] here we investigate the accuracy of the rational approximation for the two - loop evolution factor combing the approximations in eqs . ( [ eq : c2 ] , [ eq : c3 ] ) in quantity @xmath185 we obtain the accuracy better than @xmath148 for any @xmath186 ( since jp expansion contains only 13 terms for good approximation ) , in both cases of pqcd and fapt for two different ranges of energy , i.e. , low @xmath187 and high @xmath188 . the results collected in table [ table1 ] .the accuracy in per cent of the difference of the approximations @xmath189 for pqcd : @xmath190 and for fapt : @xmath191 . the results are presented in two ranges of @xmath9 : low @xmath192 and high @xmath1 . [ cols="<,<,<,<,<,<,<,<,<",options="header " , ] demonstrate that for both ranges of energy fapt has a better convergence than pqcd ; even more , the accuracy is improved for fapt at low @xmath193 ( really , pqcd must be worse but we have a low starting point @xmath49 ) . the strong hierarchy of fapt couplings , @xmath194 , remains valid even at very low @xmath195 , cf . @xcite . 99 d. v. shirkov and i. l. solovtsov , [ hep - ph/9604363 ] ; phys . * 79 * , 1209 ( 1997 ) [ hep - ph/9704333 ] . k. a. milton , i. l. solovtsov phys . d * 55 * , 5295 ( 1997 ) [ hep - ph/9611438 ] . k. a. milton , i. l. solovtsov and o. p. solovtsova , phys . b * 415 * , 104 ( 1997 ) [ arxiv : hep - ph/9706409 ] . k. a. milton , o. p. solovtsova phys . d * 57 * , 5402 ( 1998 ) [ hep - ph/9710316 ] . d. v. shirkov , theor . math . phys . * 127 * , 409 ( 2001 ) [ hep - ph/0012283 ] ; * 119 * , 438 ( 1999 ) . a. p. bakulev , s. v. mikhailov and n. g. stefanis , phys . d * 72 * , 074014 ( 2005 ) [ erratum - ibid . d * 72 * , 119908 ( 2005 ) ] [ hep - ph/0506311 ] . a. p. bakulev , s. v. mikhailov and n. g. stefanis , phys . d * 75 * , 056005 ( 2007 ) [ erratum - ibid . d * 77 * , 079901 ( 2008 ) ] [ hep - ph/0607040 ] . a. p. bakulev , s. v. mikhailov , n. g. stefanis , jhep * 1006 * , 085 ( 2010 ) [ arxiv:1004.4125 ] . a. p. bakulev , phys . nucl . * 40 * , 715 ( 2009 ) [ arxiv:0805.0829 ] ( arxiv preprint in russian ) ; n. g. stefanis , phys . * 44 * , 494 ( 2013 ) [ arxiv:0902.4805 ] . a. v. sidorov and o. p. solovtsova , mod . a 29 * ( 2014 ) 1450194 , [ arxiv:1407.6858 ] ; nonlin . phenom . complex syst . * 16 * , 397 ( 2013 ) , [ arxiv:1312.3082 ] . d. v. shirkov and i. l. solovtsov , theor . phys . * 150 * , 132 ( 2007 ) [ hep - ph/0611229 ] . a. v. nesterenko , phys . d * 62 * , 094028 ( 2000 ) ; phys . d * 64 * , 116009 ( 2001 ) ; int . j. mod . phys . a * 18 * , 5475 ( 2003 ) . a. v. nesterenko and j. papavassiliou , phys . d * 71 * , 016009 ( 2005 ) ; a. c. aguilar , a. v. nesterenko and j. papavassiliou , j. phys . g * 31 * , 997 ( 2005 ) . j. phys . g * 32 * , 1025 ( 2006 ) [ hep - ph/0511215 ] ; a. v. nesterenko , arxiv:0710.5878 . a. i. alekseev , few body syst . * 40 * , 57 ( 2006 ) [ hep - ph/0503242 ] . y. srivastava , s. pacetti , g. pancheri and a. widom , _ in the proceedings of @xmath196 physics at intermediate energies , slac , stanford , ca , usa , 30 april - 2 may 2001 , pp t19 _ [ hep - ph/0106005 ] . b. r. webber , jhep * 9810 * , 012 ( 1998 ) [ hep - ph/9805484 ] . g. cveti and c. valenzuela , j. phys . g * 32 * , l27 ( 2006 ) [ hep - ph/0601050 ] . g. cveti and c. valenzuela , phys . d * 74 * , 114030 ( 2006 ) [ hep - ph/0608256 ] . g. m. prosperi , m. raciti and c. simolo , prog . phys . * 58 * , 387 ( 2007 ) [ hep - ph/0607209 ] . g. cveti and c. valenzuela , braz . j. phys . * 38 * , 371 ( 2008 ) [ arxiv:0804.0872 ] . y. o. belyakova and a. v. nesterenko , int . a * 26 * , 981 ( 2011 ) [ arxiv:1011.1148 ] . g. cveti , r. kgerler and c. valenzuela , j. phys . g * 37 * , 075001 ( 2010 ) [ arxiv:0912.2466 ] . g. cveti , r. kgerler and c. valenzuela , phys . d * 82 , 114004 ( 2010 ) [ arxiv:1006.4199 ] . c. contreras , g. cveti , r. kgerler , p. kroger and o. orellana , arxiv:1405.5815 . c. ayala , c. contreras and g. cveti , phys . d * 85 * , 114043 ( 2012 ) [ arxiv:1203.6897 ] . a. v. kotikov , v. g. krivokhizhin and b. g. shaikhatdenov , phys . atom . nucl . * 75 * , 507 ( 2012 ) [ arxiv:1008.0545 [ hep - ph ] ] ; g. cveti , a. y. illarionov , b. a. kniehl and a. v. kotikov , phys . b * 679 * , 350 ( 2009 ) [ arxiv:0906.1925 [ hep - ph ] ] . r. s. pasechnik , d. v. shirkov and o. v. teryaev , phys . d * 78 * , 071902 ( 2008 ) [ arxiv:0808.0066 [ hep - ph ] ] ; r. s. pasechnik , d. v. shirkov , o. v. teryaev , o. p. solovtsova and v. l. khandramai , phys . d * 81 * , 016010 ( 2010 ) [ arxiv:0911.3297 [ hep - ph ] ] ; v. l. khandramai , r. s. pasechnik , d. v. shirkov , o. p. solovtsova and o. v. teryaev , phys . b * 706 * , 340 ( 2012 ) [ arxiv:1106.6352 ] . o. teryaev , nucl.phys.proc.suppl . 245 ( 2013 ) 195 [ arxiv:1309.1985 ] . p. allendes , c. ayala and g. cveti , phys . d * 89 * , 054016 ( 2014 ) [ arxiv:1401.1192 ] . v. n. gribov and l. n. lipatov , sov . j. nucl . phys . * 15 * , 438 ( 1972 ) [ yad . fiz . * 15 * , 781 ( 1972 ) ] ; l. n. lipatov , sov . . phys . * 20 * , 94 ( 1975 ) [ yad . fiz . * 20 * , 181 ( 1974 ) ] ; g. altarelli and g. parisi , nucl . b * 126 * , 298 ( 1977 ) ; y. l. dokshitzer , sov . jetp * 46 * , 641 ( 1977 ) [ zh . eksp . fiz . * 73 * , 1216 ( 1977 ) ] . v. g. krivokhizhin et al . , z. phys . c * 36 * , 51 ( 1987 ) ; v. g. krivokhizhin et al . , z. phys . c * 48 * , 347 ( 1990 ) . a. d. martin , w. j. stirling , r. s. thorne and g. watt , eur . j. * c 63 * , 189 ( 2009 ) [ arxiv:0901.0002 ] . a. j. buras , rev . phys . * 52 * , 199 ( 1980 ) . v. g. krivokhizhin and a. v. kotikov , phys . * 68 * ( 2005 ) 1873 [ yad . * 68 * ( 2005 ) 1935 ] ; v. g. krivokhizhin and a. v. kotikov , phys . nucl . * 40 * , 1059 ( 2009 ) . c. ayala and g. cveti , comput . commun . * 190 * , 182 ( 2015 ) [ arxiv:1408.6868 [ hep - ph ] ] ; arxiv:1411.1581 [ hep - ph ] . f. j. yndurain , the theory of quarks and gluons interactions ( fourth edition ) ( springer - verlag , berlin , 2006 ) . e. g. floratos , d. a. ross and c. t. sachrajda , nucl . b * 129 * , 66 ( 1977 ) [ erratum - ibid . b * 139 * , 545 ( 1978 ) ] . a. gonzalez - arroyo , c. lopez and f. j. yndurain , nucl . b * 159 * , 512 ( 1979 ) . g. curci , w. furmanski and r. petronzio , nucl . b * 175 * , 27 ( 1980 ) .

we apply ( fractional ) analytic perturbation theory ( fapt ) to the qcd analysis of the nonsinglet nucleon structure function @xmath0 in deep inelastic scattering up to the next leading order and compare the results with ones obtained within the standard perturbation qcd . based on a popular parameterization of the corresponding parton distribution we perform the analysis within the jacobi polynomial formalism and under the control of the numerical inverse mellin transform . to reveal the main features of the fapt two - loop approach , we consider a wide range of momentum transfer from high @xmath1 to low @xmath2 where the approach still works .

for a long time scattering a particle on one - dimensional ( 1d ) static potential barriers have been considered in quantum mechanics as a representative of well - understood phenomena . however , solving the so - called tunneling time problem ( ttp ) ( see reviews @xcite and references therein ) showed that this is not the case . at present there is a variety of approaches to introduce characteristic times for a 1d scattering . they are the group ( wigner ) tunneling times ( more known as the `` phase '' tunneling times ) @xcite , different variants of the dwell time @xcite , the larmor time @xcite , and the concept of the time of arrival which is based on introducing either a suitable time operator ( see , e.g. , @xcite ) or the positive operator valued measure ( see review @xcite ) . a particular class of approaches to study the temporal aspects of a 1d scattering includes the bohmian @xcite , feynman and wigner ones ( see @xcite as well as @xcite and references therein ) . one has also point out the papers @xcite to study the characteristic times of `` the forerunner preceding the main tunneling signal of the wave created by a source with a sharp onset '' . as is known ( see @xcite ) , the main question of the ttp is that of the time spent , on the average , by a particle in the barrier region in the case of a completed scattering . setting this problem implies that the particle s source and detectors are located at a considerable distance from the potential barrier . the answer to this question , for a given potential and initial state of a particle , is evident must be unique . in particular , it must not depend on the details of measurements with the removed detectors . one has to recognize that the answer has not yet been found , and this elastic scattering process looks at present like an unexplained phenomenon surrounded by paradoxes . we bear in mind , in particular , 1 ) the lack of a causal relationship between the transmitted and incident wave packets @xcite ; 2 ) a superluminal propagation of a particle through opaque potential barriers ( the hartman effect ) @xcite ; 3 ) accelerating ( on the average ) a transmitted particle , in the asymptotic region , as compared with an incident one @xcite ; 4 ) aligning the average particle s spin with the magnetic field @xcite ; 5 ) the larmor precession of the reflected particles under the non - zero magnetic field localized beyond the barrier on the side of transmission @xcite . at the first glance the bohmian mechanics provides an adequate description of the temporal aspects of a completed scattering ( see , e.g. , @xcite ) . for its `` causal '' one - particle trajectories exclude , a priory , the appearance of the above paradoxes . for example , the hartman effect does not appear in this approach : in the case of opaque rectangular barriers , the bohmian dwell time , unlike smith s and buttiker s dwell times , increases exponentially together with the barrier s width ( see also section [ a4 ] ) . it should be stressed however that the bohmian model of a 1d completed scattering is not free of paradoxes . as is well known , the region of location of the particle s source consists in this model from two parts separated by some critical point . this point is such that all particles starting from the sub - region , adjacent to the barrier region , are transmitted by the barrier ; otherwise they are reflected by it . that is , the subensembles of transmitted and reflected particles are macroscopically distinct in this model at all stages of scattering , what clearly contradicts the main principles of quantum mechanics . note , the position of the critical point depends on the barrier s shape . for a particle impinging the barrier from the left , this point approaches the left boundary of the barrier when the latter becomes less transparent . otherwise , the critical point approaches minus infinity on the ox - axis . this property means , in fact , that particles feel the barrier s shape , being however far from the barrier region . of course , this fact evidences , too , that the existing `` causal '' trajectories of the bohmian mechanics give an improper description of the scattering process . from our viewpoint , all difficulties and paradoxes to arise in studying the temporal aspects of a completed scattering result from the fact that setting this problem in the existing framework of quantum mechanics is contradictory . on the one hand , in the case of a completed scattering an observer deals only either with transmitted or reflected particles , and , consequently , all one - particle observables must be introduced individually for each sub - process . on the other hand , quantum mechanics , as it stands , does not imply the introduction of observables for the sub - processes , for its formalism does not provide the wave functions for transmission and reflection , needed for computing the expectation values of observables . so , in the case of a completed scattering , a conflicting situation arises already at the stage of setting the problem : the nature of this process requires a separate description of transmission and reflection ; while quantum mechanics , as it stands , does not allow such a description . this conflict underlies all controversy and paradoxes to arise in solving the ttp : in fact , in the existing framework of quantum theory , there are no observables which can be consistently introduced for this process . note , this concerns not only characteristic times but also all observables to have hermitian operators . for example , averaging the particle s position and momentum over the whole ensemble of particles does not give the expectation ( i.e. , most probable ) values of these quantities . as regards characteristic times , we have to stress once more that among the existing time concepts neither separate nor common times for transmission and reflection give the time spent by a particle in the barrier region . in the first case , there is no basis to distinguish ( theoretically and experimentally ) transmitted and reflected particles in the barrier region . in the second case , characteristic times introduced can not be properly interpreted ( see , e.g. , discussion of the dwell and larmor times in @xcite ) ; these times describe neither transmitted nor reflected particles ( ideal transmission and reflection are exceptional cases ) . at the same time there is a viewpoint that all the time scales introduced for a completed scattering are valid : one has only to choose a suitable clock ( operational procedure ) for each of them . this viewpoint is based on the assumption that timing a quantum particle , without influencing the scattering process , is impossible in principle . by this viewpoint the time measured should always depend on the clock used for this purpose . however , quantum phenomena , such as a completed scattering , have their own , intrinsic spatial and temporal scales , and our main task is to learn to measure these scales without influencing their values . in this paper we show that for the problem under consideration this is possible . the above conflict can be resolved in the framework of conventional quantum mechanics , and characteristic times for transmission and reflection can be introduced . for measuring these time scales without affecting the scattering process , one can exploit the larmor precession of the particle s spin under the infinitesimal magnetic field . the plan of this paper is as follows . in ( section [ a0 ] ) we introduce the concept of combined and elementary quantum processes and states . by this concept , the state of the whole quantum ensemble of particles , at the problem at hand , is a combined one to represent a coherent superposition of two ( elementary ) states of the ( to - be-)transmitted and ( to - be-)reflected subensembles of particles . in section [ a2 ] we present two solutions to the schrdinger equation to describe transmission and reflection at all stages of scattering . on their basis we define the group , dwell and larmor times for transmission and reflection ( section [ a3 ] ) . for our purposes it is relevant to address the well - known schrodinger s cat paradox which displays explicitly a principal difference between macroscopically distinct quantum states and their superpositions . as is known , macroscopically distinct quantum states are symbolized in this paradox by the dead - cat and alive - cat ones . either may be associated with a single , really existing cat which can be described in terms of one - cat observables . as regards a superposition of these two states , it can not be associated with a cat to exist really ( a cat can not be dead and alive simultaneously ) . to calculate the expectation values of one - cat observables for this state is evident to have no physical sense . as is known , quantum mechanics as it stands does not distinguish between the dead - cat and alive - cat states and their superposition . it postulates that all its rules should be equally applied to macroscopically distinct states and their superpositions . from our pint of view , the main lesson of the schrodinger s cat paradox is just that this postulate is erroneous . quantum mechanics must distinguish these two kinds of states on the conceptual level . hereinafter , any superposition of macroscopically distinct quantum states will be referred to as a combined quantum state . all quantum states , like the `` dead - cat '' and `` alive - cat '' ones , will be named here as elementary ones . thereby we emphasize that such states can not be presented as a superposition of macroscopically distinct states . note , the concepts of combined and elementary states are fully applicable to a 1d completed scattering . though we deal here with a microscopic object , at the final stage of scattering the states of the subensembles of transmitted and reflected particles are distinguished macroscopically . so that scattering a quantum particle on the potential barrier is a combined process . it consists from two alternative elementary one - particle sub - processes , transmission and reflection , evolved coherently . the main peculiarity of a time - dependent combined one - particle scattering state to describe the combination of the two elementary sub - processes is that 1 ) in the classical limit , such a state is associated with two one - particle trajectories , rather than with one ; 2 ) the squared modulus of such a state can not be interpreted as the probability density for one particle ; 3 ) for this state it is meaningless to calculate expectation values of one - particle observables , or to introduce one - particle characteristic times and trajectories . all the quantum - mechanical rules are applicable only to elementary states . neglecting this circumstance leads to paradoxes . it is useful also to note that one has to distinguish between the interference of different elementary states ( e.g. , the interference between the incident and reflected waves in the case of a non - ideal reflection ) and the self - interference of the same elementary state ( e.g. , the interference between the incident and reflected waves in the case of an ideal reflection ) . in the first case one deals with waves which are not connected causally . in the second case , interfering waves are causally connected with each other . so , to explain properly a 1d completed scattering , we have to study the behavior of the subensembles of transmitted and reflected particles at all stages of scattering . at the first glance , this programm is impracticable in principle , since quantum mechanics , as it stands , does not give the way of reconstructing the prehistory of these subensembles by their final states . however , as will be shown below ( see also @xcite ) , quantum mechanics implies such a reconstruction : we found two solutions to the schrodinger equation , which describe both the sub - processes at all stages of scattering . either consists from one incoming and only one outgoing ( transmitted or reflected ) wave . thus , though it is meaningless to say about to - be - transmitted or to - be - reflected particles , the notions of to - be - transmitted and to - be - reflected subensembles of particles are meaningful . let us consider a particle incident from the left on the static potential barrier @xmath0 confined to the finite spatial interval @xmath1 $ ] @xmath2 ; @xmath3 is the barrier width . let its in - state , @xmath4 at @xmath5 be a normalized function to belong to the set @xmath6 consisting from infinitely differentiable functions vanishing exponentially in the limit @xmath7 . the fourier - transform of such functions are known to belong to the set @xmath8 too . in this case the position , @xmath9 and momentum , @xmath10 operators both are well - defined . without loss of generality we will suppose that @xmath11 here @xmath12 is the wave - packet s half - width at @xmath5 ( @xmath13 ) . we consider a completed scattering . this means that the average velocity , @xmath14 is large enough , so that the transmitted and reflected wave packets do not overlap each other at late times . as for the rest , the relation of the average energy of a particle to the barrier s height may be any by value . we begin our analysis with the derivation of expressions for the incident , transmitted and reflected wave packets to describe , in the problem at hand , the whole ensemble of particles . for this purpose we will use the variant ( see @xcite ) of the well - known transfer matrix method @xcite . let the wave function @xmath15 to describe the stationary state of a particle in the out - of - barrier regions be written in the form @xmath16 @xmath17 here @xmath18 @xmath19 is the energy of a particle ; @xmath20 is its mass . the coefficients entering this solution are connected by the transfer matrix @xmath21 : @xmath22 @xmath23,\nonumber\\ p=\sqrt{\frac{r(k)}{t(k)}}\exp\left[i\left(\frac{\pi}{2}+ f(k)-ks\right)\right]\end{aligned}\ ] ] where @xmath24 , @xmath25 and @xmath26 are the real tunneling parameters : @xmath27 ( the transmission coefficient ) and @xmath28 ( phase ) are even and odd functions of @xmath29 , respectively ; @xmath30 ; @xmath31 ; @xmath32 . we will suppose that the tunneling parameters have already been calculated . in the case of many - barrier structures , for this purpose one may use the recurrence relations obtained in @xcite just for these real parameters . for the rectangular barrier of height @xmath33 , @xmath34^{-1},\nonumber\\ j=\arctan\left(\vartheta_{(-)}\tanh(\kappa d)\right),\\f=0,{\mbox{\hspace{3mm}}}\kappa=\sqrt{2m(v_0-e)}/\hbar,\nonumber\end{aligned}\ ] ] if @xmath35 ; and @xmath36^{-1},\nonumber\\ j=\arctan\left(\vartheta_{(+)}\tan(\kappa d)\right),\\ f=\left\{\begin{array}{c } 0,{\mbox{\hspace{3mm}}}if { \mbox{\hspace{3mm}}}\vartheta_{(-)}\sin(\kappa d)\geq 0 \\ \pi,{\mbox{\hspace{3mm}}}otherwise , \end{array } \right.\nonumber\\ \kappa=\sqrt{2m(e - v_0)}/\hbar,\nonumber\end{aligned}\ ] ] if @xmath37 ; in both cases @xmath38 ( see @xcite ) . now , taking into account exps . ( [ 50 ] ) and ( [ 500 ] ) , we can write in - asymptote , @xmath39 , and out - asymptote , @xmath40 , for the time - dependent scattering problem ( see @xcite ) : @xmath41\end{aligned}\ ] ] @xmath42 @xmath43\end{aligned}\ ] ] @xmath44;\end{aligned}\ ] ] where exps . ( [ 59 ] ) , ( [ 61 ] ) and ( [ 62 ] ) describe , respectively , the incident , transmitted and reflected wave packets . here @xmath45 is the fourier - transform of @xmath46 for example , for the gaussian wave packet to obey condition ( [ 444 ] ) , @xmath47 @xmath48 is a normalization constant . let us now show that by the final states ( [ 60])-([62 ] ) one can uniquely reconstruct the prehistory of the subensembles of transmitted and reflected particles at all stages of scattering . let @xmath49 and @xmath50 be searched - for wave functions for transmission ( twf ) and reflection ( rwf ) , respectively . by our approach their sum should give the ( full ) wave function @xmath51 to describe the whole combined scattering process . from the mathematical point of view our task is to find , for a particle impinging the barrier from the left , such two solutions @xmath49 and @xmath50 to the schrdinger equation that , for any @xmath52 , @xmath53 in the limit @xmath54 @xmath55 where @xmath56 and @xmath57 are the transmitted and reflected wave packets whose fourier - transforms presented in ( [ 61 ] ) and ( [ 62 ] ) . we begin with searching for the stationary wave functions for reflection , @xmath58 and transmission , @xmath59 let for @xmath60 @xmath61 where @xmath62 since the rwf describes only reflected particles , which are expected to be absent behind the barrier , the probability flux for @xmath63 should be equal to zero - @xmath64 and @xmath65 should be the same - @xmath66 we can exclude @xmath49 from eq . ( [ 263 ] ) . as a result , we obtain @xmath67 since @xmath68 , from eqs . ( [ 264 ] ) and ( [ 2630 ] ) it follows that @xmath69 ; @xmath70 . so , a coherent superposition of the incoming waves to describe transmission and reflection , for a given @xmath19 , yields the incoming wave of unite amplitude , that describes the whole ensemble of incident particles . in this case , not only @xmath71 , but also @xmath72 ! besides , the phase difference for the incoming waves to describe reflection and transmission equals @xmath73 irrespective of the value of @xmath19 . our next step is to show that only one root of @xmath74 gives a searched - for @xmath75 for this purpose the above solution should be extended into the region @xmath76 . to do this , we will restrict ourselves by symmetric potential barriers , though the above derivation is valid for all barriers . let @xmath0 be such that @xmath77 @xmath78 as is known , for the region of a symmetric potential barrier , one can always find odd , @xmath79 , and even , @xmath80 , solutions to the schrdinger equation . we will suppose here that these functions are known . for example , for the rectangular potential barrier ( see exps . ( [ 501 ] ) and ( [ 502 ] ) ) , @xmath81 @xmath82 note , @xmath83 is a constant , which equals @xmath84 in the case of the rectangular barrier . without loss of generality we will keep this notation for any symmetric potential barrier . before finding @xmath63 and @xmath65 in the barrier region , we have firstly to derive expressions for the tunneling parameters of symmetric barriers . let in the barrier region @xmath85 `` sewing '' this expression together with exps . ( [ 1 ] ) and ( [ 2 ] ) at the points @xmath86 and @xmath87 , respectively , we obtain @xmath88 @xmath89 as a result , @xmath90 as it follows from ( [ 50 ] ) , @xmath91 @xmath92 . hence @xmath93 @xmath94 @xmath95 . besides , for symmetric potential barriers @xmath96 when @xmath97 ; otherwise , @xmath98 then , one can show that `` sewing '' the general solution @xmath63 in the barrier region together with exp . ( [ 265 ] ) at @xmath86 , for both the roots of @xmath74 , gives odd and even functions in this region . for the problem considered , only the former has a physical meaning . the corresponding roots for @xmath99 and @xmath100 read as @xmath101 one can easily show that in this case @xmath102 for @xmath103 @xmath104 the extension of this solution onto the region @xmath105 gives @xmath106 let us now show that the searched for rwf is , in reality , zero to the right of the barrier s midpoint . indeed , as is seen from exp . ( [ 3000 ] ) , @xmath107 for all values of @xmath29 . in this case the probability flux , for any time - dependent wave function formed from @xmath63 , is equal to zero at the barrier s midpoint for any value of time . this means that reflected particles impinging the symmetric barrier from the left do not enter the region @xmath108 . thus , @xmath109 for @xmath108 . in the region @xmath110 it is described by exps . ( [ 265 ] ) and ( [ 3000 ] ) . for this solution , the probability density is everywhere continuous and the probability flux is everywhere equal to zero . as regards the searched - for twf , one can easily show that @xmath111 @xmath112 @xmath113 where @xmath114 like @xmath58 the twf is everywhere continuous and the corresponding probability flux is everywhere constant ( we have to stress once more that this flux has no discontinuity at the point @xmath115 , though the first derivative of @xmath65 on @xmath116 is discontinuous at this point ) . as in the case of the rwf , wave packets formed from @xmath65 should evolve in time with a constant norm . so , for any value of @xmath52 @xmath117 @xmath118 and @xmath119 are the average transmission and reflection coefficients , respectively . besides , @xmath120 from this it follows , in particular , that the scalar product of the wave functions for transmission and reflection , @xmath121 is a purely imagine quantity to approach zero when @xmath122 . now we are ready to proceed to the study of temporal aspects of a 1d completed scattering . the wave functions for transmission and reflection presented in the previous section permit us to introduce characteristic times for either sub - process . our main aim is to find , for each sub - process , the time spent , on the average , by a particle in the barrier region . in doing so , we have to bear in mind that there may be different approximations of this quantity . however , we have to remind that its true value must not depend , for a completed scattering , on the choice of `` clocks '' . measuring the tunneling time , under such conditions , implies that a particle has its own , internal `` clock '' to remember the time spent by the particle in the spatial region investigated . this means that the only way to measure the tunneling time for a completed scattering is to exploit the internal degrees of freedom of quantum particles . as is known , namely this idea underlies the larmor - time concept based on the larmor precession of the particle s spin under the infinitesimal magnetic field . in the above context , the concepts of the group and dwell times are rather auxiliary ones , since they can not be verified . nevertheless , they may be useful for a better understanding of the scattering process . we begin our analysis from the group time concept to give the time spent by the wave - packet s cm in the spatial regions . in other words , both for transmitted and reflected particles , we begin with timing `` mean - statistical particles '' of these subensembles ( their motion is described by the ehrenfest equations ) . in doing so , we will distinguish exact and asymptotic group times . let @xmath123 and @xmath124 be such moments of time that @xmath125 @xmath126 then , one can define the transmission time @xmath127 as the difference @xmath128 where @xmath123 is the smallest root of eq . ( [ 80 ] ) , and @xmath124 is the largest root of eq . ( [ 81 ] ) . similarly , for reflection , let @xmath129 and @xmath130 be such values of @xmath52 that @xmath131 then the exact group time for reflection , @xmath132 is @xmath133 of course , a serious shortcoming of the exact characteristic times is that they fit only for sufficiently narrow ( in @xmath116-space ) wave packets . for wide packets these times give a very rough estimation of the time spent by a particle in the barrier region . for example , one may a priory say that the exact group time for reflection , for a sufficiently narrow potential barrier and/or wide wave packet , should be equal to zero . in this case , the wave - packet s cm does not enter the barrier region . note , the potential barrier influences a particle not only when its most probable position is in the barrier region . for a completed scattering it is useful also to introduce asymptotic group times to describe the passage of the particle in the sufficiently large spatial interval @xmath134;$ ] where @xmath135 it is evident that in this case , instead of the exact wave functions for transmission and reflection , we may use the corresponding in- and out - asymptotes derived in @xmath29-representation . the `` full '' in - asymptote , like the corresponding out - asymptote , represents the sum of two wave packets : @xmath136 @xmath137;\end{aligned}\ ] ] @xmath138;\end{aligned}\ ] ] @xmath139 ( see ( [ 301 ] ) ) . one can easily show that @xmath140 ; hereinafter , the prime denotes the derivative with respect to @xmath29 . for the average wave numbers in the asymptotic spatial regions we have @xmath141 besides , at early and late times @xmath142 @xmath143 ( henceforth , angle brackets denote averaging over the corresponding in- or out - asymptotes ) . as it follows from exps . ( [ 73 ] ) and ( [ 74 ] ) , the average starting points @xmath144 and @xmath145 , for the subensembles of transmitted and reflected particles , respectively , read as @xmath146 the implicit assumption made in the standard wave - packet analysis is that transmitted and reflected particles start , on the average , from the origin ( in the above setting the problem ) . however , by our approach , just @xmath144 and @xmath145 are the average starting points of transmitted and reflected particles , respectively . they are the initial values of @xmath147 and @xmath148 , which have the status of the expectation values of the particle s position . they behave causally in time . as regards the average starting point of the whole ensemble of particles , its coordinate is the initial value of @xmath149 , which behaves non - causally in the course of scattering . this quantity has no status of the _ expectation _ value of the particle s position . let us take into account exps . ( [ 73 ] ) , ( [ 74 ] ) and analyze the motion of a particle in the spatial interval @xmath134 $ ] . in particular , let us define the transmission time for this region , making use the asymptotes of the twf . we will denote this time as @xmath150 . the equations for the arrival times @xmath123 and @xmath124 for the extreme points @xmath151 and @xmath152 , respectively , read as @xmath153 considering ( [ 73 ] ) , we obtain from here that the transmission time for this interval is @xmath154 similarly , for the reflection time @xmath155 where @xmath156 , we have @xmath157 considering ( [ 74 ] ) , one can easily show that @xmath158 the times @xmath159 ( @xmath160 ) and @xmath161 ( @xmath162 ) are , respectively , the searched - for asymptotic group times for transmission and reflection , for the barrier region : @xmath163 @xmath164 note , unlike the exact group times , the asymptotic ones may be negative by value : they do not give the time spent by a particle in the barrier region ( see also fig.1 ) . the lengths @xmath165 and @xmath166 where @xmath167 may be treated as the effective barrier s widths for transmission and reflection , respectively . let us consider the case of the rectangular barrier and obtain explicit expressions for @xmath168 ( now , both for transmission and reflection , @xmath169 since @xmath170 ) which can be treated as the effective width of the barrier for a particle with a given @xmath29 . besides , we will obtain the corresponding expressions for the expectation value , @xmath171 , of the staring point for this particle : @xmath172 . it is evident that in terms of @xmath173 the above asymptotic times for a particle with the well - defined momentum @xmath174 read as @xmath175 using exps . ( [ 501 ] ) and ( [ 502 ] ) , one can show that , for the below - barrier case ( @xmath176 ) - @xmath177 \left[\kappa_0 ^ 2\sinh(\kappa d)-k^2 \kappa d\right ] } { 4k^2\kappa^2 + \kappa_0 ^ 4\sinh^2(\kappa d)}\ ] ] @xmath178 for the above - barrier case ( @xmath179 - @xmath180\left[k^2 \kappa d-\beta \kappa_0 ^ 2\sin(\kappa d)\right ] } { 4k^2\kappa^2+\kappa_0 ^ 4\sin^2(\kappa d)}\ ] ] @xmath181 where @xmath182 @xmath183 , if @xmath184 ; otherwise , @xmath185 note , @xmath186 and @xmath187 , in the limit @xmath188 . for infinitely narrow in @xmath116-space wave packets , this property ensures the coincidence of the average starting points for both subensembles with that for all particles . for wide barriers , when @xmath189 and @xmath176 , we have @xmath190 and @xmath191 that is , the asymptotic group transmission time saturates with increasing the width of an opaque potential barrier . it is important to stress that for the @xmath192-potential , @xmath193 @xmath194 . the subensembles of transmitted and reflected particles start , on the average , from the point @xmath195 @xmath196 let us now consider the stationary scattering problem . it describes the limiting case of a scattering of wide wave packets , when the group - time concept leads to a large error in timing a particle . note , in the case of transmission the density of the probability flux , @xmath197 , for @xmath65 is everywhere constant and equal to @xmath198 . the velocity , @xmath199 , of an infinitesimal element of the flux , at the point @xmath200 equals @xmath201 outside the barrier region the velocity is everywhere constant : @xmath202 . in the barrier region it depends on @xmath116 . in the case of an opaque rectangular potential barrier , @xmath203 decreases exponentially when the infinitesimal element approaches the midpoint @xmath204 . one can easily show that @xmath205 , but @xmath206 . thus , any selected infinitesimal element of the flux passes the barrier region for the time @xmath207 , where @xmath208 by analogy with @xcite we will call this time scale the dwell time for transmission . for the rectangular barrier this time reads ( for @xmath209 and @xmath210 , respectively ) as @xmath211,\end{aligned}\ ] ] @xmath212.\end{aligned}\ ] ] in the case of reflection the situation is less simple . the above arguments are not applicable here , for the probability flux for @xmath213 is zero . as is seen , the dwell time for transmission coincides , in fact , with buttiker s dwell time introduced however on the basis of the wave function for transmission . therefore , making use of the arguments by buttiker , let us define the dwell time for reflection , @xmath214 , as @xmath215 where @xmath216 is the incident probability flux for reflection . again , for the rectangular barrier @xmath217 @xmath218 as is seen , for rectangular barriers the dwell times for transmission and reflection do not coincide with each other , unlike the asymptotic group times . we have to stress once more that exps . ( [ 4005 ] ) and ( [ 40014 ] ) , unlike smith s , buttiker s and bohmian dwell times , are defined in terms of the twf and rwf . as will be seen from the following , the dwell times introduced can be justified in the framework of the larmor - time concept . as was said above , both the group and dwell time concepts do not give the way of measuring the time spent by a particle in the barrier region . this task can be solved in the framework of the larmor time concept . as is known , the idea to use the larmor precession as clocks was proposed by baz @xcite and developed later by rybachenko @xcite and bttiker @xcite ( see also @xcite ) . however the known concept of larmor time has a serious shortcoming . it was introduced in terms of asymptotic values ( see @xcite ) . in this connection , our next step is to define the larmor times for transmission and reflection , taking into account the expressions for the corresponding wave functions in the barrier region . let us consider the quantum ensemble of electrons moving along the @xmath116-axis and interacting with the symmetrical time - independent potential barrier @xmath0 and small magnetic field ( parallel to the @xmath219-axis ) confined to the finite spatial interval @xmath1.$ ] let this ensemble be a mixture of two parts . one of them consists from electrons with spin parallel to the magnetic field . another is formed from particles with antiparallel spin . let at @xmath5 the in state of this mixture be described by the spinor @xmath220 where @xmath221 is a normalized function to satisfy conditions ( [ 444 ] ) . so that we will consider the case , when the spin coherent in state ( [ 9001 ] ) is the eigenvector of @xmath222 with the eigenvalue 1 ( the average spin of the ensemble of incident particles is oriented along the @xmath116-direction ) ; hereinafter , @xmath223 @xmath224 and @xmath225 are the pauli spin matrices . for electrons with spin up ( down ) , the potential barrier effectively decreases ( increases ) , in height , by the value @xmath226 ; here @xmath227 is the frequency of the larmor precession ; @xmath228 @xmath229 denotes the magnetic moment . the corresponding hamiltonian has the following form , @xmath230;\nonumber\\ \hat{h}=\frac{\hat{p}^2}{2 m } , { \mbox{\hspace{3mm}}}otherwise.\end{aligned}\ ] ] for @xmath231 , due to the influence of the magnetic field , the states of particles with spin up and down become different . the probability to pass the barrier is different for them . let for any value of @xmath52 the spinor to describe the state of particles read as @xmath232 in accordance with ( [ 261 ] ) , either spinor component can be uniquely presented as a coherent superposition of two probability fields to describe transmission and reflection : @xmath233 note that @xmath234 for @xmath108 . as a consequence , the same decomposition takes place for spinor ( [ 9002 ] ) : @xmath235 we will suppose that all the wave functions for transmission and reflection are known . it is important to stress here ( see ( [ 700100 ] ) that @xmath236 @xmath237 and @xmath238 are the ( real ) transmission and reflection coefficients , respectively , for particles with spin up @xmath239 and down @xmath240 . let further @xmath241 and @xmath242 be quantities to describe all particles . to study the time evolution of the average particle s spin , we have to find the expectation values of the spin projections @xmath243 , @xmath244 and @xmath245 . note , for any @xmath52 @xmath246 @xmath247 @xmath248.\end{aligned}\ ] ] similar expressions are valid for transmission and reflection : @xmath249 @xmath250 note , @xmath251 @xmath252 at @xmath253 however , this is not the case for transmission and reflection . namely , for @xmath5 we have @xmath254 @xmath255 since the norms of @xmath256 and @xmath257 are constant , @xmath258 and @xmath259 for any value of @xmath52 . for the @xmath219-components of spin we have @xmath260 so , since the operator @xmath245 commutes with hamiltonian ( [ 900200 ] ) , this projection of the particle s spin should be constant , on the average , both for transmission and reflection . from the most beginning the subensembles of transmitted and reflected particles possess a nonzero average @xmath219-component of spin ( though it equals zero for the whole ensemble of particles , for the case considered ) to be conserved in the course of scattering . by our approach it is meaningless to use the angles @xmath261 and @xmath262 as a measure of the time spent by a particle in the barrier region . as in @xcite , we will suppose further that the applied magnetic field is infinitesimal . in order to introduce characteristic times let us find the derivations @xmath263 and @xmath264 for this purpose we will use the ehrenfest equations for the average spin of particles : @xmath265dx\\ \frac{d<\hat{s}_y>_{tr}}{dt}=\frac{\hbar\omega_l}{t } \int_a^b \re[(\psi_{tr}^{(\uparrow)}(x , t))^*\psi_{tr}^{(\downarrow)}(x , t)]dx\\ \frac{d<\hat{s}_x>_{ref}}{dt}=-\frac{\hbar\omega_l}{r } \int_a^{x_c } \im[(\psi_{ref}^{(\uparrow)}(x , t))^*\psi_{ref}^{(\downarrow)}(x , t)]dx\\ \frac{d<\hat{s}_y>_{ref}}{dt}=\frac{\hbar\omega_l}{r } \int_a^{x_c } \re[(\psi_{ref}^{(\uparrow)}(x , t))^*\psi_{ref}^{(\downarrow)}(x , t)]dx.\end{aligned}\ ] ] note , @xmath266 @xmath267 hence , in the case of infinitesimal magnetic field and chosen initial conditions , when @xmath268 we have @xmath269 then , considering the above expressions for the spin projections and their derivatives on @xmath52 , we obtain @xmath270dx } { \int_{-\infty}^\infty \re[(\psi_{tr}^{(\uparrow)}(x , t))^*\psi_{tr}^{(\downarrow)}(x , t)]dx};\ ] ] @xmath271dx } { \int_{-\infty}^{x_c } \re[(\psi_{ref}^{(\uparrow)}(x , t))^*\psi_{ref}^{(\downarrow)}(x , t)]dx}.\ ] ] or , taking into account that in the first order approximation on @xmath227 , when @xmath272 and @xmath273 we have @xmath274 note , in this limit , @xmath275 and @xmath276 . as is supposed in our setting the problem , both at the initial and final instants of time , a particle does not interact with the potential barrier and magnetic field . in this case , without loss of exactness , the angles of rotation ( @xmath277 and @xmath278 ) of spin under the magnetic field , in the course of a completed scattering , can be written in the form , @xmath279 on the other hand , both the quantities can be written in the form : @xmath280 and @xmath281 where @xmath282 and @xmath283 are the larmor times for transmission and reflection . comparing these expressions with ( [ 90020 ] ) , we eventually obtain @xmath284 these are just the searched - for definitions of the larmor times for transmission and reflection . as is seen , if the state of a particle is described by a normalized wave function @xmath285 , then the time @xmath286 spent by the particle in the barrier region is @xmath287 this definition is just that introduced in @xcite ) on the basis of classical mechanics ( see also @xcite ) ; note that in both cases the integrals are calculated over the whole completed scattering . thus , on the one hand , our approach justifies the definition ( b2 ) , since this expression is obtained now as the larmor time . as a consequence , it can be verified experimentally . on the other hand , we correct the domain of the validity of this expression . by our approach , it is meaningful only in the framework of the separate description of transmission and reflection , based on the solutions @xmath288 and @xmath289 found first in the present paper . our next step is to transform exps . ( [ 922 ] ) . note , for transmission , @xmath288 reads as @xmath290 where @xmath291 is the stationary wave function for transmission ( see section [ a2 ] ) . then the integral @xmath292 in ( [ 922 ] ) can be reduced , by integrating on @xmath52 , to the form @xmath293}{e(k^\prime)-e(k)}\end{aligned}\ ] ] however , @xmath294}{e(k^\prime)-e(k)}=\frac{\pi}{\hbar}\delta[(e(k^\prime)-e(k))/\hbar]\\ = \frac{\pi m}{\hbar^2 k}\left[\delta(k^\prime - k)-\delta(k^\prime+k)\right].\end{aligned}\ ] ] making use a symmetrized expression for the real integral @xmath295 , one can show that the second term to contain @xmath296 leads to zero input into @xmath295 . as a result , for the larmor transmission time , we obtain @xmath297 or , taking into account exp . ( [ 4005 ] ) as well as the relationship @xmath298 we eventually obtain that @xmath299 where @xmath300 a similar expression takes place for @xmath283 - @xmath301 the integrands in both these expressions are evident to be non - singular at @xmath302 . thus , the larmor times for transmission and reflection are , like the local dwell time ( see @xcite ) , the average values of the corresponding dwell times . in the end of this section it is useful again to address rectangular barriers . for the stationary case , in addition to larmor times ( [ 4007 ] ) , ( [ 4009 ] ) , ( [ 40030 ] ) and ( [ 40031 ] ) ) , we present explicit expressions for the initial angles @xmath261 and @xmath303 . to the first order in @xmath227 , we have @xmath304 @xmath305 @xmath306 and @xmath307 where @xmath308 for @xmath35 and @xmath37 , respectively ; @xmath309 for @xmath35 and @xmath37 , respectively . note that @xmath310 is just the characteristic time introduced in @xcite ( see exp . ( 2.20a ) ) . however , we have to stress once more that this quantity does not describe the duration of the scattering process ( see the end of section [ a332 ] ) . as regards @xmath311 this quantity is directly associated with timing a particle in the barrier region . it describes the initial position of the `` clock - pointers '' , which they have before entering this region . let us now show that the case of tunneling a particle , with a well defined energy , through an opaque rectangular potential barrier is the most suitable one to verify our approach . let us denote the measured azimuthal angle as @xmath312 by our approach @xmath313 . that is , the final time to be registered by the particle s `` clocks '' should be equal to @xmath314 as is seen , in the general case there is a problem to distinguish the inputs @xmath315 and @xmath316 however , for a particle tunneling through an opaque rectangular barrier this problem disappears . the point is that for @xmath317 @xmath318 ( see exps . ( [ 4007 ] ) and ( [ 90028 ] ) ) . note , in the case considered , smith s dwell time ( @xmath319 ) , which coincides with the `` phase '' time , and buttiker s dwell time ( see exps . ( 3.2 ) and ( 2.20b ) in @xcite ) saturate with increasing the barrier s width . just this property of the tunneling times is interpreted as the hartman effect . at the same time , our approach denies the existence of the hartman effect : transmission time ( [ 4007 ] ) increases exponentially when @xmath320 note that the bohmian approach formally denies this effect , too . it predicts that the time , @xmath321 spent by a transmitted particle in the opaque rectangular barrier is @xmath322.\end{aligned}\ ] ] thus , for @xmath189 we have @xmath323 i.e. , @xmath324 as is seen , in comparison with our definition , @xmath325 overestimates the duration of dwelling transmitted particles in the barrier region . in the final analysis , this sharp difference between @xmath325 and @xmath207 is explained by the fact that @xmath326 to describe transmission was obtained in terms of @xmath327 one can show that the input of the to - be - reflected subensemble of particles into @xmath328 dominates inside the region of an opaque potential barrier . therefore treating this time scale as a characteristic time for transmission has no basis . as was said ( see sections [ ai ] and [ a0 ] ) , the trajectories of transmitted and reflected particles are ill - defined in the bohmian mechanics . however , we have to stress that our approach does not at all deny the bohmian one . it suggests only that causal trajectories for these particles should be redefined . an incident particle should have two possibility ( to be transmitted or to reflected by the barrier ) irrespective of the location of its starting point . this means that just two causal trajectories should evolve from each staring point : on the @xmath329-axis one should lead to plus infinity , but another should approach minus infinity . both sets of causal trajectories must be defined on the basis of @xmath288 and @xmath330 as to the rest , all mathematical tools developed in the bohmian mechanics ( see , e.g. , @xcite ) remain in force . tunneling is useful also to display explicitly the role of the exact and asymptotic group times . fig.1 shows the time dependence of the mean value of the particle s position for transmission , where @xmath331 , @xmath332 , @xmath333 . at @xmath5 the ( full ) state of the particle is described by the gaussian wave packet peaked around @xmath334 ; its half - width @xmath335 ; the average energy of the particle @xmath336 . as is seen , the exact group time gives the time spent by the cm of the transmitted wave packet in the barrier region . but the asymptotic time displays its lag , long after the scattering event , with respect to the cm of a packet , to start from the point @xmath144 and move freely with the velocity @xmath337 . in this case the exact group transmission time is equal approximately to @xmath338 , the asymptotic one is of @xmath339 , and @xmath340 . as is seen , the dwell and exact group times for transmission , both evidence that , though the asymptotic group time for transmission is small for this case , transmitted particles spend much time in the barrier region . note , also that the times spent by transmitted and reflected particles in the barrier region do not coincide even for symmetric barriers . it is shown that a 1d completed scattering is a combination of two sub - processes , transmission and reflection , evolved coherently . in the case of symmetric potential barrier we find explicitly two solutions to the schrdinger equation , which describe these sub - processes at all stages of scattering . their sum gives the wave function to describe the whole combined process . on the basis of these solutions , for either sub - process , we define the time spent , on the average , by a particle in the barrier region . for this purpose we reconsider the well - known group , dwell and larmor - time concepts . the group time concept is suitable for timing a particle in a well - localized state , when the width of a wave packet is smaller than the barrier s width . the dwell time concept is introduced for timing a particle in the stationary state . the larmor `` clock '' is the most universal instrument for timing the motion of transmitted and reflected particles , without influence on the scattering event . it is applicable for any wave packets . we found that the larmor times for transmission and reflection are the average values of the corresponding dwell times . the results of our theory can be verified experimentally . 861 e.h . hauge and j.a . stvneng , rev . phys . * 61 * , 917 ( 1989 ) . r. landauer and th . martin , rev . mod . phys . * 66 * , 217 ( 1994 ) . v.s . olkhovsky and e. recami , phys . repts . * 214 * , 339 ( 1992 ) . a.m. steinberg , phys . * 74 * , 2405 ( 1995 ) . muga , c.r . leavens , phys . repts . * 338 * , 353 ( 2000 ) . carvalho , h.m . nussenzveig , phys . repts . * 364 * , 83 ( 2002 ) . v. s. olkhovsky , e. recami , j. jakiel , phys . 398 * , 133 ( 2004 ) . wigner , phys . rev . * 98 * , 145 ( 1955 ) . hartman , j. appl . phys . * 33 * , 3427 ( 1962 ) . hauge , j.p . falck and t.a . fjeldly , phys . b * 36 * , 4203 ( 1987 ) . n. teranishi , a.m. kriman and d.k . ferry , superlatt . and microstrs . * 3 * , 509 ( 1987 ) . smith , phys . rev . * 118 * , 349 ( 1960 ) . w. jaworski and d.m . wardlaw , phys . rev . a * 37 * , 2843 ( 1988 ) . w. jaworski and d.m . wardlaw , phys . a * 38 * , 5404 ( 1988 ) . m. buttiker , phys . b * 27 * , 6178 ( 1983 ) . . leavens and g.c . aers , phys . b * 39 * , 1202 ( 1989 ) . nussenzveig , phys . a * 62 * , 042107 ( 2000 ) . mario goto , hiromi iwamoto , verissimo m. aquino , valdir c. aguilera - navarro and donald h. kobe , j. phys . a * 37 * , 3599 ( 2004 ) . muga , s. brouard and r. sala , phys . a * 167 * , 24 ( 1992 ) . christian bracher , manfred kleber , and mustafa riza , phys . a * 60 * , 1864 ( 1999 ) . baz , yad . fiz . * 4 * , 252 ( 1966 ) . rybachenko , yad . * 5 * , 895 ( 1966 ) . . leavens and g.c . aers , phys . b * 40 * , 5387 ( 1989 ) . m. buttiker , in _ time in quantum mechanics , ( lecture notes in physics vol m72 ) _ ed j.g . muga , r.s . mayato and i.l . egusquiza ( berlin : springer ) , 256 ( 2002 ) . liang , and d.h . kobe , phys . a * 64 * , 043112 ( 2001 ) . liang , y.h . nie , j.j . liang and j.q . liang , j. phys . a * 36 * , 6563 ( 2003 ) . y. aharonov , d. bohm , phys . rev . * 122 * , 1649 ( 1961 ) . s. brouard , r. sala , and j.g . muga , phys . a * 49 * , 4312 ( 1994 ) . hahne , j. phys . a * 36 * , 7149 ( 2003 ) . noh , a. fougeres , and l. mandel , phys . * 67 * , 1426 ( 1991 ) . hegerfeldt , d. seidel , and j.g . muga , phys . a * 68 * , 022111 ( 2003 ) . mckinnon and c.r . leavens , phys . a * 51 * , 2748 ( 1995 ) . c.r . leavens , phys . a * 58 * , 840 ( 1998 ) . g. grubl and k. rheinberger , j. phys . a * 35 * , 2907 ( 2002 ) . s. kreidl , g. grubl and h.g . embacher , j. phys . a * 36 * , 8851 ( 2003 ) . sabine kreidl , j. phys . a * 38 * , 5293 ( 2005 ) . d. sokolovski and l.m . baskin , phys . a * 36 * , 4604 ( 1987 ) . n. yamada , phys . * 83 * , 3350 ( 2000 ) . g. garcia - calderon , j. villavicencio , and n. yamada , phys . a * 67 * , 052106 ( 2003 ) . norifumi yamada , phys . * 93 * , 170401 ( 2004 ) . p. krekora , q. su , and r. grobe , phys . a * 64 * , 022105 ( 2001 ) . g. garcia - calderon and j. villavicencio , phys . a * 64 * , 012107 ( 2001 ) . g. garcia - calderon , j. villavicencio , f. delgado , and j.g . muga , phys . a * 66 * , 042119 ( 2002 ) . f. delgado , j.g . muga , a. ruschhaupt , g. garcia - calderon , and j. villavicencio , j. phys . a * 68 * , 032101 ( 2003 ) . m. buttiker and r. landauer , phys . * 49 * , 1739 ( 1982 ) . muga , i.l . egusquiza , j.a . damborenea , f. delgado , phys . a * 66 * , 042115 ( 2002 ) . winful , phys . * 91 * , 260401 ( 2003 ) . olkhovsky , v. petrillo , and a.k . zaichenko , phys . rev . a * 70 * , 034103 ( 2004 ) . d. sokolovski , a.z . msezane , v.r . shaginyan , phys . a * 71 * , 064103 ( 2005 ) . chuprikov , e - prints quant - ph/0405028 , quant - ph/0501067 , quant - ph/0507196 . n.l . chuprikov , sov . semicond . * 26 * , 2040 ( 1992 ) . e. merzbacher , _ quantum mechanics _ ( john wiley & sons , inc . new york ) , ( 1970 ) . taylor , _ scattering theory : the quantum theory on nonrelativistic collisions _ ( john wiley & sons , inc . new york - london - sydney ) , ( 1972 ) . chuprikov , semicond . * 31 * , 427 ( 1997 ) . fig.1 the @xmath52-dependence of the average position of transmitted particles ( solid line ) ; the initial ( full ) state vector represents the gaussian wave packet peaked around the point @xmath334 , its half - width equals to @xmath341 the average kinetic particle s energy is @xmath342 @xmath331 , @xmath332 .

a _ completed _ scattering of a particle on a static one - dimensional ( 1d ) potential barrier is a combined quantum process to consist from two elementary sub - processes ( transmission and reflection ) evolved coherently at all stages of scattering and macroscopically distinct at the final stage . the existing model of the process is clearly inadequate to its nature : all one - particle `` observables '' and `` tunneling times '' , introduced as quantities to be common for the sub - processes , can not be experimentally measured and , consequently , have no physical meaning ; on the contrary , quantities introduced for either sub - process have no basis , for the time evolution of either sub - process is unknown in this model . we show that the wave function to describe a completed scattering can be uniquely presented as the sum of two solutions to the schrdinger equation , which describe separately the sub - processes at all stages of scattering . for symmetric potential barriers such solutions are found explicitly . for either sub - process we define the time spent , on the average , by a particle in the barrier region . we define it as the larmor time . as it turned out , this time is just buttiker s dwell time averaged over the corresponding localized state . thus , firstly , we justify the known definition of the local dwell time introduced by hauge and co - workers as well by leavens and aers , for now this time can be measured ; secondly , we confirm that namely buttiker s dwell time gives the energy - distribution for the tunneling time ; thirdly , we state that all the definitions are valid only if they are based on the wave functions for transmission and reflection found in our paper . besides , we define the exact and asymptotic group times to be auxiliary in timing the scattering process .

self - organized collective dynamics is ubiquitous in the living world and emerges at all possible scales , from cell assemblies@xcite to animal groups@xcite . collective motion happens when thousands of moving individual entities coordinate with each other through local interactions such as attraction and alignment . as a result , large - scale structures of typical sizes exceeding the inter - individual distances by several orders of magnitude are formed . one of the key questions is to understand how these self - organized structures spontaneously emerge from local interactions without the intervention of any leader . with this aim , individual - based models ( ibm ) , i.e. models that describe the behavior of each individual agent have been investigated@xcite . they consist of large systems of ordinary or stochastic differential equations the numerical resolution of which is computationally intensive . to describe large - scale structures coarse - grained models such as fluid models ( fm ) are needed . fm describe the dynamics of average quantities such as the mean density or mean velocity of the individuals@xcite . attempts to derive fm from ibm of collective motion can be found in @xcite . an intermediate step in the hierarchy of models consist of kinetic models ( km)@xcite which are partial differential equations ( pde ) describing the evolution of the probability density of the particles in phase - space . fm can be obtained as singular limits of the km under the hypothesis that the individual scales are much smaller than the system scales . this pde - based derivation of fm is referred to as the hydrodynamic limit. in @xcite , the hydrodynamic limit of the vicsek ibm@xcite has been performed using an intermediate kinetic description@xcite . the vicsek ibm describes a noisy system of self - propelled particles interacting through local alignment . in @xcite , it has been shown that the absence of conservation laws ( such as momentum conservation ) resulting from self - propulsion can be overcome by introducing the new `` generalized collision invariant '' concept . the resulting model , referred to as the `` self - organized hydrodynamics ( soh ) '' is written : @xmath0 + \theta \mathcal{p}_{\omega^\perp}\nabla\rho = 0,\label{model08 - 2}\\ & & |\omega| = 1\label{model08 - 3},\end{aligned}\ ] ] where @xmath1 and @xmath2 are the density and the orientation of the mean velocity of the particles , @xmath3 , @xmath4 and @xmath5 are given parameters , and @xmath6 is the spatial dimension . we let @xmath7 be the projection matrix onto the plane orthogonal to @xmath8 . this model resembles the usual isothermal gas dynamics equations . ( [ model08 - 1 ] ) is the continuity equation expressing the conservation of mass . ( [ model08 - 2 ] ) describes how the velocity orientation evolves under transport by the flow ( the second term ) and the pressure gradient ( the third term , where @xmath9 is related to the noise in the underlying ibm and has the interpretation of a temperature ) . however , there are important differences , which arise from the fact the @xmath8 is not a true velocity but the velocity direction , i.e. it is a vector of unit norm ( which is expressed by ( [ model08 - 3 ] ) ) . to preserve this geometrical constraint , the pressure gradient has to be projected onto the normal to @xmath8 , which is the reason for the presence of @xmath10 . other differences stem from the allowed discrepancy between the two constants @xmath11 and @xmath12 . while @xmath11 fixes the material velocity to @xmath13 , the constant @xmath12 describes how @xmath8 is transported . this discrepancy originates from the lack of galilean invariance of the underlying ibm , itself resulting from self - propulsion@xcite . this model has been extended into several directions@xcite and a rigorous existence result is established in @xcite . this paper is devoted to the study of the soh model in an annular domain . annular geometries allow for simple observations of symmetry - breaking transitions induced by collective motion . when a transition from disordered to collective motion occurs , the system is set into a collective rotation in either clockwise or counter - clockwise directions . annular geometries are a traditional design for salmon cages in sea farms@xcite and for experiments with locusts@xcite , pedestrians@xcite or sperm - cell dynamics@xcite . in all these examples , a polarized motion in one direction is observed . in the sperm - cell experiments , the observation of turbulent structures that superimpose to collective rotation motivates the present work . in pure semen , sperm - cells are mostly interacting through volume exclusion . but volume exclusion interactions of rod - like self - propelled particles result in alignment@xcite . this legitimates the use of the vicsek model@xcite and of its fluid counterpart , the soh model@xcite , as models of collective sperm - cell dynamics . the vicsek model in annular geometry has been shown to exhibit polarized motion in @xcite . here , we focus on the soh model and study its normal modes in annular geometry in both the linear and nonlinear regimes . we first study the linear modes of the soh model around a perfectly polarized steady - state in sec . [ secss ] . one of the main results of this paper is that these modes are pure imaginary ( and thus , stable ) and form a countable set . in sec . [ secssnum ] , we compute the eigenmodes and eigenfunctions numerically and investigate how the eigenmodes depend on the geometry of the annulus and on the parameters of the model . we then turn towards the nonlinear model with the aims of ( i ) validating the linear analysis for small perturbations , ( ii ) investigating how the nonlinearity of the model affects the modal decomposition of the solution and ( iii ) demonstrating the capabilities of the modal decomposition to analyze the complex features of the nonlinear model . in future work , the modal decomposition will be used to calibrate the model coefficients against experimental data . we first develop the scheme in sec . [ secrelax ] and then compare the results for the linear and nonlinear models in sec . [ secnonlinearnum ] . finally we draw conclusions and perspecives in sec . [ sec : conclu ] . consider the soh model ( [ model08 - 1])-([model08 - 3 ] ) in a two - dimensional annular domain @xmath14 . we introduce polar coordinates @xmath15 $ ] where @xmath16 and @xmath17 is the angle between @xmath18 and a reference direction . we denote by @xmath19 the local basis associated to polar coordinates , i.e. @xmath20 and @xmath21 where the exponent @xmath22 indicates a rotation by an angle @xmath23 . then , we let @xmath24 and @xmath25 , where @xmath26 represents the angle between @xmath27 and @xmath8 . we recall that the constants @xmath11 , @xmath12 and @xmath9 are such that @xmath28 . for notational convenience , we introduce @xmath29 and we note that @xmath30 is of the same sign as @xmath12 and that @xmath31 . after easy algebra , the soh model ( [ model08 - 1])-([model08 - 3 ] ) is equivalent to the following system for @xmath32 and @xmath33 with @xmath34 $ ] and @xmath35 , @xmath36 = 0,\label{sys1}\\ & \rho\left[\partial_t\phi + c_2\left(\cos\phi\frac{\partial\phi}{\partial r } + \frac{\sin\phi}{r}\frac{\partial\phi}{\partial\theta}+\frac{\sin\phi}{r}\right)\right ] + \theta \left(\frac{\cos\phi}{r}\frac{\partial\rho}{\partial\theta } -\sin\phi\frac{\partial\rho}{\partial r}\right ) = 0,\label{sys2}\end{aligned}\ ] ] subject to the boundary conditions @xmath37 the first boundary condition ( [ sysbdry ] ) imposes a tangential flow to the boundary @xmath38 and consequently ensures that there is no mass flow across this boundary . now , we look for perfectly polarized steady states of the above system , i.e. steady states of the form @xmath39 where @xmath40 is independent of @xmath17 and @xmath41 in the whole domain ( we have arbitrarily chosen a rotation in the clockwise direction but of course , the results would be the same , mutatis mutandis , with the opposite choice ) . we have the the perfectly polarized steady - states form a one - parameter family of solutions given by @xmath42 where @xmath30 is given by ( [ eq : def_alpha ] ) and @xmath43 is any positive constant . inserting @xmath44 into ( [ sys2 ] ) gives @xmath45 . therefore , there exists @xmath46 such that @xmath47 . next we study the linearization of ( [ sys1 ] ) , ( [ sys2 ] ) about a perfectly polarized steady - state @xmath39 . given @xmath48 , a linear perturbation @xmath49 is given by @xmath50 expanding system ( [ sys1 ] ) , ( [ sys2 ] ) about @xmath39 and dropping terms of order @xmath51 or higher , we deduce that the system satisfied by @xmath49 is given by : @xmath52 with @xmath53 supplemented with the boundary conditions : @xmath54 and @xmath55 , @xmath56 periodic in @xmath17 . these bounday conditions are inherited from ( [ sysbdry ] ) . the second eq . in ( [ eq : bc1 ] ) is a normalization condition whose physical significance is that we are perturbing the steady - state keeping the total particle mass in the system fixed . looking for solutions @xmath49 in separation of variables form : @xmath57 we deduce that @xmath58 must satisfy the following spectral problem : @xmath59 supplemented with the boundary conditions ( [ eq : bc1 ] ) , where @xmath60 is the identity matrix . we now consider the decomposition of @xmath61 into fourier series , i.e. @xmath62where @xmath63 then , @xmath64 satisfies the following spectral problem : @xmath65 with @xmath66 supplemented with the boundary conditions : @xmath67 we first study the existence of non - trivial solutions @xmath68 to this spectral problem . we first prove the following [ thm_imaginary ] all the eigenvalues @xmath70 of ( [ eq : spect ] ) are pure imaginary . we recall that @xmath71 . we introduce the transformation : @xmath72 and find @xmath73 subject to the boundary conditions : @xmath74 let @xmath75 where @xmath76 denotes the real part of @xmath70 and @xmath77 its imaginary part . assume that @xmath78 . we divide ( [ rnsysphi ] ) by @xmath79 and use the first equation ( [ rnsysrho ] ) to get ( remembering ( [ eq : def_alpha ] ) ) : @xmath80 -\frac{1}{r^{\alpha+1}}\left[\frac{\mu}{\theta}+i\left(\frac{\nu}{\theta}-\frac{n \alpha}{r}\right)\right ] \phi_n = 0.\ ] ] multiplying ( [ phin2nd ] ) by @xmath81 ( the complex conjugate of @xmath82 ) , integrating with respect to @xmath83 , using the boundary conditions ( [ eq : fourier_bc ] ) and taking the real part of the so - obtained expression , we get @xmath84 since @xmath85 and @xmath86 , we have @xmath87 , which shows that there can not exist a non - trivial solution of the spectral problem when @xmath86 . we now determine the eigenvalues @xmath88 , @xmath89 of ( [ eq : spect ] ) . dropping the index @xmath90 for simplicity , we introduce the following transformation : @xmath91 from ( [ rnsysrho ] ) , ( [ rnsysphi ] ) , ( [ eq : fourier_bc ] ) , @xmath92 satisfies the spectral problem : @xmath93 with the operator @xmath94 acting on @xmath95 defined by @xmath96 with domain @xmath97 note that @xmath98 and that @xmath99 has real - valued coefficients . without loss of generality , we look for real - valued eigenfunctions @xmath92 . for this operator , we have the [ thm_basis ] the spectrum of @xmath94 consists of a countable set of eigenvalues @xmath100 , @xmath101 associated to a complete orthonormal system of @xmath102 of eigenfunctions @xmath103 . furthermore , @xmath104 as @xmath105 . we first assume that @xmath106 and drop the subindex @xmath90 of @xmath94 for simplicity . we will show that there exists @xmath107 such that the resolvant @xmath108 exists and is compact in @xmath102 . for this purpose , we consider @xmath109 and look for a solution @xmath110 of @xmath111 , i.e. , @xmath112 we take @xmath113 in such a way that @xmath114 , @xmath115 $ ] . multiplying ( [ uva ] ) by @xmath116 , taking its derivative with respect to @xmath83 and using ( [ uvb ] ) , we deduce that @xmath117 satisfies : @xmath118 -\frac{\alpha n^2}{r^{\frac{\alpha+3}{2 } } } \ , \big(1+\frac{\eta r}{c_2 n } \big ) \ , v_{\eta } = \tilde{h}_\eta,\ ] ] with the boundary conditions @xmath119 , where @xmath120 note that @xmath121 . now the problem consists of showing the existence of a unique weak solution @xmath122 to ( [ vtildeh ] ) , i.e. to the variational formulation : @xmath123 with @xmath124 and where the brackets at the right - hand side of ( [ varv ] ) denote duality between the distribution @xmath125 and the function @xmath126 . we introduce @xmath127 . choosing @xmath128 large enough and @xmath129 there exists @xmath130 such that @xmath131.\ ] ] therefore , the bilinear form @xmath132 where @xmath133 is the usual inner product in @xmath134 is coercive on @xmath135 . consequently , by the lax - milgram theorem , the variational formulation @xmath136 has a unique solution for any @xmath137 and the dependence of @xmath138 upon @xmath139 is continuous . this defines a continuous linear mapping @xmath140 , @xmath141 . then @xmath142 is a solution of ( [ varv ] ) if and only if @xmath143 , i.e. @xmath144 we note that @xmath145 and that @xmath146 , restricted to @xmath134 , is a bounded operator from @xmath147 to @xmath148 . after composition with the canonical imbedding of @xmath148 into @xmath147 ( still denoted by @xmath146 ) , @xmath146 is a compact operator on @xmath147 . therefore @xmath149 is a fredholm operator . in addition , @xmath150 is self - adjoint because the bilinear form @xmath151 is symmetric . thanks to the fredholm alternative , we have @xmath152 where i m denotes the range and ker denotes the null space of an operator . suppose that there exists @xmath153 such that @xmath129 and ker@xmath154 . then , @xmath155 is invertible and there exists a unique solution @xmath156 to ( [ fixed ] ) , or equivalently , to ( [ vtildeh ] ) . defining @xmath157 by ( [ uva ] ) ( remember that we suppose @xmath106 ) , then @xmath158 since @xmath159 and @xmath160 . but , since @xmath117 satisfies ( [ vtildeh ] ) in the distributional sense , @xmath157 satisfies ( [ uvb ] ) in the distributional sense . from the facts that @xmath117 and @xmath161 both belong to @xmath147 , we get that @xmath162 . therefore , @xmath163 and by ( [ uva ] ) , ( [ uvb ] ) , it satisfies @xmath164 . this shows that @xmath165 is invertible . furthermore , since @xmath166 is compactly imbedded into @xmath167 and that @xmath168 is a continuous linear map from @xmath167 into @xmath166 , the map @xmath169 is compact as an operator of @xmath167 . to prove that there exists @xmath153 such that @xmath129 and ker@xmath154 , we proceed by contradiction . we suppose that for all such @xmath153 there exists a non - trivial @xmath170 such that @xmath171 . equivalently , eq . ( [ varv ] ) with right - hand side @xmath172 has a non - trivial solution @xmath173 , which means that @xmath117 is an eigenvector for the eigenvalue @xmath174 of the variational spectral problem : to find @xmath175 and @xmath176 , @xmath177 , such that @xmath178 from the classical spectral theory of elliptic operators@xcite , we know that the eigenvalues of this problem are isolated . furthermore , @xmath174 is a simple eigenvalue . indeed , eq . ( [ vtildeh ] ) is a linear second order differential equation . for a given @xmath153 , consider two solutions @xmath179 , @xmath180 in @xmath181 of ( [ vtildeh ] ) associated to @xmath182 . the wronskian @xmath183 is zero because both @xmath179 and @xmath180 vanish at the boundaries . therefore , @xmath179 and @xmath180 are linearly dependent and consequently the dimension of the associated eigenvectors is @xmath184 . we realize that the coefficients of @xmath185 given by ( [ varveta ] ) are analytic functions of @xmath186 $ ] . then , from classical spectral theory again@xcite , one can define an analytic branch of non - zero solutions @xmath186 \to v_\eta \in h_1 ^ 0(r_1,r_2)$ ] . now , from ( [ varv ] ) with right - hand side @xmath172 , it follows that for such @xmath117 , we have @xmath187 . taking the derivative of this identity with respect to @xmath153 at @xmath188 , and using the fact that @xmath185 is a symmetric bilinear form , we get : @xmath189 now , since @xmath190 is a variational solution of ( [ vtildeh ] ) for @xmath188 with zero right - hand side , the second term is identically zero . computing the first term , we get @xmath191 the quantity inside the parentheses is a nonegative quantity which can only be @xmath174 if @xmath190 is identically zero , which contridicts the hypothesis that @xmath190 is a non - trivial solution . this shows the contradiction and proves that there exists @xmath192 small enough such that @xmath193 . in the case @xmath194 , it is an easy matter to see that the above proof can be reproduced or alternately , one can invoke directly the spectral theory of elliptic operators . details are left to the reader . now , for all @xmath195 , there exists @xmath196 such that @xmath197 exists , and is a compact self - adjoint operator of @xmath167 . by the spectral theorem for compact self - adjoint operators , there exists a hilbert basis @xmath198 of @xmath167 and a sequence @xmath199 of real numbers such that @xmath200 as @xmath105 and such that @xmath201 is an eigenfunction of @xmath202 associated to the eigenvalue @xmath203 . then , @xmath198 is a hilbert basis in @xmath167 of eigenfunctions of @xmath99 associated to the sequence of eigenvalues @xmath204 with @xmath205 . we have @xmath104 as @xmath105 , which concludes the proof . we now come back to the original spectral problem ( [ eq : spectl ] ) . we define @xmath206 where @xmath198 is the hilbert basis of eigenfunctions of @xmath99 found at theorem [ thm_basis ] , and @xmath207 is the associated sequence of eigenvalues . thanks to the change of functions ( [ eq : rhonphin ] ) , ( [ eq : defuv ] ) , @xmath208 is a hilbert basis of eigenfunctions of @xmath69 and @xmath209 is the associated sequence of eigenvalues . the system @xmath208 is orthonormal for the inner product @xmath210 furthermore , we have the following easy lemma ( whose proof is left to the reader ) : let @xmath211 be an eigenvalue of @xmath69 associated to the eigenvector @xmath212 , then @xmath213 is an eigenvalue of @xmath214 associated to the eigenvector @xmath215 . [ lem : parity ] as a consequence of this lemma the eigenvalues for @xmath194 come in opposite pairs and we number them such that @xmath216 . therefore , the sequence of eigenvalues of @xmath217 is @xmath218 . we note that @xmath174 is not an eigenvalue of @xmath217 . we now turn to the operator @xmath219 defined on @xmath220 by ( [ drdth ] ) with domain @xmath221 using theorem [ thm_basis ] and lemma [ lem : parity ] , we can state the following theorem ( the proof of which is immediate and left to the reader ) : the spectrum of @xmath219 , @xmath222 is discrete , and consists of @xmath223 associated to the following basis of eigenvectors @xmath224 which is a hilbert basis in @xmath220 for the inner product @xmath225 [ thm : specl ] from this theorem , we have the immediate [ thm_pert_sum ] let @xmath226 be the solution of the linearized model ( [ epsilonsys ] ) with initial condition @xmath227 . by standard semigroup theory , this solution belongs to @xmath228 , l^2((r_1,r_2)\times ( 0,2\pi))^2 \cap l^2([0,t ] , d({\mathcal l}))$ ] , for all time horizon @xmath229 . additionally , we assume that @xmath230 is real - valued . then , @xmath226 can be expressed as : @xmath231 where the series converges in @xmath220 and where @xmath232 and @xmath233 are given , for all @xmath234 with ( @xmath235 and @xmath236 ) or ( @xmath194 and @xmath237 ) , by : @xmath238 the mode indices @xmath90 and @xmath239 are related to the number of oscillations in the azimuthal and radial directions respectively . below , we will refer to @xmath90 as the azimuthal mode index and @xmath239 the radial mode index . first , we discuss the special case @xmath194 . from ( [ eq : spect ] ) , ( [ eq : spect2 ] ) ( with @xmath88 ) , the function @xmath240 is a solution to : @xmath241 with homogeneous boundary conditions @xmath242 . this a classical bessel equation . its solution is found e.g. in @xcite , p. 117 and is given by : @xmath243 , & \text { for integer } \tilde{n};\\ r^{\frac{\alpha+2}{2}}\left[aj_{\tilde{n}}(\beta r ) + bj_{-\tilde{n}}(\beta r)\right ] , & \text { for noninteger } \tilde{n}. \end{array } \right.\ ] ] here , @xmath244 and @xmath245 are the bessel functions of the first and second kinds respectively , @xmath246 and @xmath247 are determined from the boundary conditions . for the sake of simplicity , we focus on the case where @xmath248 is not an integer , but the extension of the considerations below to integer @xmath248 would be straightforward . the boundary conditions lead to a homogeneous linear system of two equations for @xmath247 . the existence of a non - trivial solution @xmath249 requires that the determinant of this system vanishes . this leads to the following relation : @xmath250 by finding the zeros of @xmath251 , we obtain the eigenvalues @xmath77 and then the corresponding eigenfunctions @xmath249 . for @xmath235 , we introduce the following numerical scheme . given an integer @xmath252 , we define a uniform meshsize @xmath253 on the interval @xmath254 $ ] and discretization points @xmath255 , where @xmath256 and @xmath257 . for each @xmath90 , @xmath258 and @xmath259 denote the numerical approximation of @xmath260 and @xmath261 on grid points @xmath262 s and @xmath263 s respectively . the numerical scheme is [ rnsysnum ] _ j+12 - = _ j+12 , & @xmath264 , + + _ j = _ j , & @xmath265 . no boundary condition is imposed on @xmath266 . concerning @xmath267 , we have @xmath268 . [ remfdn0 ] we can modify the scheme ( [ rnsysnum ] ) and use it to compute the solution in the case @xmath194 by adding @xmath269 and @xmath267 on both sides of the equations respectively . in the numerical tests , we choose a set of parameter values given by @xmath270 accuracy tests ( not reported here ) have demonstrated that the numerical scheme is of order @xmath271 for any value of @xmath90 . in the case @xmath194 , we can illustrate the good accuracy of the scheme by comparing the computed value of the eigenvalue to its analytic expression ( [ bdryn0 ] ) . the comparision is given in table [ n0nu ] . with @xmath272 mesh points , the scheme mentioned in remark [ remfdn0 ] gives almost the exact eigenvalues . .the eigenvalues @xmath77 for azimuthal mode @xmath194 and various values of the radial mode index @xmath239 . comparison between the method using the bessel functions ( formula ( [ bdryn0 ] ) ) and the scheme mentioned in remark [ remfdn0 ] with @xmath272 mesh points . the parameter values are given by ( [ eq : parval ] ) . [ cols="^,^,^,^,^,^,^",options="header " , ] [ nnu ] for illustration purposes , we plot some of the eigenmodes @xmath273 for a better interpretation of the results , we plot the perturbation density @xmath274 and the orientation vector @xmath275 while the former corresponds to the perturbation only , the latter corresponds to the total solution ( steady - state plus perturbation ) . we use a fairly large value of @xmath276 in order to magnify the influence of the perturbation . since the chosen annular domain is rather thin , we rescale the plot onto an artificially wider annulus . again , the chosen set of parameters is given by ( [ eq : parval ] ) . [ n0t1 ] displays the modes @xmath277 ( figs . [ n0t1 ] ( a , b ) ) and @xmath278 ( figs . [ n0t1 ] ( c , d ) ) . the left figures ( figs . [ n0t1 ] ( a , c ) ) show the color - coded values of the density perturbations @xmath274 ( [ eq : rhoperturb ] ) as functions of the two - dimensional coordinates @xmath279 in the annulus . the right figures ( figs . ( b , d ) ) provide a representation of the orientation vector field @xmath280 ( [ eq : omperturb ] ) . in the case of mode @xmath277 ( figs . [ n0t1 ] ( a , b ) ) , since @xmath194 , the solution does not vary in the @xmath17 direction and the density perturbation @xmath274 has two zeros in the @xmath83 direction . in the case of mode @xmath278 [ n0t1 ] ( c , d ) ) , the solution displays four periods in the @xmath17 direction and has only one zero of the density perturbation @xmath274 in the @xmath83 direction . + @xmath281 + we numerically investigate the influence of the parameters @xmath282 and @xmath9 on the eigenvalues . we take the parameter values ( [ eq : parval ] ) as references . we vary one of the five parameters @xmath283 at a time , fixing the other values to those of ( [ eq : parval ] ) . [ n2para ] ( a ) shows the eigenvalues @xmath77 as functions of the parameters @xmath284 . the inserted - inside fig . [ n2para ] ( a ) display how the eigenvalues depend on @xmath11 . [ n2para ] ( b ) shows how the eigenvalues depend on @xmath12 . four eigenvalues corresponding to the modes @xmath285 are displayed we observe that when the annular domain becomes narrower , i.e. @xmath284 is larger and closer to @xmath286 , the absolute value of @xmath77 is getting larger . the influence of @xmath286 ( not displayed ) is similar . as a result , the phase velocities of the modes become faster in a thinner domain , except for @xmath287 , which corresponds to no oscillation in the radial direction . as a function of @xmath11 and @xmath9 , @xmath288 is monotonically increasing for all values of @xmath90 ( see insert inside fig . [ n2para ] ( a ) for @xmath11 . the behavior as a function of @xmath9 is similar and not displayed ) . the effect of a variation of @xmath12 is different : @xmath77 itself ( instead of @xmath288 ) is increasing with respect to @xmath12 . in this section , we discuss the numerical resolution of the nonlinear soh model ( [ model08 - 1])-([model08 - 3 ] ) , subject to the boundary conditions ( [ sysbdry ] ) . its numerical solution will be compared with the solution of the linearized problem found in sec . [ secssnum ] . we will further analyze how the nonlinear model departs from its linearization when the perturbation of the steady - state becomes large . one of the difficulties in solving the nonlinear model is the geometric constraint @xmath289 ( [ model08 - 3 ] ) and the resulting non - conservativity of the model , arising from the presence of the projection operator @xmath290 in ( [ model08 - 2 ] ) . we rely on a method proposed in @xcite where the soh model is approximated by a relaxation problem consisting of an unconstrained conservative hyperbolic system supplemented with a relaxation operator onto vector fields satisfying the constraint ( [ model08 - 3 ] ) . in this section , we introduce this relaxation system in cylindrical coordinates in the annular domain . the relaxation model is given by : [ relaxmodel ] where @xmath291 and @xmath292 is not constrained to be of unit norm . the relaxation term at the right - hand side of ( [ relaxmodel-2 ] ) contributes to making @xmath293 . in cylindrical coordinates , let @xmath294 , @xmath295 . dropping the superindex @xmath153 for simplicity , ( [ relaxmodel ] ) can be written as @xmath296 of course , we request that @xmath297 are @xmath298-periodic with respect to @xmath17 . we supplement the relaxation system with similar boundary conditions as ( [ sysbdry ] ) . first , we request that the mass flux vanishes on @xmath38 , implying that @xmath299 , \quad \forall t \in { \mathbb r}_+.\ ] ] when @xmath291 , the relaxation term forces @xmath300 . therefore , we assume the same boundary condition ( [ sysbdry ] ) as for the soh model , supplemented with the condition that @xmath301 , namely @xmath302 , \quad \forall t \in { \mathbb r}_+ . \label{eq : relbdryphi}\ ] ] we have the following theorem , whose proof is analogous to that of proposition 3.1 in @xcite and is omitted . the relaxation model ( [ relaxcylin1])-([relaxcylin3 ] ) with boundary conditions ( [ eq : relbdryphi ] ) converges to the original model ( [ sys1 ] ) , ( [ sys2 ] ) with boundary conditions ( [ sysbdry ] ) as @xmath153 goes to @xmath174 . the scheme developed in @xcite relies on writing the hyperbolic part of the relaxation system in conservative form . indeed , the use of a non - conservative form may lead to unphysical solutions , which are not valid approximations of the underlying particle system@xcite . introducing @xmath303 defined by @xmath304 eqs . ( [ relaxcylin1 ] ) , ( [ relaxcylin2 ] ) can be rewritten in terms of the vector function @xmath305 as follows : @xmath306 where @xmath307 and @xmath308 of course , we request that @xmath309 is @xmath298 periodic in @xmath17 . the boundary conditions ( [ eq : relbdryphi ] ) translate into : @xmath310 we apply the method proposed in @xcite , which consists in splitting ( [ relaxvec ] ) into a conservative step and a relaxation step . in the conservative step , we solve ( [ relaxvec ] ) with @xmath311 . in the relaxation step , we solve ( [ relaxvec ] ) with @xmath312 . when @xmath291 this last step can be replaced by a mere normalization of @xmath8 i.e. changing @xmath92 into @xmath313 . the conservative step is solved by classical shock - capturing schemes ( see @xcite for details ) . we take uniform meshes for @xmath83 and @xmath17 . careful accuracy tests ( not reported here ) have demonstrated that this method is of order @xmath184 . we take a pure eigenmode as initial condition and compare the numerical solution of the nonlinear model to that of the linearized model . we take an initial condition given by @xmath314 with @xmath315 given by ( [ eq : rhoperturb ] ) . let @xmath316 denote the exact solution of the nonlinear model ( [ sys1 ] ) , ( [ sys2 ] ) with boundary conditions ( [ sysbdry ] ) , @xmath317 the solution of the linearized system given by theorem [ thm_pert_sum ] , and @xmath318 the numerical solution of the nonlinear model computed thanks to the method summarized at sec . [ subsec_nummet ] . consider @xmath319 for example . formally , we have @xmath320 ( we neglect the errors due to the numerical computation of the functions @xmath321 which are small ) , while @xmath322 ( since the scheme is of order @xmath184 ) . consequently , we have @xmath323 fig . [ l2mode_multi_n3 ] ( a ) shows the @xmath324-distance ( below referred to as the `` error '' ) between the numerical solution of the nonlinear model and that of the linearized system at time @xmath325 , as a function of the meshsize @xmath326 for an initial condition ( [ eq : inicond ] ) corresponding to mode @xmath327 and @xmath328 . different perturbation magnitudes @xmath329 ( red squares ) , @xmath330 ( green triangles ) , @xmath331 ( blue crosses ) are used . the parameter values are those of ( [ eq : parval ] ) . we notice that for a given value of @xmath276 , the error decreases with decreasing values of @xmath326 until @xmath326 reaches the approximate values @xmath332 ( for @xmath333 and @xmath330 ) and @xmath334 ( for @xmath331 ) . when @xmath326 is decreased further , the error stays constant but this constant is smaller for smaller @xmath276 . this suggests that , consistently with ( [ eq : error_linear ] ) , the error is dominated by the linearization error for small values of @xmath326 . this interpretation is also consistent with the observation that the threshold value of @xmath326 under which the error saturates decreases when @xmath276 becomes smaller . however , the decay of the error seems to be first order in @xmath276 instead of being second order as inferred from ( [ eq : error_linear ] ) . this suggests that nonlinear effects are rapidly moving the solution away from the linear regime . however , other diagnostics discussed in the section below show that the linearized model actually provides a very good approximation of the nonlinear model in practical situations . in this section , we take larger values of @xmath276 and quantify the difference between the solutions of the nonlinear and linearized models . due to nonlinear mode coupling , it is expected that , even with a pure mode initial condition , new modes will be gradually turned on by the nonlinearity . let @xmath335 denote the difference between the numerical solution of the nonlinear model and the steady - state , rescaled by the factor @xmath336 . we define the energy @xmath337 of the perturbation as @xmath338 where the double bracket refers to the inner product ( [ eq : innerprod2 ] ) and @xmath339 is given by ( [ eq : defknm ] ) with @xmath340 replaced by @xmath341 . the quantity @xmath342 ( respectively @xmath343 ) represents the energy stored in the modes @xmath344 ( respectively in the modes @xmath345 and @xmath346 ) at time @xmath347 . in the purely linear case , @xmath339 is independent of @xmath347 . in the nonlinear case , its variation with @xmath347 provides a measure of how the nonlinearity affects the amplitude of the corresponding modes . the initial data is a perturbation of the steady - state by a pure eigenmode , i.e. @xmath348 with @xmath349 , @xmath350 . we test different values of @xmath276 . for this initial condition , [ l2mode_multi_n3 ] ( b , c ) show @xmath339 as a function of @xmath347 in log - log scale for @xmath351 and @xmath352 respectively . the initial mode @xmath353 is represented with blue x s . in fig . [ l2mode_multi_n3 ] ( b ) corresponding to a moderate perturbation @xmath351 , only modes @xmath277 ( red squares ) and @xmath354 ( purple triangles ) appear . mode @xmath355 appears first but saturates while mode @xmath356 appears later but reaches higher intensities . both modes eventually saturate . likewise , the initial mode decays as higher order modes ( not represented in the figure ) are turned on by the nonlinearity . the initial growth of modes @xmath355 and @xmath356 is linear in log - log scale , which corresponds to a power law growth in time . the two modes have comparable growth rates ( the two increasing parts of the curves are parallel straight lines ) . in the case of a larger perturbation @xmath357 displayed in fig . [ l2mode_multi_n3 ] ( c ) the situation is strikingly more complex , with a wealth of other modes appearing . in addition to modes @xmath358 ( red squares ) and @xmath356 ( purple triangle ) , we notice mode @xmath359 ( cyan diamonds ) and @xmath360 ( green circles ) . mode @xmath359 which was absent from fig . [ l2mode_multi_n3 ] ( b ) now overtakes mode @xmath356 at the beginning , but the latter reaches a higher intensity after some time . the decay of the initial mode @xmath361 is also more pronounced . it should be noted that some modes stay extinct all the time . this shows that some pairs of modes are only weakly coupled by the nonlinearity . in order to illustrate the successive turn on of the various modes , we have arbitrarily fixed a threshold value @xmath362 ( represented by the horizontal dashed blue lines on figs . [ l2mode_multi_n3 ] ( b , c ) ) . in fig . [ l2mode_multi_n3 ] ( d ) , we have reported the first time @xmath363 at which @xmath339 reaches the values @xmath364 and plotted it as a function of @xmath276 in log - log scale , for modes @xmath365 ( blue x s ) , @xmath358 ( blue squares ) and @xmath366 ( red circles ) . the corresponding times @xmath363 are also indicated explicitly on figs . [ l2mode_multi_n3 ] ( b , c ) ) . [ l2mode_multi_n3 ] ( d ) shows that for small @xmath276 , mode @xmath356 is the earliest one to turn on . but as @xmath276 increases , this feature changes and mode @xmath355 ( which was extinct for smaller value of @xmath276 ) appears earlier . when @xmath276 is increased further , mode @xmath359 also appears , later than @xmath355 but earlier than @xmath356 . this illustrates that the nonlinear mode coupling can exhibit rather complex features and non - monotonic behavior as a function of the perturbation intensity @xmath276 . however , even for these large perturbation cases , the amplitude of the initial mode always remains one order of magnitude larger than those of the successively excited modes . this shows that the linear model still provides a fairly good approximation of the solution of the nonlinear model . + we now investigate the qualitative features of the solution in a large amplitude case . figs . [ n4nu1t0_2 ] shows the numerical solution corresponding to a pure mode initial data ( [ eq : modeinitial ] ) with @xmath367 , @xmath368 and @xmath369 . it displays the density @xmath319 at times @xmath370 ( left ) and @xmath371 ( right ) as a function of the two - dimensional position coordinates @xmath279 in the annulus , in color code ( color bar to the right of the figure ) . we observe that the solution remains @xmath372-periodic in the @xmath17-direction ( as the linear mode would be ) but the density contours have lost their sinusoidal shape . instead , oblique shock waves have formed and are reflected by the boundary . these simulations suggest the existence of unsmooth periodic solutions of the nonlinear soh model in this geometric configuration . + we now investigate a large perturbation amplitude case with a random intitial data . more precisely , the initial data is given by a random combination of eigenmodes such that @xmath373 and @xmath374 as follows : @xmath375 where @xmath232 and @xmath233 are randomly sampled in the intervals @xmath376 $ ] and @xmath377 $ ] respectively , according to the uniform distribution . the numerical simulation is performed with @xmath378 , @xmath379 and @xmath380 . [ fig : random ] shows the numerical solution at time @xmath371 ( which approximately corresponds to the rotation of the fluid by a quater of a circle ) . it displays the density @xmath319 as a function of the two - dimensional position coordinates @xmath279 in the annulus , in color code ( color bar to the right of the figure ) . [ fig : random ] ( a ) shows the solution of the linearized model , obtained by summation of the corresponding eigenmodes , while fig . [ fig : random ] ( b ) displays the numerical solution of the nonlinear model with the same initial condition . we observe a very good agreement between the linearized and nonlinear solutions , in spite of a fairly large perturbation amplitude . by looking carefully , one notices that the nonlinear solution has slightly lower maxima and larger minima , due to the action of numerical diffusion ( which is absent from the linearized solution ) . the nonlinear solution also exhibits steeper gradients due to nonlinear shock formation . the use of the linearized solution results in considerable computational speed - up compared to that of the nonlinear one . indeed , the computation of the eigenmodes and their summation to construct the solution is almost instantaneous on a standard laptop . by comparison , the computation of the nonlinear solutions takes of the order of an hour . therefore , given the considerable computation speed - up , we consider that the performances of the linearized model are excellent . these performances make the linearized model a model of choice to perform parameter calibration on experimental data . indeed , parameter calibration involves the iterative resolution of a minimization problem which consists of finding the set of parameters which minimize the distance between the solution and the data . with the linearized model , this calibration phase can be expected to require very little computational time . this is important , since this set of parameters is expected to change from one experiment to the next and consequently , the calibration phase must be performed for each experiment . a real - time analysis of an experiment therefore requires a very efficient algorithm . in this paper , we have studied the soh model on an annular domain . we have linearized the system about perfectly polarized steady - states . and shown that the resulting system has are only pure imaginary eigenvalues and that they form a countable set associated to an ortho - normal basis of eigenvectors . a numerical scheme for the fully nonlinear system has been proposed . its results are consistent with the modal analysis for small perturbations of polarized steady - states . for large perturbations , nonlinear mode - coupling has been shown to result in the progressive turn - on of new modes in a complex fashion . finally , we have assessed the efficiency of the modal decomposition to analyze the complex patterns of the solution . in future work , we will gradually include more physical effects in the model such as adding a repulsive force between the particles to prevent the formation of large concentrations , or immersing the particles in a surrounding fluid to give a better account of the dynamics of active particle suspensions like sperm . finally , we plan to use the modal analysis to accurately calibrate the model against experimental observations of collective motion . chuang , m. r. dorsogna , d. marthaler , a. l. bertozzi and l. s. chayes , state transitions and the continuum limit for a 2d interacting , self - propelled particle system , _ physica d _ * 232 * ( 2007 ) 3347 . p. degond and j - g . liu , hydrodynamics of self - alignment interactions with precession and derivation of the landau - lifschitz - gilbert equation , _ math . models methods appl . sci . _ * 22 suppl . 1 * ( 2012 ) 1140001 . d. johansson , f. laursen , a. fern , j. e. fosseidengen , p. klebert , l. h. stien , t. vgseth and f. oppedal , the interaction between water currents and salmon swimming behaviour in sea cages , _ plos one _ * 9 * ( 2014 ) e97635 . m. moussad , e. g. guillot , m. moreau , j. fehrenbach , o. chabiron , s. lemercier , j. pettr , c. appert - rolland , p. degond and g. theraulaz , traffic instabilities in self - organized pedestrian crowds , _ plos computational biology _ * 8 * ( 2012 ) e1002442 . v. i. ratushnaya , d. bedeaux , v. l. kulinskii and a. v. zvelindovsky , collective behavior of self - propelling particles with kinematic constraints : the relation between the discrete and the continuous description , _ physica a _ * 381 * ( 2007 ) 3946 . k. tunstrom , y. katz , c. c. ioannou , c. huepe , m. j. lutz and i. d. couzin , collective states , multistability and transitional behavior in schooling fish , _ plos computational biology _ * 9 * ( 2013 ) e1002915 .

the self - organized hydrodynamics model of collective behavior is studied on an annular domain . a modal analysis of the linearized model around a perfectly polarized steady - state is conducted . it shows that the model has only pure imaginary modes in countable number and is hence stable . numerical computations of the low - order modes are provided . the fully non - linear model is numerically solved and nonlinear mode - coupling is then analyzed . finally , the efficiency of the modal decomposition to analyze the complex features of the nonlinear model is demonstrated . \1 . department of mathematics , imperial college london + london , sw7 2az , united kingdom + pdegond@imperial.ac.uk + 2 . universit de toulouse ; ups , insa , ut1 , utm + institut de mathmatiques de toulouse , france + and cnrs ; institut de mathmatiques de toulouse , umr 5219 , france + hyu@math.univ-toulouse.fr * acknowledgements : * this work was supported by the anr contract motimo ( anr-11-monu-009 - 01 ) . the first author is on leave from cnrs , institut de mathmatiques , toulouse , france . he acknowledges support from the royal society and the wolfson foundation through a royal society wolfson research merit award and by nsf grant rnms11 - 07444 ( ki - net ) . the second authors wishes to acknowledge the hospitality of the department of mathematics , imperial college london , where this research was conducted . both authors wish to thank f. plourabou ( imft , toulouse , france ) for enlighting discussions . * key words : * collective dynamics ; self - organization ; emergence ; fluid model ; hydrodynamic limit ; symmetry - breaking ; alignment interaction ; polarized motion ; spectral analysis ; relaxation model ; splitting scheme ; conservative form ; nonlinear mode - coupling . * ams subject classification : * 35l60 , 35l65 , 35p10 , 35q80 , 82c22 , 82c70 , 82c80 , 92d50 . 0.4 cm

few quantities in physics play such a singular role in their theories as the entropy in the boltzmann - gibbs ( bg ) formulation of statistical mechanics : it provides , in an elegant and simple way , the fundamental link between the microscopic structure of a system and its macroscopic behavior . almost two decades ago , tsallis @xcite proposed the following entropy expression as a nonextensive generalization of the boltzmann - gibbs ( bg ) formalism for statistical mechanics , @xmath0^q } { q-1},\ ] ] where @xmath1 is a constant , @xmath2 a parameter and @xmath3 a probability distribution over the phase space variables @xmath4 . instead of the usual bg exponential , upon maximization with a fixed average energy constraint , the above entropy gives a power - law distribution , @xmath5^{\frac{1}{q-1}},\ ] ] where @xmath6 is a temperature - like parameter and @xmath7 is the energy of the system . in the limit @xmath8 , the above expressions reduce to the familiar bg forms of entropy , @xmath9 , and exponential distribution @xmath10 . the consequences of this generalization are manifold and far - reaching ( for a recent review see e.g. @xcite ) , but its most notorious one is certainly the fact that @xmath11 , along with other thermodynamical quantities , are _ nonextensive _ for @xmath12 . whether such formalism can be verified experimentally or elucidated from any previously established theoretical framework are obvious questions that arise naturally . the answer to the first one seems to be affirmative , as there are currently numerous references to systems that are better described within the generalized approach of tsallis than with the traditional bg formalism @xcite , the underlying argument being usually based on a best fit to either numerical or experimental data by choosing an appropriate value for the parameter @xmath2 . this is , by far , the most frequent approach towards tsallis distribution and is surely not conclusive . its justification from first principles , although already addressed at different levels in previous studies @xcite , is nevertheless controversial and still represents an open research issue . for example , it has been recently suggested that systems in contact with finite heat baths should follow the thermostatistics of tsallis @xcite . this derivation , however , relies on a particular _ ansatz _ for the density of states of the bath and lacks a more direct connection with the entropy , as pointed out in ref . also , it has been shown in ref . @xcite that the tsallis statistics , together with the biased averaging scheme , can be mapped into the conventional boltzmann - gibbs statistics by a redefinition of variables that results from the scaling properties of the tsallis entropy . in a recent study @xcite , a derivation of the generalized canonical distribution is presented from first principles statistical mechanics . it is shown that the particular features of a macroscopic subunit of the canonical system , namely , the heat bath , determines the nonextensive signature of its thermostatistics . more precisely , it is exactly demonstrated that if one specifies the heat bath to satisfy the relation @xmath13 where @xmath14 is a temperature - like parameter and @xmath15 is the thermodynamic temperature , the form of the distribution eq . ( [ phasedist ] ) is recovered @xcite . equation ( [ q - eq ] ) is essentially equivalent to eq . ( [ phasedist ] ) . however , it reveals a direct connection between the finite aspect of the many - particle system and the generalized @xmath2-statistics @xcite . it is analogous to state that , if the condition of an infinite heat bath capacity is violated , the resulting canonical distribution can no longer be of the exponential type and therefore should not follow the traditional bg thermostatistics . inspired by these results , we propose here a theoretical approach for the thermostatistics of tsallis that is entirely based on standard methods of statistical mechanics . subsequently , we will not only recover the previous observation that an adequate physical setting for the tsallis formalism should be found in the physics of finite systems , but also derive a novel and exact correspondence between the hamiltonian structure of a system and its closed - form @xmath2-distribution , supporting our findings through a specific numerical experiment . we start by considering , in a shell of constant energy , a system whose hamiltonian can be written as a sum of two parts , viz . @xmath16 where @xmath17 , with @xmath18 , @xmath19 , @xmath20 and so on . the fact that tsallis distribution is a power - law instead of exponential strongly suggests us to look for scale - invariant forms of hamiltonians @xcite . furthermore , since scale - invariant hamiltonians constitute a particular case of homogeneous functions @xcite , our approach here is to show that , if @xmath21 satisfies a generalized homogeneity relation of the type @xmath22 where @xmath23 are non - null real constants , then the correct statistics for @xmath24 is the one proposed by tsallis . the foregoing derivation is based on a simple scaling argument , but we shall draw parallels to ref . @xcite whenever appropriate . the structure function ( density of states ) for @xmath21 at the energy level @xmath7 is given by @xmath25 where @xmath26 is the volume element in the subspace spanned by @xmath27 . for systems satisfying equation ( [ homog - rel ] ) , this function can be evaluated taking @xmath28 and computing @xmath29 where we define @xmath30 and utilize the notation @xmath31 , @xmath32 , etc . , and @xmath33 hence , if @xmath34 is defined at a value @xmath35 , it is also defined at every @xmath36 , with @xmath28 . we can then write @xmath37 and express the canonical distribution law over the phase space of @xmath38 as @xmath39 where @xmath40 is the total energy of the joint system composed by @xmath38 and @xmath21 , and @xmath41 is its structure function , @xmath42 where @xmath43 and @xmath44 are the infinitesimal volume elements of the phase spaces of @xmath38 and @xmath21 , respectively . comparing eq . ( [ phasedist - new ] ) with the distribution in the form of eq . ( [ phasedist ] ) , we get the following relation between @xmath2 , @xmath6 and @xmath40 : @xmath45 notice that one could reach exactly the same result using the methodology proposed in ref . @xcite , i.e. by evaluating @xmath46 at @xmath47 , calculating @xmath48 through eq . ( [ q - eq ] ) and inserting these quantities back in eq . ( [ phasedist ] ) . as already mentioned , previous studies have shown that the distribution of tsallis eq . ( [ phasedist ] ) is compatible with some anomalous `` canonical '' configurations where the heat bath is finite @xcite or composes a peculiar type of extended phase - space dynamics @xcite . in our approach , the observation of tsallis distribution simply reflects the finite size of a thermal environment with the property ( [ homog - rel ] ) , the thermodynamical limit corresponding to @xmath49 in eq . ( [ q - micro ] ) . we emphasize that , although similar conclusions could be drawn from refs . @xcite , the theoretical framework introduced here permits us to put forward a rigorous realization of the @xmath2-thermostatistics : it stems from the _ weak _ coupling of a system to a `` heat bath '' whose hamiltonian is a homogeneous function of its coordinates , _ the value of @xmath2 being completely determined by its degree of homogeneity , eq . ( [ q - micro])_. this provides also a direct correspondence between the parameter @xmath2 and the hamiltonian structure through geometrical elements of its phase space , viz . the surfaces of constant energy @xmath50 . as a specific application of the above results , we investigate the form of the momenta distribution law for a classical @xmath51-body problem in @xmath52-dimensions . the hamiltonian of such a system can be written as @xmath53 where we define @xmath54 , @xmath55 is the linear momentum vector of an arbitrary particle @xmath56 ( hence the number of degrees of freedom of the system @xmath57 is @xmath58 ) , and @xmath21 ( the `` bath '' ) is due to a homogeneous potential @xmath59 of degree @xmath60 , i.e. , @xmath61 with @xmath62 . at this point , we emphasize that the distinction between `` system '' and `` bath '' is merely formal and does not necessarily involve a physical boundary . it relies solely on the fact that we can decompose the total hamiltonian in two parts @xcite . by making the correspondences @xmath63 , @xmath64 , @xmath58 , @xmath65 , @xmath66 and @xmath67 , the homogeneity relation ( [ homog - rel ] ) is satisfied . from eq . ( [ phasedist - new ] ) it then follows that @xmath68^{\frac{1}{q-1}},\ ] ] where the nonextensivity measure @xmath2 is given by @xmath69 it is often argued that the range of the forces should play a fundamental role in deciding between the bg or tsallis formalisms to describe the thermostatistics of an @xmath51-body system @xcite . for example , the scaling properties of the one - dimensional ising model with long - range interaction has been investigated analytically @xcite and numerically @xcite in the context of tsallis thermostatistics , whereas in ref . @xcite a rigorous approach was adopted to study the nonextensivity of a more general class of long - range systems in the thermodynamic limit ( see below ) . recall that , for a @xmath52-dimensional system , an interaction is said to be long - ranged if @xmath70 . within this regime , the thermostatistics of tsallis is expected to apply , while for @xmath71 the system should follow the standard bg behavior @xcite . this conjecture is not confirmed by the results of the problem at hand . indeed , eq . ( [ phasedist - f ] ) is consistent with the generalized @xmath2-distribution eq . ( [ phasedist ] ) no matter what the value of @xmath60 is , as long as it is non - null and @xmath51 is finite . in the limit @xmath72 , however , we always get @xmath73 , with the value of @xmath60 determining the shape of the curve @xmath74 . if @xmath75 , @xmath2 approaches the value @xmath57 from above , while for @xmath76 the value of @xmath2 is always less than @xmath57 . therefore , for ( ergodic ) classical systems with @xmath51 particles interacting through a homogeneous potential , the _ equilibrium _ distribution of momenta always goes to the boltzmann distribution , @xmath77 $ ] , when @xmath72 . this observation should be confronted with the recent results of vollmayr - lee and luijten @xcite , who investigated the nonextensivity of long - range ( therein `` nonintegrable '' ) systems with algebraically decaying interactions through a rigorous kac - potential technique . contrary to the trend established by the practitioners of tsallis formalism , those authors argue that it is possible to obtain the nonextensive scaling relations of tsallis without resorting to an a priori @xmath2-statistics , the boltzmann - gibbs prescription ( @xmath78 ) being sufficient for describing long - range systems of the type above . even though our findings embody partially the same message ( we are not yet concerned about scaling relations ) , there are some caveats that prevent their results from being straightforwardly applicable to our problem : neither a system - size regulator for the energy nor a cutoff function is present in our treatment . this is immediately in contrast with their observation that the `` bulk '' thermodynamics strongly depends on the functional form of the regulator . moreover , by not addressing the distribution function explicitly at finite system sizes , that work has very little in common with the most interesting part of our study , which might in fact explain some observations of the @xmath2-distribution . notwithstanding these differences , we believe that an investigation of the scaling properties of the system studied here would elucidate from a different perspective the connection of tsallis thermostatistics with nonextensivity and is certainly a very welcome endeavor . it is important to stress here that the essential feature determining the canonical distribution is the geometry of the phase space region that is effectively visited by the system . in a previous work by latora @xcite , the dynamics of a classical system of @xmath51 spins with infinitely long - range interaction is investigated through numerical simulations , and the results indicate that if the thermodynamic limit ( @xmath79 ) is taken before the infinite - time limit ( @xmath80 ) , the system does not relax to the boltzmann - gibbs equilibrium . instead , it displays anomalous behavior characterized by stable non - gaussian velocity distributions and dynamical correlation in phase space . this might be due to the appearance of metastable state regions that have a fractal nature with low dimension . in our theoretical approach , however , it is assumed that _ the infinite - time limit is taken before the thermodynamic limit_. as a consequence , metastable or quasi - stationary states like the ones observed by latora _ @xcite with a particular long - range hamiltonian system can not be predicted within the framework of our methodology . whether this type of dynamical behavior can be generally and adequately described in term of the nonextensive thermostatistics of tsallis still represents an open question of great scientific interest . in order to corroborate our method , we investigate through numerical simulation the statistical properties of a linear chain of anharmonic oscillators . besides the kinetic term , the hamiltonian includes both on - site and nearest - neighbors quartic potentials , i.e. @xmath81 the choice of this system is inspired by the so - called fermi - pasta - ulam ( fpu ) problem , originally devised to test whether statistical mechanics is capable or not to describe dynamical systems with a small number of particles @xcite . from eq . ( [ hamilt - osc ] ) , we obtain the equations of motion and integrate them numerically together with the following set of initial conditions : @xmath82 where @xmath83 is a random number within @xmath84 . undoubtedly , a rigorous analysis concerning the ergodicity of this dynamical system would be advisable before adopting the fpu chain as a plausible case study . this represents a formidable task , even for such a simple problem @xcite . for our practical purposes , it suffices , however , to test if the system displays equipartition among its linear momentum degrees of freedom , since this is one of the main signatures of ergodic systems . indeed , one can show from the so - called birkhoff - khinchin ergodic theorem that , for ( almost ) all trajectories @xcite , @xmath85 where @xmath86 is the volume of the phase space with @xmath87 , @xmath88 denotes the the usual time average of an observable @xmath89 , and @xmath90 stands for the absolute temperature of the _ whole system _ , @xmath91 ( cf . we then follow the time evolution of the quantities @xmath92 to check if they approach unity as @xmath93 increases . from the results of our simulations with different values of @xmath51 and several sets of initial conditions , we observe in all cases the asymptotic behavior , @xmath94 as @xmath95 . this procedure also indicates a good estimate for the relaxation time of the system , @xmath96 , so we shall consider our statistical data only for @xmath97 , with a typical observation time in the range @xmath98 , after thermalization . as a function of the transformed variable @xmath99 for @xmath100 ( circles ) , @xmath101 ( triangles up ) , @xmath102 ( squares ) , and @xmath103 ( triangles down ) anharmonic oscillators . the solid straight lines are the best fit to the simulation data of the expected power - law behavior eq . ( [ phasedist - f ] ) . the slopes are 1.0068 ( 1.0 ) , 3.07 ( 3.0 ) , 7.21 ( 7.0 ) , and 15.27 ( 15.0 ) for @xmath100 , @xmath101 , @xmath102 , and @xmath103 , respectively , and the numbers in parentheses indicate the expected values obtained from eq . ( [ q - nbody ] ) . the departure from the power - law behavior at the extremes of the curves is due to finite - time sampling . ] in fig . 1 we show the logarithmic plot of the distribution @xmath104 against the transformed variable @xmath99 for systems with @xmath100 , @xmath101 , @xmath102 , and @xmath103 oscillators . based on the above result , we assume ergodicity and compute the distribution of momenta from the fluctuations in time of @xmath105 through the relation @xmath106 where the @xmath107 factor accounts for the degeneracy of the momenta consistent with the magnitude of @xmath38 ( cf . indeed , we observe in all cases that the fluctuations in @xmath105 follow very closely the prescribed power - law behavior eq . ( [ phasedist - f ] ) , with exponents given by eq . ( [ q - nbody ] ) . these results , therefore , provide clear evidence for the validity of our dynamical approach to the generalized thermostatistics . in conclusion , we have shown that the generalized formalism of tsallis can be applied to homogeneous hamiltonian systems to engender an adequate theoretical framework for the statistical mechanics of finite systems . of course , we do not expect that our approach can explain the whole spectrum of problems in which tsallis statistics can be applied . however , our exact results clearly indicate that , as far as homogeneous hamiltonian systems are concerned , the range of the interacting potential should play no role in the equilibrium statistical properties of a system in the thermodynamic limit @xcite . under these conditions , the conventional bg thermostatistics remains valid and general , i.e. , for the specific class of homogeneous hamiltonians investigated here , the thermodynamic limit ( @xmath108 ) leads always to bg distributions . a. b. adib thanks the departamento de fsica at universidade federal do cear for the kind hospitality during most part of this work and dartmouth college for the financial support . we also thank the brazilian agencies cnpq and funcap for financial support . c. tsallis , `` nonextensive statistical mechanics and thermodynamics : historical background and present status , '' in _ nonextensive statistical mechanics and its applications , _ s. abe and y. okamoto ( eds . ) ( springer - verlag , berlin , 2001 ) ; also in braz . j. phys . * 29 * , 1 ( 1999 ) . observing that @xmath109 , where @xmath110 is the structure function of the heat bath , and @xmath111 is its derivative , and integrating eq . ( [ q - eq ] ) with the initial condition @xmath112 we get that @xmath113 , where @xmath114 is a constant . this implies that the structure function is a finite power of @xmath7 for @xmath115 , and therefore the phase space is finite dimensional . it is worth mentioning that a related connection between scale - invariant thermodynamics and tsallis statistics was also proposed by p. a. alemany , phys . lett . a , * 235 * 452 ( 1997 ) , although the approach adopted by the author is not based on the more fundamental ergodic arguments presented here . e. fermi , j. pasta , s. ulam and m. tsingou , _ studies of nonlinear problems i_. los alamos preprint la-1940 ( 7 november 1955 ) ; reprinted in e. fermi , _ collected papers _ , vol . ii ( univ . of chicago press , chicago , 1965 ) p. 978 .

we show that finite systems whose hamiltonians obey a generalized homogeneity relation rigorously follow the nonextensive thermostatistics of tsallis . in the thermodynamical limit , however , our results indicate that the boltzmann - gibbs statistics is always recovered , regardless of the type of potential among interacting particles . this approach provides , moreover , a one - to - one correspondence between the generalized entropy and the hamiltonian structure of a wide class of systems , revealing a possible origin for the intrinsic nonlinear features present in the tsallis formalism that lead naturally to power - law behavior . finally , we confirm these exact results through extensive numerical simulations of the fermi - pasta - ulam chain of anharmonic oscillators .

studies on collective motion of coupled oscillators have attracted considerable attention over the last few decades@xcite . it is commonly seen that a population of autonomous elements performs certain biological functions by behaving collectively@xcite . it has in fact been pointed out that collective motion is crucial to information processing and transmission in living organisms@xcite . in the brain , the neurons are exclusively coupled through chemical synapses , i.e. , the neurons communicate by pulses of transmitter@xcite . chemical synapses commonly form dense and complex networks . for mathematical modeling of neuronal networks , homogeneous all - to - all ( or , global ) coupling is often adopted . although the global coupling may be a little too idealistic , the corresponding networks share a lot of properties in common with systems with complex and dense networks . in the present paper , we consider a population of neural oscillators with delayed , all - to - all pulse - coupling . the oscillator we use is called the leaky integrate - and - fire ( lif ) model . there are a large amount of papers concerning lif in physics and neuroscience , e.g. , see @xcite . this is because lif is a quite simple model still capturing some essential characteristics of neuronal dynamics , i.e. , it represents an integrator with relaxation , and resets after it fires . though our population model is commonly used , ( e.g. , see @xcite ) , its collective dynamics does not seem to have been studied so carefully . we are particularly concerned with peculiar collective dynamics called slow switching@xcite . the study of collective dynamics in the original form of the model is not easy to handle because the coupling involves a long term memory . we thus develop an asymptotic theory and reduce our model into a form without memory , by which an analytical study of collective dynamics becomes possible . the population model we consider consists of @xmath0 identical elements with delayed , all - to - all pulse - coupling . the dynamics of each elements is described by a single variable @xmath1 ( @xmath2 ) which corresponds to the membrane potential of a neuron . the equation for @xmath1 is given by @xmath3 the parameter @xmath4 is the so - called resting potential to which @xmath1 relaxes when the coupling is absent . it is assumed that when @xmath1 reaches a threshold value which is set to @xmath5 , it is instantaneously reset to zero . this event is interpreted as a spiking event . the dynamics is thus called lif . when a neuron spikes , it emits a pulse toward each neuron coupled to it , and the latter receives the pulse with some delay called a synaptic delay . the coupling is assumed to be homogeneous and all - to - all , so that its effect can be represented by one global variable @xmath6 , given by @xmath7 here , @xmath8 represents a series of times at which the @xmath9-th neuron spikes and @xmath10 denotes a summation over the series of such spikes ; @xmath11 is the synaptic delay , and @xmath12 is a _ pulse function _ , given by @xmath13 where @xmath14 is the heaviside function ; @xmath15 and @xmath16 are constants satisfying @xmath17 . in the limit @xmath18 , @xmath12 becomes @xmath19 , which is called the alpha function@xcite . @xmath20 is called the reversal potential to which @xmath1 relaxes when @xmath21 is positive , i.e. , while the neuron receives the pulses . @xmath22 is a positive constant characterizing the strength of the coupling . the coupling assumed above is characteristic to the synaptic coupling . the coupling become excitatory ( epsp ) if @xmath23 , and inhibitory ( ipsp ) if @xmath24 . if @xmath25 , lif becomes an excitable neuron , while if @xmath26 , it repeats periodic spikes , namely , it represents a neural oscillator . we assume @xmath26 throughout the present paper , and call each element an _ oscillator_. then , we can define a variable @xmath27 varying smoothly with time , which turns out useful in the following discussion . we call @xmath27 the _ phase _ of the @xmath28-th oscillator , and define it by @xmath29 which varies between @xmath30 and the intrinsic period of oscillation @xmath31 given by @xmath32 note that @xmath27 satisfies @xmath33 in the absence of coupling . by numerically integrating our model under random initial distributions of @xmath1 , we find various types of collective behavior . among them , we are particularly interested in the slow switching phenomenon , which can arise when @xmath34 and @xmath35 . as displayed in fig . [ fig : slowswitching ] , the whole population , which was initially distributed almost uniformly , splits into two subpopulations , each of which converges almost to a point cluster . however , after some time the phase - advanced cluster starts to scatter . then , this scattered group starts to converge again as it comes behind the preexisting cluster . in this way , the preexisting cluster becomes a phase - advanced cluster . after some time , again , this phase - advanced cluster begins to scatter , and a similar process repeats again and again . in other words , the system switches back and forth between a pair of two - cluster states . for larger times , the system comes closer to each of well - defined two - cluster states and stays near the state longer . theoretically , these switchings repeats indefinitely , although in numerical integrations the system converges at one of the two - cluster states in a finite time and stops switching due to numerical round - off errors@xcite . the slow switching phenomenon occurs within a broad range of parameter values provided that @xmath22 is small , and the time constants @xmath36 and @xmath11 are small compared with @xmath31 . for larger @xmath36 and @xmath11 , the slow switching phenomenon becomes less probable , and the appearance of steady multi - cluster states becomes more probable instead . for @xmath37 , we find no two - cluster states involving slow switching , while steady multi - cluster states are observed in most cases . the corresponding phase diagram will be presented in sec . [ sec : hetero ] ( see fig . [ fig : stability ] ) . our model given by eq . ( [ model ] ) is relatively simple , still it would be difficult to get some insight into its collective dynamics analytically . fortunately , however , our main results given in sec . [ sec : results ] do not change qualitatively in the weak coupling limit , i.e. , @xmath38 . in this limit , our model is reduced to a much simpler form with which we can study the existence and stability of various cluster states analytically . derivation of the reduced model is given as follows . substituting @xmath39 into eq . ( [ model ] ) , we obtain @xmath40 where @xmath41 it is convenient in the following calculation to redefine @xmath42 as a @xmath31-periodic function , or , @xmath43 ( @xmath44 ) . note that sudden drop of @xmath45 at @xmath46 is due to our rule employed , i.e. , the membrane potential is instantaneously reset at @xmath47 . we also define a residual phase @xmath48 by @xmath49 substituting eq . ( [ psi ] ) into eq . ( [ gomi2 ] ) , we obtain @xmath50 we now assume that @xmath22 is sufficiently small so that the r.h.s of eq . ( [ model2 ] ) is sufficiently smaller than the intrinsic frequency @xmath51 . this allow us to make averaging of the r.h.s of eq . ( [ model2 ] ) over the period @xmath31 . the zeroth order approximation with respect to the smallness of @xmath22 , which corresponds to the free oscillations , is given by @xmath52 and @xmath53 where @xmath54 is the latest time at which the @xmath9-th neuron spikes . in the first order approximation , we may time - average eq . ( [ model2 ] ) over the range between @xmath55 and @xmath56 using eqs . ( [ kinji1 ] ) and ( [ kinji2 ] ) : @xmath57 where @xmath58 @xmath59 \exp[{-\alpha \lambda } ] d\lambda \nonumber \\ & = & \frac{(e^t-1)\exp[{\alpha ( x\ { \rm mod } \ t)}]-(e^{\alpha t}-1)\exp[{x\ { \rm mod } \ t } ] } { t(1-\alpha)(e^{\alpha t}-1)}.\end{aligned}\ ] ] note that @xmath60 and @xmath61 are @xmath31-periodic functions . figure [ fig : gamma ] illustrates a typical shape of the coupling function given by eq . ( [ gamma ] ) . furthermore , using the identity @xmath62 and the zeroth order approximation @xmath63 , we may replace @xmath54 by @xmath64 in eq . ( [ pm2 ] ) in the first order approximation . thus , we finally obtain @xmath65 where @xmath66 and @xmath67 . equation ( [ pm ] ) is the standard form of the phase model . note that the error involved in eq . ( [ kinji2 ] ) may look to diverge as @xmath68 , still the final error vanishes in the first order approximation due to the decay of @xmath12 . it should be noted that the reduced model is free from memory effects , but the effect of delay has been converted to a phase shift in the coupling function . similar form of the phase model is generally obtained in delayed coupled oscillators when the coupling is sufficiently weak@xcite . hereafter , we ignore the degree of freedom associated with the dynamics of the center of mass ( or , mean phase ) which can be decoupled in the phase model . important parameters of our phase model given by eq . ( [ pm ] ) with eq . ( [ gamma ] ) are @xmath69 and the sign of @xmath70 ( i.e. , the sign of @xmath71 ) . the reason is the following . we may take @xmath72 by properly rescaling of @xmath56 and @xmath73 , while its sign is crucial because the local stability of any solution depends on it . @xmath73 gives the frequency of steady rotation of the whole system , which is irrelevant to collective dynamics . we choose @xmath31 as an independent parameter by which @xmath4 becomes dependent through eq . ( [ period ] ) . it is remarkable that our coupling function is independent of @xmath20 . in fact , change in @xmath20 causes no qualitative change in our result as far as the sign of @xmath71 remains the same . interestingly , even if we replace the term @xmath74 by a constant @xmath75 in eq . ( [ model ] ) , i.e. , @xmath76 then we can reduce this model similarly and obtain the same coupling function as in eq.([gamma ] ) . we have checked that eq . ( [ model3 ] ) actually reproduces qualitatively the same results as those given in sec . [ sec : results ] . in that case , negative @xmath75 corresponds to the case @xmath37 in eq . ( [ model ] ) . in the following section , we assume @xmath34 and @xmath18 unless stated otherwise . in this section , we study a two - oscillator system , or , @xmath77 . although the two - oscillator system is not directly related to the main subject of the present paper , one may learn some basic properties of our phase model from this simple case . defining @xmath78 , we obtain @xmath79 phase locking solutions are obtained by putting @xmath80 , and the associated eigenvalues are given by @xmath81 . figure [ fig : diagram ] shows a bifurcation diagram of the phase locking solutions , in which we take @xmath11 as a control parameter . we find that for small @xmath11 the trivial solutions @xmath82 ( in - phase locking ) and @xmath83 ( anti - phase locking ) are unstable , while there are a pair of stable branches of non - trivial solutions . the point @xmath84 is close to the bifurcation point where the in - phase state loses stability . the bifurcation occurs at @xmath85 , where @xmath86 corresponds to the minimum of @xmath60 ( see fig . [ fig : gamma ] ) . because @xmath86 is negative , the in - phase state can not be stable for small or vanishing delays ( while it can be stable for delays comparable to @xmath31 due to the @xmath31-periodic nature of our phase model ) . @xmath87 is extremely small , which is due to the sudden drop of @xmath45 at @xmath46 and the particular rule employed in our model , i.e. , a neuron is assumed to spike and reset simultaneously . the width of the stable branches of the trivial solutions is the same as that of the decreasing part of @xmath60 . owing to the peculiar shape of @xmath45 , the width is of the same order as the width of @xmath12 , which is @xmath88 . the stability of the in - phase state is identical with that of the state of perfect synchrony . in terms of the original model , we now present a qualitative interpretation of why the in - phase locking state is unstable for small or vanishing delays . we consider the dynamics of two neurons which are initially very close in phase . the effect of a pulse on the phase @xmath27 is larger for smaller @xmath89 . @xmath89 is monotonously decreasing except when it is reset ( which reflects on the property of @xmath45 that it is increasing except @xmath46 ) . thus , the neuron with larger @xmath1 makes a larger jump in phase when it receives a pulse , by which the phase difference between the two neurons becomes larger when they receive a pulse . on the other hand , the situation becomes different if two neurons lie before and after the resetting point , i.e. , if the phase - advanced neuron has smaller @xmath1 . in that case , the phase difference becomes smaller when they receive a pulse . according to our dynamical rule , however , resetting and spiking occur simultaneously , so that they receive pulses when the phase - advanced neuron has larger @xmath1 . therefore , the in - phase state becomes inevitably unstable even without delay . if we want to obtain a stable in - phase state for small delays , we should employ a rule such that a neuron spikes before it is reset , which would be more physiologically plausible than the rule employed here . the trivial in - phase solution and the non - trivial solutions of the two - oscillator system correspond respectively to the state of perfect synchrony and two - cluster states when we go over to a large population . in this section , we study local stability of the two - cluster states . although non - trivial solutions are stable for small or vanishing delays in the two - oscillator system , the corresponding two - cluster states turn out unstable . we consider a steadily oscillating two - cluster state in which the two clusters consist of @xmath90 and @xmath91 oscillators , respectively . the oscillators inside each cluster are completely phase - synchronized , and the phase - difference between the clusters is constant in time , which is denoted by @xmath92 . from eq . ( [ pm ] ) , the phase difference obeys the equation @xmath93 when @xmath92 is constant , we obtain a relation between @xmath94 and @xmath92 as @xmath95 we designate a two - cluster state satisfying eq . ( [ p - delta ] ) as @xmath96 . the eigenvalues of the stability matrix are calculated as @xmath97 @xmath98 @xmath99 where @xmath100 means @xmath101 . the multiplicities of @xmath102 and @xmath103 are @xmath104,@xmath105 and @xmath5 , respectively . @xmath106 and @xmath107 correspond to fluctuations in phase of the two oscillators inside the phase - advanced and phase - retarded cluster , respectively . @xmath108 corresponds to fluctuations in the phase difference @xmath92 between the clusters . substituting eq . ( [ gamma ] ) into eq . ( [ p - delta ] ) , we obtain a relation between @xmath94 and @xmath92 , the corresponding curve being shown in fig . [ fig : p - delta](a ) . by using this relation , the eigenvalues of @xmath96 can be obtained , which are displayed in fig . [ fig : p - delta](b ) as a function of @xmath92 . it is found that no two - cluster states are stable , and there is a set of @xmath96 for which @xmath109 and @xmath110 . positive @xmath106 means that the two - cluster state is unstable with respect to perturbations inside a phase - advanced cluster . on the other hand , @xmath110 means that it is _ stable _ against perturbations inside a phase - retarded cluster as far as the perfect phase - synchrony of the phase - advanced cluster is maintained . within a certain range of @xmath94 , there are pairs of two - cluster states represented by @xmath96 and @xmath111 with the same stability property , and they appear as the solid lines in fig [ fig : p - delta](a ) . in the next section , we explain how a _ heteroclinic loop _ between the @xmath96 and @xmath111 is stably formed in our model . similarly to the discussion in sec . [ sec : two - oscillator ] , the stability property mentioned above can also be understood in terms of the original model . every neuron inside the phase - advanced cluster always receives pulses when its membrane potential is increasing . then , the phase - difference between two neurons inside the cluster , if any , always increases , so that the phase - advanced cluster is inevitably unstable . on the other hand , the neurons inside the phase - retarded cluster can receive pulses ( emitted by the phase - advanced cluster ) during their resetting . then , the phase - differences between neurons inside the phase - retarded cluster , if any , become smaller , so that the phase - retarded cluster can be stable . we first note that there is a particular symmetry of our model which turns out crucial to the persistent formation of the heteroclinic loop . the symmetry is given by @xmath112 due to this symmetry , the units which have the same membrane potential at a given time behave identically thereafter . in other words , once a point cluster is formed , it remain a point cluster forever . we assume that a pair of two - cluster states ( called a and b ) exists and has the same stability property as that discussed in sec . [ sec : stability ] , i.e. , the phase - advanced cluster is unstable , and the phase - retarded clusters is stable . suppose that our system is in a two - cluster state a initially . when the oscillators inside the phase - advanced cluster are perturbed while the phase - retarded cluster is kept unperturbed [ see fig . [ fig : zukai](a ) ] , the former begins to disintegrate while the latter remains a point cluster . then , the group of dispersed oscillators and the point cluster coexist in the system [ see fig . [ fig : zukai](b ) ] . we know , however , that in the presence of a point cluster , there exists a stable two - cluster state in which the existing point cluster is phase - advanced . from this fact , the dispersed oscillators are expected to converge to form a point cluster coming behind the preexisting point cluster . in this way , the system relaxes to another two - cluster state b [ see fig . [ fig : zukai](c ) ] . from the above statement , it is implied that in our high - dimensional phase space there exists a saddle connection from the state a to the state b. the existence of a saddle connection from the state b to the state a can be argued similarly . a heteroclinic loop is thus formed between the pair of the two cluster states a and b. in terms of the phase model , the above argument can be reinterpreted in a little more precise language . in the phase model given by eq . ( [ pm ] ) , a symmetry property similar to eq . ( [ identity_v ] ) also holds : @xmath113 our argument will be based on the following assumptions : * @xmath96 with @xmath109 and @xmath110 exits , * @xmath111 with @xmath114 and @xmath115 exits , where we define @xmath116 and @xmath117 , and @xmath118 and @xmath119 ( @xmath120 ) are the eigenvalues of @xmath96 and @xmath111 , respectively . note that if @xmath121 , the two clusters in question are identical , or @xmath122 , so that ( a ) and ( b ) are identical . figure [ fig : hetero ] illustrates a schematic presentation of the @xmath123 dimensional phase space structure , where we ignore the degree of freedom associated with the dynamics of the center of mass . @xmath124 and @xmath125 are identical with the subspaces where the phase - advance and phase - retarded clusters of @xmath96 remain point clusters , respectively . by considering the direction of eigenvectors , one can easily confirm that @xmath124 and @xmath125 are identical with the stable subspaces of @xmath96 and @xmath111 , respectively . furthermore , because @xmath124 and @xmath125 are _ invariant _ subspaces due to the symmetry given by eq . ( [ identity ] ) , @xmath96 and @xmath111 are attractors within @xmath124 and @xmath125 , respectively . thus , a heteroclinic loop between @xmath96 and @xmath111 should necessarily exits . the saddle connections in question are stably formed through the invariant subspaces which exist for the symmetry of equations of motion given by eq . ( [ identity ] ) . the heteroclinic loop is thus robust against small perturbations to the system unless the symmetry is broken . whether the resulting heteroclinic loop is attracting or not depends on the following quantity : @xmath126 it was argued in ref . @xcite that if @xmath127 , the system can approach the heteroclinic loop and come to move along it . in that case , the time interval during which the system is trapped to in the vicinity of one of the two - cluster states increases exponentially with time . substituting the eigenvalues obtained from eqs . ( [ l1 ] ) and ( [ l2 ] ) using eq . ( [ gamma ] ) into eq . ( [ exponent ] ) , we find that the heteroclinic loops within a certain range of @xmath94 are in fact attracting for small @xmath36 and @xmath11 . phase diagrams of the heteroclinic loops and symmetric multi - cluster states is shown in fig . [ fig : stability ] , where we choose @xmath11 as a control parameter ( see appendix for the stability of the symmetric multi - cluster states ) . in this section , we concentrate on the vicinity of the bifurcation point where the state of perfect synchrony loses stability . as noted in sec . [ sec : two - oscillator ] , the bifurcation occurs at @xmath85 . then , for small @xmath128 , the coupling function can be expanded as @xmath129 suppose that @xmath130 and @xmath131 are positive . we further put @xmath132 by properly rescaling @xmath70 in eq . ( [ pm ] ) . in order to find possible two - cluster states , we solve eq . ( [ p - delta ] ) using eq . ( [ gamma - appro ] ) . we then obtain three solutions for @xmath92 as a function of @xmath94 and @xmath11 . one is the trivial solution @xmath82 ( the perfect synchrony ) , and the others are given by @xmath133 where @xmath134 . note that the expression above using the approximate @xmath135 given by eq . ( [ gamma - appro ] ) is valid only for small @xmath92 , which is actually the case if @xmath94 is close to @xmath136 and @xmath137 is small . substituting the expressions in eq . ( [ delta ] ) into eqs . ( [ l1])-([l3 ] ) , we obtain eigenvalues associated with the two - cluster states . the resulting bifurcation diagram for given @xmath94 is shown in fig . [ fig : bunki ] . the solid and broken lines give the branches of negative and positive @xmath103 , respectively . two solid branches exist for @xmath138 , which are represented by @xmath96 and @xmath111 with @xmath116 and @xmath117 . one can easily confirm that the eigenvalues of these states satisfy @xmath139 and @xmath140 for arbitrary @xmath94 and small @xmath137 , which agree with the condition for the existence of a heteroclinic loop . the quantity @xmath141 defined by eq . ( [ exponent ] ) can also be calculated and turns out to be larger than @xmath5 . thus , all the local stability conditions for the existence of an attracting heteroclinic loop are generally satisfied just above the bifurcation point provided @xmath142 . it is also possible that a heteroclinic loop is formed when @xmath143 . in that case , it is expected to arise _ subcritically _ , so that both the heteroclinic loop and the state of perfect synchrony may be stable over some region of negative @xmath144 . in fact , we found that such bistability arises in a population of the morris - leccar oscillators@xcite with the same coupling form as in eq . ( [ model ] ) , and an analysis by means of the phase dynamics actually shows that @xmath131 is negative . to confirm the corresponding bifurcation structure , we have to consider higher orders of @xmath128 in the coupling function . the details of this issue are omitted here . we have discussed the slow switching phenomenon in a population of delayed pulse - coupled oscillators . we found that the phenomenon is caused by the formation of an attracting heteroclinic loop between a pair of two - cluster states . a particular stability property of the two - cluster states and a certain symmetry of our model are responsible for its formation . our original model given by eq . ( [ model ] ) is reduced to the standard phase model in the weak coupling limit , by which we succeeded in studying the stability of the two - cluster states analytically , and confirming the structure of the heteroclinic loop . it was also argued that under the mild condition of the coupling function all the local stability conditions for the existence of an attracting heteroclinic loop are generally satisfied just above the bifurcation point . the physical mechanism of the formation of a heteroclinic loop we describe in sec . [ sec : hetero ] does not depend on the nature of elements ( e.g. , phase oscillator , limit - cycle oscillator , excitable elements , chaotic elements ) and couplings ( e.g. , diffusive coupling , pulse coupling ) . it is expected , therefore , that a heteroclinic loop arises in a wide class of models of coupled elements . according to ref.@xcite , we summarize here the existence and the stability analysis of _ symmetric multi - cluster states _ in the phase model given by eq . ( [ pm ] ) . in the symmetric @xmath145-cluster state , it is assumed that each cluster consists of @xmath146 oscillators . we denote the phase of cluster @xmath147 as @xmath148 ( @xmath149 ) . there always exists the following solution : @xmath150 with @xmath151 which corresponds to the state in which the @xmath145 clusters are equally separated in phase and rotate at a constant frequency @xmath152 . the associated eigenvalues are calculated as @xmath153 @xmath154).\ ] ] @xmath155 is a intra - cluster eigenvalue with multiplicity of @xmath156 . @xmath157 ( p=1, ,n-1 ) are associated with inter - cluster fluctuations . if all of these eigenvalues have negative real part , the symmetric @xmath145-cluster state is stable .

we show that peculiar collective dynamics called slow switching arises in a population of leaky integrate - and - fire oscillators with delayed , all - to - all pulse - couplings . by considering the stability of cluster states and symmetry possessed by our model , we argue that saddle connections between a pair of the two - cluster states are formed under general conditions . slow switching appears as a result of the system s approach to the saddle connections . it is also argued that such saddle connections easy to arise near the bifurcation point where the state of perfect synchrony loses stability . we develop an asymptotic theory to reduce the model into a simpler form , with which an analytical study of cluster states becomes possible .

a hamilton decomposition of a graph or digraph @xmath1 is a set of edge - disjoint hamilton cycles which together cover all the edges of @xmath1 . the topic has a long history but some of the main questions remain open . in 1892 , walecki showed that the edge set of the complete graph @xmath4 on @xmath2 vertices has a hamilton decomposition if @xmath2 is odd ( see e.g. @xcite for the construction ) . if @xmath2 is even , then @xmath2 is not a factor of @xmath5 , so clearly @xmath4 does not have such a decomposition . walecki s result implies that a complete digraph @xmath1 on @xmath2 vertices has a hamilton decomposition if @xmath2 is odd . more generally , tillson @xcite proved that a complete digraph @xmath1 on @xmath2 vertices has a hamilton decomposition if and only if @xmath6 . a tournament is an orientation of a complete graph . we say that a tournament is _ regular _ if every vertex has equal in- and outdegree . thus regular tournaments contain an odd number @xmath2 of vertices and each vertex has in- and outdegree @xmath7 . the following beautiful conjecture of kelly ( see e.g. @xcite ) , which has attracted much attention , states that every regular tournament has a hamilton decomposition : [ kelly ] every regular tournament on @xmath2 vertices can be decomposed into @xmath7 edge - disjoint hamilton cycles . in this paper we prove an approximate version of kelly s conjecture . [ main1 ] for every @xmath0 there exists an integer @xmath8 so that every regular tournament on @xmath9 vertices contains at least @xmath3 edge - disjoint hamilton cycles . in fact , we prove the following stronger result , where we consider orientations of almost complete graphs which are almost regular . an _ oriented graph _ is obtained from an undirected graph by orienting its edges . so it has at most one edge between every pair of vertices , whereas a digraph may have an edge in each direction . [ main ] for every @xmath10 there exist @xmath11 and @xmath12 such that the following holds . suppose that @xmath1 is an oriented graph on @xmath13 vertices such that every vertex in @xmath1 has in- and outdegree at least @xmath14 . then @xmath1 contains at least @xmath15 edge - disjoint hamilton cycles . the _ minimum semidegree _ @xmath16 of an oriented graph @xmath1 is the minimum of its minimum outdegree and its minimum indegree . so the minimum semidegree of a regular tournament on @xmath2 vertices is @xmath7 . most of the previous partial results towards kelly s conjecture have been obtained by giving bounds on the minimum semidegree of an oriented graph which guarantees a hamilton cycle . this approach was first used by jackson @xcite , who showed that every regular tournament on at least 5 vertices contains a hamilton cycle and a hamilton path which are edge - disjoint . zhang @xcite then showed that every such tournament contains two edge - disjoint hamilton cycles . improved bounds on the value of @xmath16 which forces a hamilton cycle were then found by thomassen @xcite , hggkvist @xcite , hggkvist and thomason @xcite as well as kelly , khn and osthus @xcite . finally , keevash , khn and osthus @xcite showed that every sufficiently large oriented graph @xmath1 on @xmath2 vertices with @xmath17 contains a hamilton cycle . this bound on @xmath18 is best possible and confirmed a conjecture of hggkvist @xcite . note that this result implies that every sufficiently large regular tournament on @xmath2 vertices contains at least @xmath19 edge - disjoint hamilton cycles . this was the best bound so far towards kelly s conjecture . kelly s conjecture has also been verified for @xmath20 by alspach ( see the survey @xcite ) . a result of frieze and krivelevich @xcite states that theorem [ main ] holds for ` quasi - random ' tournaments . as indicated below , we will build on some of their ideas in the proof of theorem [ main ] . it turns out that theorem [ main ] can be generalized even further : any large almost regular oriented graph on @xmath2 vertices whose in- and outdegrees are all a little larger than @xmath21 can almost be decomposed into hamilton cycles . the corresponding modifications to the proof of theorem [ main ] are described in section [ 38 ] . we also discuss some further open questions in that section . jackson @xcite also introduced the following bipartite version of kelly s conjecture ( both versions are also discussed e.g. in the handbook article by bondy @xcite ) . bipartite tournament _ is an orientation of a complete bipartite graph . [ kellybip ] every regular bipartite tournament has a hamilton decomposition . an undirected version of conjecture [ kellybip ] was proved independently by auerbach and laskar @xcite , as well as hetyei @xcite . however , a bipartite version of theorem [ main ] does not hold , because there are almost regular bipartite tournaments which do not even contain a single hamilton cycle . ( consider for instance the following ` blow - up ' of a 4-cycle : the vertices are split into 4 parts @xmath22 whose sizes are almost but not exactly equal , and we have all edges from @xmath23 to @xmath24 , with indices modulo 4 . ) kelly s conjecture has been generalized in several directions . for instance , given an oriented graph @xmath1 , define its _ excess _ by @xmath25 where @xmath26 denotes the number of outneighbours of the vertex @xmath27 , and @xmath28 the number of its inneighbours . pullman ( see e.g. conjecture 8.25 in @xcite ) conjectured that if @xmath1 is an oriented graph such that @xmath29 for all vertices @xmath27 of @xmath1 , where @xmath30 is odd , then @xmath1 has a decomposition into @xmath31 directed paths . to see that this would imply kelly s conjecture , let @xmath1 be the oriented graph obtained from a regular tournament by deleting a vertex . another generalization was made by bang - jensen and yeo @xcite , who conjectured that every @xmath32-edge - connected tournament has a decomposition into @xmath32 spanning strong digraphs . in @xcite , thomassen also formulated the following weakening of kelly s conjecture . [ thomconj ] if @xmath1 is a regular tournament on @xmath33 vertices and @xmath34 is any set of at most @xmath35 edges of @xmath1 , then @xmath36 has a hamilton cycle . in @xcite , we proved a result on the existence of hamilton cycles in ` robust expander digraphs ' which implies conjecture [ thomconj ] for large tournaments ( see @xcite for details ) . @xcite also contains the related conjecture that for any @xmath37 , there is an @xmath38 so that every strongly @xmath38-connected tournament contains @xmath39 edge - disjoint hamilton cycles . further support for kelly s conjecture was also provided by thomassen @xcite , who showed that the edges of every regular tournament on @xmath2 vertices can be covered by @xmath40 hamilton cycles . in @xcite the first two authors observed that one can use theorem [ main ] to reduce this to @xmath41 hamilton cycles . a discussion of further recent results about hamilton cycles in directed graphs can be found in the survey @xcite . it seems likely that the techniques developed in this paper will also be useful in solving further problems . in fact , christofides , khn and osthus @xcite used similar ideas to prove approximate versions of the following two long - standing conjectures of nash - williams @xcite : let @xmath1 be a @xmath42-regular graph on at most @xmath43 vertices , where @xmath44 . then @xmath1 has a hamilton decomposition . [ nwconj2 ] let @xmath1 be a graph on @xmath2 vertices with minimum degree at least @xmath45 . then @xmath1 contains @xmath46 edge - disjoint hamilton cycles . ( actually , nash - williams initially formulated conjecture [ nwconj2 ] with the term @xmath19 replaced by @xmath47 , but babai found a counterexample to this . ) another related problem was raised by erds ( see @xcite ) , who asked whether almost all tournaments @xmath1 have at least @xmath18 edge - disjoint hamilton cycles . note that an affirmative answer would not directly imply that kelly s conjecture holds for almost all regular tournaments , which would of course be an interesting result in itself . there are also a number of corresponding questions for random undirected graphs ( see e.g. @xcite ) . after giving an outline of the argument in the next section , we will state a directed version of the regularity lemma and some related results in section [ 3 ] . section [ 4 ] contains statements and proofs of several auxiliary results , mostly on ( almost ) @xmath48-factors in ( almost ) regular oriented graphs . the proof of theorem [ main ] is given in section [ 5 ] . a generalization of theorem [ main ] to oriented graphs with smaller degrees is discussed in section [ 38 ] . suppose we are given a regular tournament @xmath1 on @xmath2 vertices and our aim is to ` almost ' decompose it into hamilton cycles . one possible approach might be the following : first remove a spanning regular oriented subgraph @xmath49 whose degree @xmath50 satisfies @xmath51 . let @xmath52 be the remaining oriented subgraph of @xmath1 . now consider a decomposition of @xmath52 into @xmath48-factors @xmath53 ( which clearly exists ) . next , try to transform each @xmath54 into a hamilton cycle by removing some of its edges and adding some suitable edges of @xmath49 . this is of course impossible if many of the @xmath54 consist of many cycles . however , an auxiliary result of frieze and krivelevich in @xcite implies that we can ` almost ' decompose @xmath52 so that each @xmath48-factor @xmath54 consists of only a few cycles . if @xmath49 were a ` quasi - random ' oriented graph , then ( as in @xcite ) one could use it to successively ` merge ' the cycles of each @xmath54 into hamilton cycles using a ` rotation - extension ' argument : delete an edge of a cycle @xmath55 of @xmath54 to obtain a path @xmath56 from @xmath57 to @xmath58 , say . if there is an edge of @xmath49 from @xmath58 to another cycle @xmath59 of @xmath54 , then extend @xmath56 to include the vertices of @xmath59 ( and similarly for @xmath57 ) . continue until there is no such edge . then ( in @xmath49 ) the current endvertices of the path @xmath56 have many neighbours on @xmath56 . one can use this together with the quasi - randomness of @xmath49 to transform @xmath56 into a cycle with the same vertices as @xmath56 . now repeat this , until we have merged all the cycles into a single ( hamilton ) cycle . of course , one has to be careful to maintain the quasi - randomness of @xmath49 in carrying out this ` rotation - extension ' process for the successive @xmath54 ( the fact that @xmath54 contains only few cycles is important for this ) . the main problem is that @xmath1 need not contain such a spanning ` quasi - random ' subgraph @xmath49 . so instead , in section [ applydrl ] we use szemerdi s regularity lemma to decompose @xmath1 into quasi - random subgraphs . we then choose both our @xmath48-factors @xmath54 and the graph @xmath49 according to the structure of this decomposition . more precisely , we apply a directed version of szemerdi s regularity lemma to obtain a partition of the vertices of @xmath1 into a bounded number of clusters @xmath60 so that almost all of the bipartite subgraphs spanned by ordered pairs of clusters are quasi - random ( see section [ 3.3 ] for the precise statement ) . this then yields a reduced digraph @xmath61 , whose vertices correspond to the clusters , with an edge from one cluster @xmath62 to another cluster @xmath63 if the edges from @xmath62 to @xmath63 in @xmath1 form a quasi - random graph . ( note that @xmath61 need not be oriented . ) we view @xmath61 as a weighted digraph whose edge weights are the densities of the corresponding ordered pair of clusters . we then obtain an unweighted multidigraph @xmath64 from @xmath61 as follows : given an edge @xmath65 of @xmath61 joining a cluster @xmath62 to @xmath63 , replace it with @xmath66 copies of @xmath65 , where @xmath67 is approximately proportional to the density of the ordered pair @xmath68 . it is not hard to show that @xmath64 is approximately regular ( see lemma [ multimin ] ) . if @xmath64 were regular , then it would have a decomposition into @xmath48-factors , but this assumption may not be true . however , we can show that @xmath64 can ` almost ' be decomposed into ` almost ' @xmath48-factors . in other words , there exist edge - disjoint collections @xmath69 of vertex - disjoint cycles in @xmath64 such that each @xmath70 covers almost all of the clusters in @xmath64 ( see lemma [ multifactor1 ] ) . now we choose edge - disjoint oriented spanning subgraphs @xmath71 of @xmath1 so that each @xmath72 corresponds to @xmath70 . for this , consider an edge @xmath65 of @xmath61 from @xmath62 to @xmath63 and suppose for example that @xmath73 , @xmath74 and @xmath75 are the only @xmath70 containing copies of @xmath65 in @xmath64 . then for each edge of @xmath1 from @xmath62 to @xmath63 in turn , we assign it to one of @xmath76 , @xmath77 and @xmath78 with equal probability . then with high probability , each @xmath72 consists of bipartite quasi - random oriented graphs which together form a disjoint union of ` blown - up ' cycles . moreover , we can arrange that all the vertices have degree close to @xmath79 ( here @xmath80 is the cluster size and @xmath81 a small parameter which does not depend on @xmath82 ) . we now remove a small proportion of the edges from @xmath1 ( and thus from each @xmath72 ) to form oriented subgraphs @xmath83 of @xmath1 , where @xmath84 . ideally , we would like to show that each @xmath72 can almost be decomposed into hamilton cycles . since the @xmath72 are edge - disjoint , this would yield the required result . one obvious obstacle is that the @xmath72 need not be spanning subgraphs of @xmath1 ( because of the exceptional set @xmath85 returned by the regularity lemma and because the @xmath70 are not spanning . ) so in section [ sec : incorp ] we add suitable edges between @xmath72 and the leftover vertices to form edge - disjoint oriented spanning subgraphs @xmath86 of @xmath1 where every vertex has degree close to @xmath79 . ( the edges of @xmath87 and @xmath88 are used in this step . ) but the distribution of the edges added in this step may be somewhat ` unbalanced ' , with some vertices of @xmath72 sending out or receiving too many of them . in fact , as discussed at the beginning of section [ skel ] , we can not even guarantee that @xmath86 has a single @xmath48-factor . we overcome this new difficulty by adding carefully chosen further edges ( from @xmath89 this time ) to each @xmath86 which compensate the above imbalances . once these edges have been added , in section [ nicefactor ] we can use the max - flow min - cut theorem to almost decompose each @xmath86 into @xmath48-factors @xmath90 . ( this is one of the points where we use the fact that the @xmath72 consist of quasi - random graphs which form a union of blown - up cycles . ) moreover , ( i ) the number of cycles in each of these @xmath48-factors is not too large and ( ii ) most of the cycles inherit the structure of @xmath70 . more precisely , ( ii ) means that most vertices @xmath91 of @xmath72 have the following property : let @xmath62 be the cluster containing @xmath91 and let @xmath92 be the successor of @xmath62 in @xmath70 . then the successor @xmath93 of @xmath91 in @xmath90 lies in @xmath92 . in section [ 4.6 ] we can use ( i ) and ( ii ) to merge the cycles of each @xmath90 into a @xmath48-factor @xmath94 consisting only of a bounded number of cycles for each cycle @xmath95 of @xmath70 , all the vertices of @xmath86 which lie in clusters of @xmath95 will lie in the same cycle of @xmath96 . we will apply a rotation - extension argument for this , where the additional edges ( i.e. those not in @xmath90 ) come from @xmath97 . finally , in section [ merging ] we will use the fact that @xmath64 contains many short paths to merge each @xmath94 into a single hamilton cycle . the additional edges will come from @xmath98 and @xmath99 this time . throughout this paper we omit floors and ceilings whenever this does not affect the argument . given a graph @xmath1 , we denote the degree of a vertex @xmath100 by @xmath101 and the maximum degree of @xmath1 by @xmath102 . given two vertices @xmath103 and @xmath104 of a digraph @xmath1 , we write @xmath105 for the edge directed from @xmath103 to @xmath104 . we denote by @xmath106 the set of all outneighbours of @xmath103 . so @xmath106 consists of all those @xmath107 for which @xmath108 . we have an analogous definition for @xmath109 given a multidigraph @xmath1 , we denote by @xmath110 the _ multiset _ of vertices where a vertex @xmath107 appears @xmath32 times in @xmath111 if @xmath1 contains precisely @xmath32 edges from @xmath103 to @xmath104 . again , we have an analogous definition for @xmath112 . we will write @xmath113 for example , if this is unambiguous . given a vertex @xmath103 of a digraph or multidigraph @xmath1 , we write @xmath114 for the outdegree of @xmath103 , @xmath115 for its indegree and @xmath116 for its degree . the maximum of the maximum outdegree @xmath117 and the maximum indegree @xmath118 is denoted by @xmath119 . the _ minimum semidegree _ @xmath16 of @xmath1 is the minimum of its minimum outdegree @xmath120 and its minimum indegree @xmath121 . throughout the paper we will use @xmath122 , @xmath123 and @xmath124 as ` shorthand ' notation . for example , @xmath125 is read as @xmath126 and @xmath127 . a _ 1-factor _ of a multidigraph @xmath1 is a collection of vertex - disjoint cycles in @xmath1 which together cover all the vertices of @xmath1 . given @xmath128 , we write @xmath129 to denote the number of edges in @xmath1 with startpoint in @xmath34 and endpoint in @xmath130 . similarly , if @xmath1 is an undirected graph , we write @xmath129 for the number of all edges between @xmath34 and @xmath130 . given a multiset @xmath131 and a set @xmath132 we define @xmath133 to be the multiset where @xmath103 appears as an element precisely @xmath32 times in @xmath133 if @xmath134 , @xmath135 and @xmath103 appears precisely @xmath32 times in @xmath131 . we write @xmath136 for @xmath137 $ ] . we will often use the following chernoff bound for binomial and hypergeometric distributions ( see e.g. ) . recall that the binomial random variable with parameters @xmath138 is the sum of @xmath2 independent bernoulli variables , each taking value @xmath48 with probability @xmath139 or @xmath140 with probability @xmath141 . the hypergeometric random variable @xmath131 with parameters @xmath142 is defined as follows . we let @xmath143 be a set of size @xmath2 , fix @xmath144 of size @xmath145 , pick a uniformly random @xmath146 of size @xmath147 , then define @xmath148 . note that @xmath149 . [ chernoff ] suppose @xmath131 has binomial or hypergeometric distribution and @xmath150 . then @xmath151 . in the proof of theorem [ main ] we will use the directed version of szemerdi s regularity lemma . before we can state it we need some more notation and definitions . density _ of an undirected bipartite graph @xmath1 with vertex classes @xmath34 and @xmath130 is defined to be @xmath152 we will write @xmath153 if this is unambiguous . given any @xmath154 , we say that @xmath1 is _ @xmath155$]-regular _ if for all sets @xmath156 and @xmath157 with @xmath158 and @xmath159 we have @xmath160 . in the case when @xmath161 we say that @xmath1 is _ @xmath162-regular_. given @xmath163 we say that @xmath1 is _ @xmath164-super - regular _ if all sets @xmath156 and @xmath157 with @xmath158 and @xmath159 satisfy @xmath165 and , furthermore , if @xmath166 for all @xmath167 and @xmath168 for all @xmath169 . note that this is a slight variation of the standard definition . given disjoint vertex sets @xmath34 and @xmath130 in a digraph @xmath1 , we write @xmath170 for the oriented bipartite subgraph of @xmath1 whose vertex classes are @xmath34 and @xmath130 and whose edges are all the edges from @xmath34 to @xmath130 in @xmath1 . we say @xmath170 is _ @xmath171$]-regular and has density @xmath172 _ if this holds for the underlying undirected bipartite graph of @xmath170 . ( note that the ordering of the pair @xmath170 is important here . ) in the case when @xmath173 we say that _ @xmath170 is @xmath162-regular and has density @xmath172_. similarly , given @xmath163 we say @xmath170 is _ @xmath174-super - regular _ if this holds for the underlying undirected bipartite graph . the diregularity lemma is a variant of the regularity lemma for digraphs due to alon and shapira @xcite . its proof is similar to the undirected version . we will use the degree form of the diregularity lemma which can be derived from the standard version in the same manner as the undirected degree form ( see @xcite for a sketch of the latter ) . [ dilemma ] for every @xmath175 and every integer @xmath176 there are integers @xmath177 and @xmath8 such that if @xmath1 is a digraph on @xmath178 vertices and @xmath179 $ ] is any real number , then there is a partition of the vertex set of @xmath1 into @xmath180 and a spanning subdigraph @xmath52 of @xmath1 such that the following holds : * @xmath181 , * @xmath182 , * @xmath183 , * @xmath184 for all vertices @xmath185 , * for all @xmath186 the digraph @xmath187 $ ] is empty , * for all @xmath188 with @xmath189 the pair @xmath190 is @xmath191-regular and has density either @xmath140 or at least @xmath30 . we call @xmath192 _ clusters _ , @xmath85 the _ exceptional set _ and the vertices in @xmath85 _ exceptional vertices_. we refer to @xmath52 as the _ pure digraph_. the last condition of the lemma says that all pairs of clusters are @xmath162-regular in both directions ( but possibly with different densities ) . the _ reduced digraph @xmath61 of @xmath1 with parameters @xmath191 , @xmath30 and @xmath176 _ is the digraph whose vertices are @xmath193 and in which @xmath194 is an edge precisely when @xmath190 is @xmath191-regular and has density at least @xmath30 . the next result shows that we can partition the set of edges of an @xmath162-(super)-regular pair into edge - disjoint subgraphs such that each of them is still ( super)-regular . [ split ] let @xmath195 and suppose @xmath196 . then there exists an integer @xmath197 such that for all @xmath198 the following holds . * suppose that @xmath199 is an @xmath162-regular pair of density @xmath30 where @xmath200 . then there are @xmath201 edge - disjoint spanning subgraphs @xmath202 of @xmath1 such that each @xmath203 is @xmath204$]-regular of density @xmath205 . * if @xmath206 and @xmath199 is @xmath164-super - regular with @xmath200 . then there are two edge - disjoint spanning subgraphs @xmath207 and @xmath208 of @xmath1 such that each @xmath203 is @xmath209-super - regular . we first prove ( i ) . suppose we have chosen @xmath210 sufficiently large . initially set @xmath211 for each @xmath212 . we consider each edge of @xmath1 in turn and add it to each @xmath213 with probability @xmath214 , independently of all other edges of @xmath1 . so the probability that @xmath105 is added to none of the @xmath203 is @xmath215 . moreover , @xmath216 . given @xmath156 and @xmath157 with @xmath217 we have that @xmath218 . thus @xmath219 for each @xmath82 . proposition [ chernoff ] for the binomial distribution implies that with high probability @xmath220 for each @xmath221 and every @xmath156 and @xmath157 with @xmath217 . such @xmath203 are as required in ( i ) . the proof of ( ii ) is similar . indeed , as in ( i ) one can show that with high probability any @xmath222 and @xmath157 with @xmath217 satisfy @xmath223 ( for @xmath224 ) . moreover , each vertex @xmath225 satisfies @xmath226 ( for @xmath224 ) and similarly for the vertices in @xmath130 . so again proposition [ chernoff ] for the binomial distribution implies that with high probability @xmath227 for all @xmath225 and @xmath228 for all @xmath229 . altogether this shows that with high probability both @xmath207 and @xmath208 are @xmath209-super - regular . suppose @xmath230 and let @xmath1 be a digraph . let @xmath61 and @xmath52 denote the reduced digraph and pure digraph respectively , obtained by applying lemma [ dilemma ] to @xmath1 with parameters @xmath231 and @xmath176 . for each edge @xmath232 of @xmath61 we write @xmath233 for the density of @xmath234 . ( so @xmath235 . ) the _ reduced multidigraph _ @xmath64 of @xmath1 with parameters @xmath236 and @xmath176 is obtained from @xmath61 by setting @xmath237 and adding @xmath238 directed edges from @xmath60 to @xmath239 whenever @xmath240 . we will always consider the reduced multidigraph @xmath64 of a digraph @xmath1 whose order is sufficiently large in order to apply lemma [ split ] to any pair @xmath234 of clusters with @xmath240 . let @xmath241 and @xmath242 be the spanning subgraphs of @xmath190 obtained from lemma [ split ] . ( so each @xmath243 is @xmath162-regular of density @xmath244 . ) let @xmath245 denote the directed edges from @xmath60 to @xmath239 in @xmath64 . we associate each @xmath246 with the edges in @xmath243 . [ multimin ] let @xmath247 and let @xmath1 be a digraph of sufficiently large order @xmath2 with @xmath248 and @xmath249 . apply lemma [ dilemma ] with parameters @xmath231 and @xmath176 to obtain a pure digraph @xmath52 and a reduced digraph @xmath61 of @xmath1 . let @xmath64 denote the reduced multidigraph of @xmath1 with parameters @xmath236 and @xmath176 . then @xmath250 note the corresponding upper bound would not hold if we considered @xmath61 instead of @xmath64 here . given any @xmath251 , let @xmath233 denote the density of @xmath190 . then @xmath252 by lemma [ dilemma ] . thus @xmath253 so indeed @xmath254 . similar arguments can be used to show that @xmath255 and @xmath256 . we will also need the well - known fact that for any cycle @xmath55 of the reduced multigraph @xmath64 we can delete a small number of vertices from the clusters in @xmath55 in order to ensure that each edge of @xmath55 corresponds to a super - regular pair . we include a proof for completeness . [ superreg ] let @xmath257 be a cycle in the reduced multigraph @xmath64 as in lemma [ multimin ] . for each @xmath258 let @xmath259 denote the edge of @xmath55 which joins @xmath260 to @xmath261 ( where @xmath262 ) . then we can choose subclusters @xmath263 of size @xmath264 such that @xmath265 is @xmath266-super - regular ( for each @xmath258 ) . recall that for each @xmath267 the digraph @xmath268 corresponding to the edge @xmath259 of @xmath55 is @xmath162-regular and has density @xmath244 . so @xmath260 contains at most @xmath269 vertices whose outdegree in @xmath268 is either at most @xmath270 or at least @xmath271 . similarly , there are at most @xmath269 vertices in @xmath260 whose indegree in @xmath272 is either at most @xmath270 or at least @xmath271 . let @xmath273 be a set of size @xmath274 obtained from @xmath260 by deleting all these vertices ( and some additional vertices if necessary ) . it is easy to check that @xmath275 are subclusters as required . finally , we will use the following crude version of the fact that every @xmath155$]-regular pair contains a subgraph of given maximum degree @xmath276 whose average degree is close to @xmath276 . [ boundmax ] suppose that @xmath277 and that @xmath278 is an @xmath155$]-regular pair of density @xmath279 with @xmath2 vertices in each class . then @xmath278 contains a subgraph @xmath49 whose maximum degree is at most @xmath280 and whose average degree is at least @xmath281 . let @xmath282 be the set of vertices of degree at least @xmath283 and define @xmath284 similarly . then @xmath285 . let @xmath286 and @xmath287 . then @xmath288 is still @xmath289$]-regular of density at least @xmath290 . now consider a spanning subgraph @xmath49 of @xmath288 which is obtained from @xmath288 by including each edge with probability @xmath291 . so the expected degree of every vertex is at most @xmath292 and the expected number of edges of @xmath49 is at least @xmath293 . now apply the chernoff bound on the binomial distribution in proposition [ chernoff ] to each of the vertex degrees and to the total number of edges in @xmath49 to see that with high probability @xmath49 has the desired properties . our main aim in this subsection is to show that the reduced multidigraph @xmath64 contains a collection of ` almost ' 1-factors which together cover almost all the edges of @xmath64 ( see lemma [ multifactor1 ] ) . to prove this we will need the following result which implies @xmath64 contains many edges between any two sufficiently large sets . the second part of the lemma will be used in section [ sec : shifted ] . [ keevashmult ] let @xmath294 . suppose that @xmath1 is an oriented graph of order @xmath2 with @xmath295 . let @xmath61 and @xmath64 denote the reduced digraph and the reduced multidigraph of @xmath1 obtained by applying lemma [ dilemma ] ( with parameters @xmath296 and @xmath297 respectively ) . let @xmath298 . then the following properties hold . * let @xmath299 be such that @xmath300 ) \geq ( 1/2-c)|x|/\beta$ ] . then for all ( not necessarily disjoint ) subsets @xmath34 and @xmath130 of @xmath131 of size at least @xmath301 there are at least @xmath302 directed edges from @xmath34 to @xmath130 in @xmath64 . * let @xmath303 denote the spanning subdigraph of @xmath61 obtained by deleting all edges which correspond to pairs of density at most @xmath172 ( in the pure digraph @xmath52 ) . then @xmath304 and for all ( not necessarily disjoint ) subsets @xmath34 and @xmath130 of @xmath305 of size at least @xmath306 there are at least @xmath307 directed edges from @xmath34 to @xmath130 in @xmath303 . we first prove ( i ) . recall that for every edge @xmath232 of @xmath61 there are precisely @xmath238 edges from @xmath308 to @xmath239 in @xmath64 , where @xmath233 denotes the density of @xmath309 . but @xmath310 since @xmath1 is oriented and so @xmath64 contains at most @xmath311 edges between @xmath60 and @xmath239 ( here we count the edges in both directions ) . by deleting vertices from @xmath34 and @xmath130 if necessary we may assume that @xmath312 . we will distinguish two cases . suppose first that @xmath313 and let @xmath314 . define @xmath315 and @xmath316 . then @xmath317}(v)-e(y,\overline{y})-e(\overline{y},y)\\ & { \ge } & |y|(1 - 2c)|x|/\beta -|y|(|x|-|y|)/\beta = |y|(|y|-2c|x|)/\beta\ge |x|^2/(30\beta).\end{aligned}\ ] ] so suppose next that @xmath318 . then @xmath319 . therefore , @xmath320}(v)-e(a,\overline{a\cup b})-e(a)\\ & { \ge } & & \ge & |a|[(1/2-c)-(1/5 + 2c)-(1/2-c)/2]|x|/\beta\ge |x|^2/(60\beta),\end{aligned}\ ] ] as required . to prove ( ii ) we consider the weighted digraph @xmath321 obtained from @xmath303 by giving each edge @xmath232 of @xmath303 weight @xmath233 . given a cluster @xmath60 , we write @xmath322 for the sum of the weights of all edges sent out by @xmath60 in @xmath321 . we define @xmath323 similarly and write @xmath324 for the minimum of @xmath325 over all clusters @xmath60 . note that @xmath326 . moreover , lemma [ dilemma ] implies that @xmath327 . thus each @xmath328 satisfies @xmath329 and so @xmath330 . arguing in the same way for inweights gives us @xmath331 let @xmath332 be as in ( ii ) . similarly as in ( i ) ( setting @xmath333 and @xmath334 in the calculations ) one can show that the sum of all weights of the edges from @xmath34 to @xmath130 in @xmath321 is at least @xmath307 . but this implies that @xmath303 contains at least @xmath307 edges from @xmath34 to @xmath130 . [ multifactor1 ] let @xmath335 . suppose that @xmath1 is an oriented graph of order @xmath2 with @xmath295 . let @xmath64 denote the reduced multidigraph of @xmath1 with parameters @xmath336 and @xmath176 obtained by applying lemma [ dilemma ] . let @xmath337 . then there exist edge - disjoint collections @xmath69 of vertex - disjoint cycles in @xmath64 such that each @xmath70 covers all but at most @xmath338 of the clusters in @xmath64 . let @xmath339 . since @xmath340 , lemma [ multimin ] implies that @xmath341 first we find a set of clusters @xmath342 with the following properties : * @xmath343 , * @xmath344 for all @xmath345 . we obtain @xmath131 by choosing a set of @xmath346 clusters uniformly at random . then each cluster @xmath60 satisfies @xmath347 proposition [ chernoff ] for the hypergeometric distribution now implies that with nonzero probability @xmath131 satisfies our desired conditions . ( recall that @xmath348 is a multiset . formally proposition [ chernoff ] does not apply to multisets . however , for each @xmath349 we can apply proposition [ chernoff ] to the set of all those clusters which appear at least @xmath350 times in @xmath351 , and similarly for @xmath352 . ) note that @xmath353 for each @xmath354 . we now add a small number of _ temporary edges _ to @xmath355 in order to turn it into an @xmath356-regular multidigraph where @xmath357 . we do this as follows . as long as @xmath355 is not @xmath356-regular there exist @xmath358 such that @xmath60 has outdegree less than @xmath356 and @xmath239 has indegree less than @xmath356 . in this case we add an edge from @xmath60 to @xmath239 . ( note we may have @xmath359 , in which case we add a loop . ) we decompose the edge set of @xmath360 into @xmath356 1-factors @xmath361 . ( to see that we can do this , consider the bipartite multigraph @xmath49 where both vertex classes @xmath362 consist of a copy of @xmath363 and we have @xmath364 edges between @xmath167 and @xmath169 if there are precisely @xmath364 edges from @xmath57 to @xmath58 in @xmath355 , including the temporary edges . then @xmath49 is regular and so has a perfect matching . this corresponds to a @xmath48-factor @xmath365 . now remove the edges of @xmath365 from @xmath49 and continue to find @xmath366 in the same way . ) since at each cluster we added at most @xmath367 temporary edges , all but at most @xmath368 of the @xmath369 contain at most @xmath370 temporary edges . by relabeling if necessary we may assume that @xmath371 are such @xmath48-factors . we now remove the temporary edges from each of these @xmath48-factors , though we still refer to the digraphs obtained in this way as @xmath371 . so each @xmath369 spans @xmath355 and consists of cycles and at most @xmath372 paths . our aim is to use the clusters in @xmath131 to piece up these paths into cycles in order to obtain edge - disjoint directed subgraphs @xmath373 of @xmath64 where each @xmath70 is a collection of vertex - disjoint cycles and @xmath374 . let @xmath375 denote all the paths lying in one of @xmath371 ( so @xmath376 ) . our next task is to find edge - disjoint paths and cycles @xmath377 of length @xmath378 in @xmath64 with the following properties . * if @xmath379 consists of a single cluster @xmath380 then @xmath381 is a cycle consisting of @xmath382 clusters in @xmath131 as well as @xmath383 . * if @xmath379 is a path of length @xmath384 then @xmath381 is a path whose startpoint is the endpoint of @xmath379 . similarly the endpoint of @xmath381 is the startpoint of @xmath379 . * if @xmath379 is a path of length @xmath385 then the internal clusters in the path @xmath381 lie in @xmath131 . * if @xmath386 and @xmath387 lie in the same @xmath369 then @xmath388 and @xmath389 are vertex - disjoint . so conditions ( i)(iii ) imply that @xmath390 is a directed cycle for each @xmath391 . assuming we have found such paths and cycles @xmath377 , we define @xmath392 as follows . suppose @xmath393 are the paths in @xmath369 . then we obtain @xmath70 from @xmath369 by adding the paths and cycles @xmath394 to @xmath369 . condition ( iv ) ensures that the @xmath70 are indeed collections of vertex - disjoint cycles . it remains to show the existence of @xmath377 . suppose that for some @xmath395 we have already found @xmath396 and now need to define @xmath381 . consider @xmath379 and suppose it lies in @xmath369 . let @xmath397 denote the startpoint of @xmath379 and @xmath398 its endpoint . we call an edge @xmath399 in @xmath64 _ free _ if it has not been used in one of @xmath400 . let @xmath130 be the set of all those clusters @xmath401 for which at least @xmath402 of the edges at @xmath403 in @xmath404 $ ] are not free . our next aim is to show that @xmath130 is small . more precisely , @xmath405 to see this , note that @xmath406 edges of @xmath407 $ ] lie in one of @xmath408 thus , @xmath409 . ( the extra factor of 2 comes from the fact that we may have counted edges at the vertices in @xmath130 twice . ) since @xmath410 this implies that @xmath411 , as desired . we will only use clusters in @xmath412 when constructing @xmath381 . note that @xmath397 receives at most @xmath413 edges from @xmath130 in @xmath64 . since we added at most @xmath414 temporary edges to @xmath355 per cluster , @xmath397 can be the startpoint or endpoint of at most @xmath414 of the paths @xmath415 . thus @xmath397 lies in at most @xmath416 of the paths and cycles @xmath408 . in particular , at most @xmath417 edges at @xmath418 in @xmath64 are not free . we will avoid such edges when constructing @xmath381 . for each of @xmath408 we have used @xmath382 clusters in @xmath131 . let @xmath419 denote the paths which lie in @xmath369 ( so @xmath420 ) . thus at most @xmath421 clusters in @xmath131 already lie in the paths and cycles @xmath394 . so for @xmath381 to satisfy ( iv ) , the inneighbour of @xmath397 on @xmath381 must not be one of these clusters . note that @xmath397 receives at most @xmath422 edges in @xmath64 from these clusters . thus in total we can not use @xmath423 of the edges which @xmath397 receives from @xmath131 in @xmath64 . but @xmath424 and so we can still choose a suitable cluster @xmath425 in @xmath426 which will play the role of the inneighbour of @xmath397 on @xmath381 . let @xmath427 denote the corresponding free edge in @xmath64 which we will use in @xmath381 . a similar argument shows that we can find a cluster @xmath428 to play the role of the outneighbour of @xmath398 on @xmath381 . so @xmath429 , @xmath430 does not lie on any of @xmath431 and there is a free edge @xmath432 in @xmath64 . we need to choose the outneighbour @xmath433 of @xmath430 on @xmath381 such that @xmath434 , @xmath433 has not been used in @xmath435 and there is a free edge from @xmath430 to @xmath433 in @xmath64 . let @xmath436 denote the set of all clusters in @xmath437 which satisfy these conditions . since @xmath438 at most @xmath439 edges at @xmath430 in @xmath404 $ ] are not free . so @xmath430 sends out at least @xmath440 free edges to @xmath441 in @xmath64 . on the other hand , as before one can show that @xmath430 sends at most @xmath422 edges to clusters in @xmath437 which already lie in @xmath431 . hence , @xmath442\geq ( 1/2 - 3c)|x|$ ] . similarly we need to choose the inneighbour @xmath443 of @xmath425 on @xmath381 such that @xmath444 , @xmath443 has not been used in @xmath431 and so that @xmath64 contains a free edge from @xmath443 to @xmath425 . let @xmath445 denote the set of all clusters in @xmath437 which satisfy these conditions . as before one can show that @xmath446 . recall that @xmath300)\geq ( 1/2 - 5d)|x|/\beta$ ] by our choice of @xmath131 . thus lemma [ keevashmult](i ) implies that @xmath404 $ ] contains at least @xmath447 edges from @xmath436 to @xmath445 . since all but at most @xmath448 edges of @xmath64 are free , there is a free edge @xmath449 from @xmath436 to @xmath445 . let @xmath450 be a free edge from @xmath430 to @xmath433 in @xmath64 and let @xmath451 be a free edge from @xmath443 to @xmath452 ( such edges exist by definition of @xmath436 and @xmath445 ) . we take @xmath381 to be the directed path or cycle which consists of the edges @xmath432 , @xmath453 , @xmath454 , @xmath455 and @xmath456 . frieze and krivelevich @xcite showed that every @xmath457-super - regular pair @xmath458 contains a regular subgraph @xmath459 whose density is almost the same as that of @xmath458 . the following lemma is an extension of this , where we can require @xmath459 to have a given degree sequence , as long as this degree sequence is almost regular . [ fandk ] let @xmath460 . suppose that @xmath461 is an @xmath462-super - regular pair where @xmath463 . define @xmath464 . suppose we have a non - negative integer @xmath465 associated with each @xmath466 and a non - negative integer @xmath467 associated with each @xmath468 such that @xmath469 . then @xmath458 contains a spanning subgraph @xmath470 in which @xmath471 is the degree of @xmath466 and @xmath472 is the degree of @xmath468 . we first obtain a directed network @xmath143 from @xmath458 by adding a source @xmath364 and a sink @xmath473 . we add an edge @xmath474 of capacity @xmath475 for each @xmath466 and an edge @xmath476 of capacity @xmath477 for each @xmath468 . we give all the edges in @xmath458 capacity @xmath48 and direct them from @xmath62 to @xmath403 . our aim is to show that the capacity of any cut is at least @xmath478 . by the max - flow min - cut theorem this would imply that @xmath143 admits a flow of value @xmath479 , which by construction of @xmath143 implies the existence of our desired subgraph @xmath459 . so consider any @xmath480-cut @xmath481 where @xmath482 with @xmath483 and @xmath484 . let @xmath485 and @xmath486 the capacity of this cut is @xmath487 and so our aim is to show that @xmath488 now @xmath489 by ( [ aim1 ] ) we may assume that @xmath490 . ( since otherwise @xmath491 and thus ( [ aim ] ) is satisfied . ) similarly by ( [ aim2 ] ) we may assume that @xmath492 . let @xmath493 . we now consider several cases . * case 1 . * @xmath494 and @xmath495 since @xmath458 is @xmath496-super - regular we have that @xmath497 ( the last inequality follows since @xmath498 . ) together with ( [ aim1 ] ) this implies ( [ aim ] ) . * @xmath494 , @xmath499 and @xmath500 again since @xmath458 is @xmath496-super - regular we have that @xmath501 as before , to prove ( [ aim ] ) we will show that @xmath502 thus by ( [ eqaim ] ) it suffices to show that @xmath503 . we know that @xmath504 since @xmath505 hence , @xmath506 . so @xmath503 as @xmath500 so indeed ( [ aim ] ) is satisfied . * * @xmath494 , @xmath499 and @xmath507 by ( [ aim2 ] ) in order to prove ( [ aim ] ) it suffices to show that @xmath508 since ( [ eqaim ] ) also holds in this case , this means that it suffices to show that @xmath509 . since @xmath510 and @xmath511 we have that @xmath512 . thus @xmath513 and so indeed ( [ aim ] ) holds . * @xmath514 since @xmath490 we have that @xmath515 . hence , @xmath516 and so by ( [ aim1 ] ) we see that ( [ aim ] ) is satisfied , as desired . * @xmath517 . similarly as in case 4 it follows that @xmath518 . indeed , if @xmath519 then @xmath520 and as @xmath521 this implies @xmath522 , a contradiction . it is easy to see that every regular oriented graph @xmath1 contains a @xmath48-factor . the following result states that if @xmath1 is also dense , then ( i ) we can guarantee a @xmath48-factor with few cycles . such @xmath48-factors have the advantage that we can transform them into a hamilton cycle by adding / deleting a comparatively small number of edges . ( ii ) implies that even if @xmath1 contains a sparse ` bad ' subgraph @xmath49 , then there will be a @xmath48-factor which does not contain ` too many ' edges of @xmath49 . [ 1factororiented ] let @xmath523 and @xmath524 . let @xmath1 be a @xmath525-regular oriented graph whose order @xmath2 is sufficiently large and where @xmath526 . suppose @xmath527 are sets of vertices in @xmath1 with @xmath528 . let @xmath49 be an oriented subgraph of @xmath1 such that @xmath529 for all @xmath530 ( for each @xmath82 ) . then @xmath1 has a @xmath48-factor @xmath531 such that * @xmath531 contains at most @xmath532 cycles ; * for each @xmath82 , at most @xmath533 edges of @xmath534 are incident to @xmath23 . to prove this result we will use ideas similar to those used by frieze and krivelevich @xcite . in particular , we will use the following bounds on the number of perfect matchings in a bipartite graph . [ matchingbounds ] suppose that @xmath130 is a bipartite graph whose vertex classes have size @xmath2 and @xmath535 are the degrees of the vertices in one of these vertex classes . let @xmath536 denote the number of perfect matchings in @xmath130 . then @xmath537 furthermore , if @xmath130 is @xmath525-regular then @xmath538 the upper bound in theorem [ matchingbounds ] was proved by brgman @xcite . the lower bound is a consequence of the van der waerden conjecture which was proved independently by egorychev @xcite and falikman @xcite . we will deduce ( i ) from the following result in @xcite , which in turn is similar to lemma 2 in @xcite . [ 2factor ] for all @xmath539 there exists @xmath540 such that the following holds . let @xmath130 be a @xmath541-regular bipartite graph whose vertex classes @xmath62 and @xmath63 satisfy @xmath542 . let @xmath543 be any perfect matching from @xmath62 to @xmath63 which is disjoint from @xmath130 . let @xmath544 be a perfect matching chosen uniformly at random from the set of all perfect matchings in @xmath130 . let @xmath545 be the resulting @xmath546-factor . then the probability that @xmath531 contains more than @xmath532 cycles is at most @xmath547 . * proof of lemma [ 1factororiented ] . * consider the @xmath525-regular bipartite graph @xmath130 whose vertex classes @xmath548 are copies of @xmath549 and where @xmath550 is joined to @xmath551 if @xmath105 is a directed edge in @xmath1 . note that every perfect matching in @xmath130 corresponds to a @xmath48-factor of @xmath1 and vice versa . let @xmath536 denote the number of perfect matchings of @xmath130 . then @xmath552 by theorem [ matchingbounds ] . here we have also used stirling s formula which implies that for sufficiently large @xmath80 , @xmath553 we now count the number @xmath554 of @xmath48-factors of @xmath1 which contain more than @xmath533 edges of @xmath49 which are incident to @xmath23 . note that @xmath555 indeed , the term @xmath556 in ( [ mured1 ] ) gives an upper bound for the number of ways we can choose @xmath533 edges from @xmath49 which are incident to @xmath23 such that no two of these edges have the same startpoint and no two of these edges have the same endpoint . the term @xmath557 in ( [ mured1 ] ) uses the upper bound in theorem [ matchingbounds ] to give a bound on the number of @xmath48-factors in @xmath1 containing @xmath558 fixed edges . now @xmath559 since @xmath560 and @xmath561 since @xmath562 . furthermore , @xmath563 so by ( [ mured1 ] ) we have that @xmath564 since @xmath565 , @xmath566 and @xmath2 is sufficiently large . now we apply lemma [ 2factor ] to @xmath130 where @xmath543 is the identity matching ( i.e. every vertex in @xmath567 is matched to its copy in @xmath568 ) . then a cycle of length @xmath569 in @xmath570 corresponds to a cycle of length @xmath39 in @xmath1 . so , since @xmath2 is sufficiently large , the number of @xmath48-factors of @xmath1 containing more than @xmath532 cycles is at most @xmath571 . so there exists a @xmath48-factor @xmath531 of @xmath1 which satisfies ( i ) and ( ii ) . the following lemma will be a useful tool when transforming @xmath48-factors into hamilton cycles . given such a @xmath48-factor @xmath531 , we will obtain a path @xmath56 by cutting up and connecting several cycles in @xmath531 ( as described in the proof sketch in section [ sketch ] ) . we will then apply the lemma to obtain a cycle @xmath55 containing precisely the vertices of @xmath56 . [ rotationlemma ] let @xmath572 . let @xmath1 be an oriented graph on @xmath573 vertices . suppose that @xmath62 and @xmath403 are disjoint subsets of @xmath549 of size @xmath80 with the following property : @xmath574 suppose that @xmath575 is a directed path in @xmath1 where @xmath576 and @xmath577 . let @xmath131 denote the set of inneighbours @xmath578 of @xmath579 which lie on @xmath56 so that @xmath466 and @xmath580 . similarly let @xmath132 denote the set of outneighbours @xmath578 of @xmath581 which lie on @xmath56 so that @xmath582 and @xmath583 . suppose that @xmath584 . then there exists a cycle @xmath55 in @xmath1 containing precisely the vertices of @xmath56 such that @xmath585 . furthermore , @xmath586 consists of edges from @xmath131 to @xmath587 and edges from @xmath588 to @xmath132 . ( here @xmath587 is the set of successors of vertices in @xmath131 on @xmath56 and @xmath588 is the set of predecessors of vertices in @xmath132 on @xmath56 . ) clearly we may assume that @xmath589 . let @xmath590 denote the set of the first @xmath591 vertices in @xmath131 along @xmath56 and @xmath592 the set of the last @xmath593 vertices in @xmath131 along @xmath56 . we define @xmath594 and @xmath595 analogously . so @xmath596 and @xmath597 . we have two cases to consider . * case 1 . * all the vertices in @xmath590 precede those in @xmath595 along @xmath56 . partition @xmath598 where @xmath599 denotes the set of the first @xmath600 vertices in @xmath590 along @xmath56 . we partition @xmath595 into @xmath601 and @xmath602 analogously . let @xmath603 denote the set of successors on @xmath56 of the vertices in @xmath604 and @xmath605 the set of predecessors of the vertices in @xmath601 . so @xmath606 and @xmath607 . further define * @xmath608 and * @xmath609 . so @xmath610 and @xmath611 . from ( [ label ] ) it follows that @xmath612 and similarly @xmath613 . since @xmath610 and @xmath611 , by ( [ label ] ) @xmath1 contains an edge @xmath614 from @xmath615 to @xmath616 . since @xmath617 , by definition of @xmath616 it follows that @xmath1 contains an edge @xmath618 for some @xmath619 . likewise , since @xmath620 , there is an edge @xmath621 for some @xmath622 . furthermore , @xmath623 and @xmath624 are edges of @xmath1 by definition of @xmath625 and @xmath605 . it is easy to check that the cycle @xmath626 has the required properties ( see figure 1 ) . for example , @xmath586 consists of the edges @xmath627 , @xmath628 , @xmath629 and @xmath630 . the former two edges go from @xmath131 to @xmath587 and the latter two from @xmath588 to @xmath132 . [ fig : rotation ] [ ] [ ] @xmath579 [ ] [ ] @xmath631 [ ] [ ] @xmath632 [ ] [ ] @xmath633 [ ] [ ] @xmath634 [ ] [ ] @xmath635 [ ] [ ] @xmath636 [ ] [ ] @xmath637 [ ] [ ] @xmath638 [ ] [ ] @xmath581 from case 1,title="fig : " ] * case 2 . * all the vertices in @xmath594 precede those in @xmath592 along @xmath56 . let @xmath639 be the predecessors of the vertices in @xmath594 and @xmath640 the successors of the vertices in @xmath592 on @xmath56 . so @xmath641 and @xmath642 and @xmath643 . thus by ( [ label ] ) there exists an edge @xmath644 from @xmath645 to @xmath646 . again , it is easy to check that the cycle @xmath647 has the desired properties . suppose @xmath61 is a digraph and @xmath531 is a collection of vertex - disjoint cycles with @xmath648 . a _ closed shifted walk @xmath63 in @xmath61 with respect to @xmath531 _ is a walk in @xmath649 of the form @xmath650 where * @xmath651 is the set of all cycles in @xmath531 ; * @xmath475 lies on @xmath72 and @xmath652 is the successor of @xmath475 on @xmath72 for each @xmath653 ; * @xmath654 is an edge of @xmath61 ( here @xmath655 ) . note that the cycles @xmath656 are not necessarily distinct . if a cycle @xmath72 in @xmath531 appears exactly @xmath473 times in @xmath63 we say that @xmath72 is _ traversed @xmath473 times_. note that a closed shifted walk @xmath63 has the property that for every cycle @xmath55 of @xmath531 , every vertex of @xmath55 is visited the same number of times by @xmath63 . the next lemma will be used in section [ merging ] to combine cycles of @xmath1 which correspond to different cycles of @xmath531 into a single ( hamilton ) cycle . shifted walks were introduced in @xcite , where they were used for a similar purpose . [ shiftedwalk ] let @xmath657 . suppose that @xmath1 is an oriented graph of order @xmath2 with @xmath295 . let @xmath61 denote the reduced digraph of @xmath1 with parameters @xmath658 and @xmath176 obtained by applying lemma [ dilemma ] . let @xmath659 . let @xmath303 denote the spanning subgraph of @xmath61 obtained by deleting all edges which correspond to pairs of density at most @xmath172 in the pure digraph @xmath52 . let @xmath531 be a collection of vertex - disjoint cycles with @xmath660 and @xmath661 . then @xmath303 contains a closed shifted walk with respect to @xmath531 so that each cycle @xmath55 in @xmath531 is traversed at most @xmath662 times . let @xmath663 denote the cycles of @xmath531 . we construct our closed shifted walk @xmath63 as follows : for each cycle @xmath72 , choose an arbitrary vertex @xmath664 lying on @xmath72 and let @xmath665 denote its successor on @xmath72 . let @xmath666 and let @xmath667 be the set of predecessors of @xmath668 on @xmath531 . similarly , let @xmath669 and let @xmath670 be the set of successors of @xmath60 on @xmath531 . since @xmath304 by lemma [ keevashmult](ii ) , we have @xmath671 and @xmath672 . so by lemma [ keevashmult](ii ) there is an edge @xmath673 from @xmath667 to @xmath674 in @xmath303 . then we obtain a walk @xmath675 from @xmath676 to @xmath677 by first traversing @xmath72 to reach @xmath664 , then use the edge from @xmath664 to the successor @xmath578 of @xmath678 , then traverse the cycle in @xmath531 containing @xmath578 as far as @xmath679 , then use the edge @xmath673 , then traverse the cycle in @xmath531 containing @xmath680 as far as @xmath681 , and finally use the edge @xmath682 . ( here @xmath683 . ) @xmath63 is obtained by concatenating the @xmath675 . without loss of generality we may assume that @xmath684 . define further constants satisfying @xmath685 let @xmath1 be an oriented graph of order @xmath686 such that @xmath687 . apply the diregularity lemma ( lemma [ dilemma ] ) to @xmath1 with parameters @xmath231 and @xmath176 to obtain clusters @xmath688 of size @xmath80 , an exceptional set @xmath85 , a pure digraph @xmath52 and a reduced digraph @xmath61 ( so @xmath689 ) . let @xmath303 be the spanning subdigraph of @xmath61 whose edges correspond to pairs of density at least @xmath172 . so @xmath232 is an edge of @xmath303 if @xmath190 has density at least @xmath172 . let @xmath64 denote the reduced multidigraph of @xmath1 with parameters @xmath236 and @xmath176 . for each edge @xmath232 of @xmath61 let @xmath233 denote the density of the @xmath162-regular pair @xmath190 . recall that each edge @xmath690 is associated with the @xmath32th spanning subgraph @xmath243 of @xmath190 obtained by applying lemma [ split ] with parameters @xmath691 and @xmath692 . each @xmath243 is @xmath162-regular with density @xmath244 . lemma [ multimin ] implies that @xmath693 ( the second inequality holds since @xmath694 . ) apply lemma [ multifactor1 ] to @xmath64 in order to obtain @xmath695 edge - disjoint collections @xmath69 of vertex - disjoint cycles in @xmath64 such that each @xmath70 contains all but at most @xmath346 of the clusters in @xmath64 . let @xmath696 denote the set of all those vertices in @xmath1 which do not lie in clusters covered by @xmath70 . so @xmath697 for all @xmath84 and @xmath698 . we now apply lemma [ superreg ] to each cycle in @xmath70 to obtain subclusters of size @xmath264 such that the edges of @xmath70 now correspond to @xmath266-super - regular pairs . by removing one extra vertex from each cluster if necessary we may assume that @xmath274 is even . all vertices not belonging to the chosen subclusters of @xmath70 are added to @xmath696 . so now latexmath:[\ ] ] we proceed similarly for all vertices in @xmath760 , with the random choices being independent for different vertices @xmath781 . ( note that every edge of @xmath1 is free with respect to at most one vertex in @xmath760 . ) then using the lower bound on @xmath757 for all @xmath781 we have @xmath790 for each @xmath711 and all @xmath752 . as before , applying the chernoff type bound in proposition [ chernoff ] for each @xmath82 and summing up the failure probabilities over all @xmath82 shows that with nonzero probability the following properties hold : * ( [ freedeg])([freein ] ) imply that @xmath791 for each @xmath792 . * ( [ inc ] ) implies that @xmath793 for each @xmath752 . together with the properties of @xmath86 established after choosing the edges at the vertices in @xmath768 it follows that @xmath794 for every @xmath795 and @xmath796 for every @xmath752 . furthermore , @xmath769 are still edge - disjoint since when dealing with the vertices in @xmath760 we only added free edges . by discarding any edges assigned to @xmath86 which lie entirely in @xmath696 we can ensure that ( i ) holds . so altogether ( i)(iii ) are satisfied , as desired . as mentioned in the previous section we will use each of the @xmath86 to piece together roughly @xmath748 hamilton cycles of @xmath1 . we will achieve this by firstly adding some more special edges to each @xmath86 ( see section [ skel ] ) and then almost decomposing each @xmath86 into @xmath48-factors . however , in order to use these @xmath48-factors to create hamilton cycles we will need to ensure that no @xmath48-factor contains a @xmath546-path with start- and endpoint in @xmath696 , and midpoint in @xmath72 . unfortunately @xmath86 might contain such paths . to avoid them , we will ` randomly split ' each @xmath86 . we start by considering a random partition of each @xmath797 . using the chernoff bound in proposition [ chernoff ] for the hypergeometric distribution one can show that there exists a partition of @xmath403 into subclusters @xmath798 and @xmath799 so that the following conditions hold : * @xmath800 for each @xmath797 . * @xmath801 and @xmath802 for each @xmath750 . ( here @xmath803 and @xmath804 . ) recall that each edge @xmath805 corresponds to the @xmath739-super - regular pair @xmath737 . let @xmath806 . so @xmath807 apply lemma [ split](ii ) to obtain a partition @xmath808 of the edge set of @xmath737 so that the following condition holds : * the edges of @xmath809 and @xmath810 both induce an @xmath811-super - regular pair which spans @xmath737 . we now partition @xmath86 into two oriented spanning subgraphs @xmath812 and @xmath813 as follows . * the edge set of @xmath812 is the union of all @xmath809 ( over all edges @xmath701 of @xmath70 ) together with all the edges in @xmath86 from @xmath696 to @xmath814 , and all edges in @xmath86 from @xmath815 to @xmath696 . * the edge set of @xmath813 is the union of all @xmath810 ( over all edges @xmath701 of @xmath70 ) together with all the edges in @xmath86 from @xmath696 to @xmath815 , and all edges in @xmath86 from @xmath814 to @xmath696 . note that neither @xmath812 nor @xmath813 contains the type of @xmath546-paths we wish to avoid . for each @xmath711 we use lemma [ split](ii ) to partition the edge set of each @xmath97 to obtain edge - disjoint oriented spanning subgraphs @xmath816 and @xmath817 so that the following condition holds : * for each edge @xmath701 in @xmath70 , both @xmath816 and @xmath817 contain a spanning oriented subgraph of @xmath722 which is @xmath818-super - regular . moreover , all edges in @xmath816 and @xmath817 belong to one of these pairs . similarly we partition the edge set of each @xmath99 to obtain edge - disjoint oriented spanning subgraphs @xmath819 and @xmath820 so that the following condition holds : * for each edge @xmath701 in @xmath70 , both @xmath819 and @xmath820 contain a spanning oriented subgraph of @xmath722 which is @xmath821-super - regular . moreover , all edges in @xmath819 and @xmath820 belong to one of these pairs . we pair @xmath816 and @xmath819 with @xmath812 and pair @xmath817 and @xmath820 with @xmath813 . we now have @xmath822 edge - disjoint oriented subgraphs of @xmath1 , namely @xmath823 . to simplify notation , we relabel these oriented graphs as @xmath824 where @xmath825 we similarly relabel the oriented graphs @xmath826 as @xmath827 and relabel @xmath828 as @xmath829 in such a way that @xmath97 and @xmath99 are the oriented graphs which we paired with @xmath86 . for each @xmath82 we still use the notation @xmath70 , @xmath72 and @xmath696 in the usual way . now ( i ) from section [ sec : incorp ] becomes * @xmath830 where @xmath103 has neighbours only in @xmath72 , for all @xmath750 , while ( ii ) and ( iii ) remain valid . note that all vertices ( including the vertices of @xmath696 ) in each @xmath86 now have in- and outdegree close to @xmath831 . in section [ nicefactor ] our aim is to find a @xmath832-regular oriented subgraph of @xmath86 , where @xmath833 however , this may not be possible : suppose for instance that @xmath696 consists of a single vertex @xmath103 , @xmath70 consists of 2 cycles @xmath55 and @xmath59 and that all outneighbours of @xmath103 lie on @xmath55 and all inneighbours lie on @xmath59 . then @xmath86 does not even contain a @xmath48-factor . a similar problem arises if for example @xmath696 consists of a single vertex @xmath103 , @xmath70 consists of a single cycle @xmath834 , all outneighbours of @xmath103 lie in the cluster @xmath568 and all inneighbours in the cluster @xmath835 . note that in both situations , the edges between @xmath696 and @xmath72 are not ` well - distributed ' or ` balanced ' . to overcome this problem , we add further edges to @xmath72 which will ` balance out ' the edges between @xmath72 and @xmath696 which we added previously . these edges will be part of the skeleton walks which we define below . to motivate the definition of the skeleton walks it may be helpful to consider the second example above : suppose that we add an edge @xmath65 from @xmath567 to @xmath836 . then @xmath86 now has a @xmath48-factor . in general , we can not find such an edge , but it will turn out that we can find a collection of 5 edges fulfilling the same purpose . a _ skeleton walk _ @xmath837 in @xmath1 with respect to @xmath86 is a collection of distinct edges @xmath838 , @xmath839 , @xmath840 , @xmath841 and @xmath842 of @xmath1 with the following properties : * @xmath843 and all vertices in @xmath844 lie in @xmath72 . * given some @xmath845 , let @xmath846 denote the cluster in @xmath70 containing @xmath847 and let @xmath55 denote the cycle in @xmath70 containing @xmath403 . then @xmath848 , where @xmath849 is the predecessor of @xmath403 on @xmath55 . note that whenever @xmath850 is a union of edge - disjoint skeleton walks and @xmath851 is a cluster in @xmath70 , the number of edges in @xmath850 whose endpoint is in @xmath403 is the same as the number of edges in @xmath850 whose startpoint is in @xmath852 . as indicated above , this ` balanced ' property will be crucial when finding a @xmath832-regular oriented subgraph of @xmath86 in section [ nicefactor ] . the 2nd , 3rd and 4th edge of each skeleton walk @xmath853 with respect to @xmath86 will lie in the ` random ' graph @xmath89 chosen in section [ applydrl ] . more precisely , each of these three edges will lie in a ` slice ' @xmath854 of @xmath89 assigned to @xmath86 . we will now partition @xmath89 into these ` slices ' @xmath855 . to do this , recall that any edge @xmath701 in @xmath64 corresponds to an @xmath162-regular pair of density at least @xmath718 in @xmath89 . here @xmath705 and @xmath706 are viewed as clusters in @xmath64 , so @xmath856 . apply lemma [ split](i ) to each such pair of clusters to find edge - disjoint oriented subgraphs @xmath857 of @xmath89 so that for each @xmath854 all the edges @xmath701 in @xmath64 correspond to @xmath858$]-regular pairs with density at least @xmath859 in @xmath854 . recall that by ( i@xmath860 ) in section [ randomsplit ] each vertex @xmath861 has at least @xmath862 outneighbours in @xmath72 and at least @xmath863 inneighbours in @xmath72 . we pair @xmath832 of these outneighbours @xmath864 with distinct inneighbours @xmath865 . for each of these @xmath832 pairs @xmath866 we wish to find a skeleton walk with respect to @xmath86 whose @xmath48st edge is @xmath867 and whose @xmath378th edge is @xmath868 . we denote the union of these @xmath832 pairs @xmath869 of edges over all @xmath795 by @xmath870 . in section [ randomsplit ] we partitioned each cluster @xmath797 into subclusters @xmath798 and @xmath799 . we next show how to choose the skeleton walks for all those @xmath86 for which each edge in @xmath86 with startpoint in @xmath696 has its endpoint in @xmath814 ( and so each edge in @xmath86 with endpoint in @xmath696 has startpoint in @xmath815 ) . the other case is similar , one only has to interchange @xmath814 and @xmath815 . [ skelg ] we can find a set @xmath871 of @xmath872 skeleton walks with respect to @xmath86 , one for each pair of edges in @xmath870 , such that @xmath871 has the following properties : * for each skeleton walk in @xmath871 , its @xmath546nd , @xmath873rd and @xmath382th edge all lie in @xmath854 and all these edges have their startpoint in @xmath815 and endpoint in @xmath814 . * any two of the skeleton walks in @xmath871 are edge - disjoint . * every @xmath752 is incident to at most @xmath874 edges belonging to the skeleton walks in @xmath871 . note that @xmath875 by ( [ v0 ] ) and ( [ tau ] ) . to find @xmath871 , we will first find so - called shadow skeleton walks ( here the internal edges are edges of @xmath64 instead of @xmath1 ) . more precisely , a _ shadow skeleton walk _ @xmath876 with respect to @xmath86 is a collection of two edges @xmath838 , @xmath842 of @xmath1 and three edges @xmath877 , @xmath878 , @xmath879 of @xmath64 with the following properties : * @xmath838 , @xmath842 is a pair in @xmath870 . * @xmath880 , @xmath881 and each @xmath882 is a vertex of a cycle in @xmath70 and @xmath883 is the predecessor of @xmath882 on that cycle . note that in the second condition we slightly abused the notation : as @xmath882 is a cluster in @xmath64 , it only corresponds to a cluster in @xmath70 ( which has size @xmath274 and is a subcluster of the one in @xmath64 ) . however , in order to simplify our exposition , we will use the same notation for a cluster in @xmath64 as for the cluster in @xmath70 corresponding to it . we refer to the edge @xmath884 as the @xmath350th edge of the shadow skeleton walk @xmath876 . given a collection @xmath885 of shadow skeleton walks ( with respect to @xmath86 ) we say an edge of @xmath64 is _ bad _ if it is used at least @xmath886 times in @xmath885 , and _ very bad _ if it is used at least @xmath887 times in @xmath885 . we say an edge from @xmath403 to @xmath62 in @xmath64 is _ @xmath888-bad _ if it is used at least @xmath130 times as a @xmath546nd edge in the shadow skeleton walks of @xmath885 . an edge from @xmath63 to @xmath403 in @xmath64 is _ @xmath889-bad _ if it is used at least @xmath130 times as a @xmath382th edge in the shadow skeleton walks of @xmath885 . to prove claim [ skelg ] we will first prove the following result . [ shadow ] we can find a collection @xmath890 of @xmath872 shadow skeleton walks with respect to @xmath86 , one for each of pair in @xmath870 , such that the following condition holds : * for each @xmath891 , every edge in @xmath64 is used at most @xmath130 times as a @xmath350th edge of some shadow skeleton walk in @xmath890 . in particular no edge in @xmath64 is very bad . suppose that we have already found @xmath892 of our desired shadow skeleton walks for @xmath86 . let @xmath869 be a pair in @xmath870 for which we have yet to define a shadow skeleton walk . we will now find such a shadow skeleton walk @xmath876 . suppose @xmath893 and @xmath894 , where @xmath895 . let @xmath403 denote the predecessor of @xmath896 in @xmath897 and @xmath63 the successor of @xmath898 in @xmath70 . we define @xmath899 to consist of all those clusters @xmath900 for which there exists an edge from @xmath403 to @xmath62 in @xmath64 which is not @xmath888-bad . by definition of @xmath86 ( condition ( ii ) in section [ sec : incorp ] ) , each @xmath752 has at most @xmath901 inneighbours in @xmath696 in @xmath86 . so the number of @xmath888-bad edges is at most @xmath902 together with ( [ rmdeg ] ) this implies that @xmath903 similarly we define @xmath904 to consist of all those clusters @xmath900 for which there exists an edge from @xmath62 to @xmath63 in @xmath64 which is not @xmath905-bad . again , @xmath906 . let @xmath907 denote the set of those clusters which are the predecessors in @xmath70 of a cluster in @xmath899 . similarly let @xmath908 denote the set of those clusters which are the successors in @xmath70 of a cluster in @xmath904 . so @xmath909 and @xmath910 . by lemma [ keevashmult](i ) applied with @xmath911 there exist at least @xmath912 edges in @xmath64 from @xmath907 to @xmath908 . on the other hand , the number of bad edges is at most @xmath913 so we can choose an edge @xmath914 from @xmath907 to @xmath908 in @xmath64 which is not bad . let @xmath587 denote the successor of @xmath131 in @xmath70 and @xmath588 the predecessor of @xmath132 in @xmath70 . thus @xmath915 and @xmath916 and so there is an edge @xmath917 in @xmath64 which is not @xmath888-bad and an edge @xmath918 which is not @xmath905-bad . let @xmath876 be the shadow skeleton walk consisting of the edges @xmath867 , @xmath917 , @xmath914 , @xmath918 , and @xmath868 . then we can add @xmath876 to our collection of @xmath39 skeleton walks that we have found already . we now use claim [ shadow ] to prove claim [ skelg ] . 55*proof of claim [ skelg ] . * we apply claim [ shadow ] to obtain a collection @xmath890 of shadow skeleton walks . we will replace each edge of @xmath64 in these shadow skeleton walks with a distinct edge of @xmath854 to obtain our desired collection @xmath871 of skeleton walks . recall that each edge @xmath919 in @xmath64 corresponds to an @xmath920$]-regular pair of density at least @xmath921 in @xmath854 . thus in @xmath854 the edges from @xmath799 to @xmath922 induce a @xmath923$]-regular pair of density @xmath924 . ( here @xmath925 and @xmath926 are the partitions of @xmath403 and @xmath63 chosen in section [ randomsplit ] . ) let @xmath927 and note that @xmath928 . so we can now apply lemma [ boundmax ] to @xmath929 to obtain a subgraph @xmath930 $ ] with maximum degree at most @xmath931 and at least @xmath932 edges . we do this for all those edges in @xmath64 which are used in a shadow skeleton walk in @xmath890 . since no edge in @xmath64 is very bad , for each @xmath933 we can replace an edge @xmath919 in @xmath876 with a distinct edge @xmath65 from @xmath799 to @xmath922 lying in @xmath930 $ ] . thus we obtain a collection @xmath871 of skeleton walks which satisfy properties ( i ) and ( ii ) of claim [ skelg ] . note that by the construction of @xmath871 every vertex @xmath752 is incident to at most @xmath934 edges which play the role of a @xmath546nd , @xmath873rd or @xmath382th edge in a skeleton walk in @xmath871 . condition ( ii ) in section [ sec : incorp ] implies that @xmath104 is incident to at most @xmath935 edges which play the role of a 1st or 5th edge in a skeleton walk in @xmath871 . so in total @xmath104 is incident to at most @xmath936 edges of the skeleton walks in @xmath871 . hence ( iii ) and thus the entire claim is satisfied . we now add the edges of the skeleton walks in @xmath871 to @xmath86 . moreover , for each @xmath795 we delete all those edges at @xmath103 which do not lie in a skeleton walk in @xmath871 . our aim in this section is to find a suitable collection of 1-factors in each @xmath86 which together cover almost all the edges of @xmath86 . in order to do this , we first choose a @xmath832-regular spanning oriented subgraph @xmath937 of @xmath86 and then apply lemma [ 1factororiented ] to @xmath937 . we will refer to all those edges in @xmath86 which lie in a skeleton walk in @xmath871 as _ red _ , and all other edges in @xmath86 as _ white_. given @xmath797 and @xmath938 , we denote by @xmath939 the set of all those vertices which receive a white edge from @xmath103 in @xmath86 . similarly we denote by @xmath940 the set of all those vertices which send out a white edge to @xmath103 in @xmath86 . so @xmath941 and @xmath942 , where @xmath896 and @xmath852 are the successor and the predecessor of @xmath403 in @xmath70 . note that @xmath86 has the following properties : * @xmath943 for each @xmath750 . moreover , @xmath103 does not have any in- or outneighbours in @xmath696 . * every path in @xmath86 consisting of two red edges has its midpoint in @xmath696 . * for each @xmath944 the white edges in @xmath86 from @xmath239 to @xmath945 induce a @xmath946-super - regular pair @xmath947 . * every vertex @xmath948 receives at most @xmath949 red edges and sends out at most @xmath950 red edges in @xmath86 . * in total , the vertices in @xmath86 lying in a cluster @xmath951 send out the same number of red edges as the vertices in @xmath945 receive . in order to find our @xmath832-regular spanning oriented subgraph of @xmath86 , consider any edge @xmath952 . given any @xmath953 , let @xmath954 denote the number of red edges sent out by @xmath955 in @xmath86 . similarly given any @xmath956 , let @xmath957 denote the number of red edges received by @xmath958 in @xmath86 . by @xmath959 we have that @xmath960 and by @xmath961 we have that @xmath962 thus we can apply lemma [ fandk ] to obtain an oriented spanning subgraph of @xmath947 in which each @xmath955 has outdegree @xmath963 and each @xmath958 has indegree @xmath964 . we apply lemma [ fandk ] to each @xmath952 . the union of all these oriented subgraphs together with the red edges in @xmath86 clearly yield a @xmath832-regular oriented subgraph @xmath965 of @xmath86 , as desired . we will use the following claim to almost decompose @xmath965 into @xmath48-factors with certain useful properties . [ nice1factors ] let @xmath966 be a spanning @xmath525-regular oriented subgraph of @xmath86 where @xmath967 . then @xmath966 contains a @xmath48-factor @xmath968 with the following properties : * @xmath968 contains at most @xmath532 cycles . * for each @xmath951 , @xmath968 contains at most @xmath969 red edges incident to vertices in @xmath239 . * let @xmath970 denote the set of vertices which are incident to a red edge in @xmath968 . then @xmath971 for each @xmath972 . * @xmath973 for each @xmath972 . a direct application of lemma [ 1factororiented ] to @xmath966 proves the claim . indeed , we apply the lemma with @xmath974 , @xmath975 , @xmath976 and with the oriented spanning subgraph of @xmath966 whose edge set consists precisely of the red edges in @xmath966 playing the role of @xmath49 . furthermore , the clusters in @xmath977 together with the sets @xmath978 and @xmath979 ( for each @xmath972 ) play the role of the @xmath980 . repeatedly applying claim [ nice1factors ] we obtain edge - disjoint @xmath48-factors @xmath981 of @xmath86 satisfying conditions ( i)(iv ) of the claim , where @xmath982 our aim is now to transform each of the @xmath90 into a hamilton cycle using the edges of @xmath97 , @xmath98 and @xmath99 . let @xmath983 denote the cycles in @xmath70 and define @xmath984 to be the set of vertices in @xmath86 which lie in clusters in the cycle @xmath985 . in this subsection , for each @xmath82 and @xmath350 we will merge the cycles in @xmath90 to obtain a @xmath48-factor @xmath94 consisting of at most @xmath986 cycles . recall from section [ nicefactor ] that we call the edges of @xmath86 which lie on a skeleton walk in @xmath871 red and the non - red edges of @xmath86 white . we call the edges of the ` random ' oriented graph @xmath97 defined in section [ applydrl ] _ green_. ( recall that @xmath97 was modified in section [ randomsplit ] . ) we will use the edges from @xmath97 to obtain @xmath48-factors @xmath987 for each @xmath86 with the following properties : * if @xmath988 or @xmath989 then @xmath94 and @xmath990 are edge - disjoint . * for each @xmath991 all @xmath992 which send out a white edge in @xmath90 lie on the same cycle @xmath55 in @xmath94 . * @xmath993 for all @xmath82 and @xmath350 . moreover , @xmath994 consists of green and white edges only . * for every edge in @xmath90 both endvertices lie on the same cycle in @xmath94 . * all the red edges in @xmath90 still lie in @xmath94 . before showing the existence of @xmath48-factors satisfying ( @xmath740)(@xmath995 ) , we will derive two further properties ( @xmath996 ) and ( @xmath997 ) from them which we will use in the next subsection . so suppose that @xmath94 is a @xmath48-factor satisfying the above conditions . consider any cluster @xmath846 . claim [ nice1factors](ii ) implies that @xmath90 contains at most @xmath998 red edges with startpoint in @xmath403 . so the cycle @xmath55 in @xmath94 which contains all vertices @xmath999 sending out a white edge in @xmath90 must contain at least @xmath1000 such vertices @xmath103 . in particular there are at least @xmath1001 vertices @xmath1002 which lie on @xmath55 . so some of these vertices @xmath104 send out a white edge in @xmath90 . but by @xmath1003 this means that @xmath55 contains all those vertices @xmath1002 which send out a white edge in @xmath90 . repeating this argument shows that @xmath55 contains all vertices in @xmath1004 which send out a white edge in @xmath90 ( here @xmath985 is the cycle on @xmath70 that contains @xmath403 ) . furthermore , by property @xmath1005 , @xmath55 contains all vertices in @xmath1004 which receive a white edge in @xmath90 . by property @xmath1006 in section [ nicefactor ] no vertex of @xmath72 is both the a startpoint of a red edge in @xmath86 and an endpoint of a red edge in @xmath86 . so this implies that all vertices in @xmath984 lie on @xmath55 . thus if we obtain @xmath48-factors @xmath1007 satisfying @xmath1008@xmath1009 then the following conditions also hold : * for each @xmath1010 and each @xmath1011 all the vertices in @xmath984 lie on the same cycle in @xmath94 . * for each @xmath846 and each @xmath1012 at most @xmath1013 vertices in @xmath403 lie on a red edge in @xmath94 . ( condition @xmath1014 follows from claim [ nice1factors](ii ) and the ` moreover ' part of @xmath1015 . ) for every @xmath82 , we will define the 1-factors @xmath987 sequentially . initially , we let @xmath1016 . so the @xmath94 satisfy all conditions except @xmath1017 . next , we describe how to modify @xmath1018 so that it also satisfies @xmath1019 ) . recall from section [ randomsplit ] that for each edge @xmath1020 of @xmath70 the pair @xmath1021 is @xmath1022-super - regular and thus @xmath1023 . furthermore , whenever @xmath797 and @xmath938 , the outneighbourhood of @xmath103 in @xmath97 lies in @xmath896 and the inneighbourhood of @xmath103 in @xmath97 lies in @xmath852 . let @xmath816 denote the oriented spanning subgraph of @xmath97 whose edge set consists of those edges @xmath105 of @xmath97 for which @xmath103 is not a startpoint of a red edge in our current @xmath48-factor @xmath1018 and @xmath104 is not an endpoint of a red edge in @xmath1018 . consider a white edge @xmath105 in @xmath1018 . claim [ nice1factors](iii ) implies that @xmath103 sends out at most @xmath1024 green edges @xmath1025 in @xmath97 which do not lie in @xmath816 . so @xmath1026 . similarly , @xmath1027 . ( however , if @xmath1028 is a red edge in @xmath1018 then @xmath1029 . ) thus we have the following properties of @xmath97 and @xmath816 : * for each @xmath846 all the edges in @xmath97 sent out by vertices in @xmath403 go to @xmath896 . * if @xmath105 is a white edge in @xmath1018 then @xmath1030 . * consider any @xmath797 . let @xmath1031 and @xmath1032 be such that @xmath1033 . then @xmath1034 . if @xmath1018 does not satisfy ( @xmath1035 ) , then it contains cycles @xmath1036 such that there is a cluster @xmath797 and white edges @xmath105 on @xmath55 and @xmath1037 on @xmath1038 with @xmath1039 and @xmath1040 . we have 3 cases to consider . firstly , we may have a green edge @xmath1041 such that @xmath1042 lies on a cycle @xmath1043 in @xmath1018 . then @xmath1044 and @xmath1042 is the endpoint of a white edge in @xmath1018 ( by @xmath1045 and the definition of @xmath816 ) . secondly , there may be a green edge @xmath1046 such that @xmath1047 lies on a cycle @xmath1048 in @xmath1018 . so here @xmath1049 is the startpoint of a white edge in @xmath1018 . if neither of these cases hold , then @xmath1050 lies on @xmath55 and @xmath1051 lies on @xmath1038 . since @xmath1052 by ( @xmath1053 ) , we can use ( @xmath1054 ) to find a green edge @xmath1055 from @xmath1051 to @xmath1050 . then @xmath1056 , @xmath1057 , @xmath1058 is the startpoint of a white edge in @xmath1018 and @xmath1059 is the endpoint of a white edge in @xmath1018 . we will only consider the first of these 3 cases . the other cases can be dealt with analogously : in the second case @xmath1047 plays the role of @xmath103 and @xmath1060 plays the role of @xmath1042 . in the third case @xmath1058 plays the role of @xmath103 and @xmath1059 plays the role of @xmath1042 . so let us assume that the first case holds , i.e. there is a green edge @xmath1041 such that @xmath1042 lies on a cycle @xmath1043 in @xmath1018 and @xmath1042 lies on a white edge @xmath1061 on @xmath59 . let @xmath56 denote the directed path @xmath1062 from @xmath1063 to @xmath1064 . suppose that the endpoint @xmath1047 of @xmath56 lies on a green edge @xmath1065 such that @xmath27 lies outside @xmath56 . then @xmath1066 is the endpoint of a white edge @xmath1028 lying on the cycle @xmath1067 in @xmath1018 which contains @xmath27 . we extend @xmath56 by replacing @xmath56 and @xmath1067 with @xmath1068 . we make similar extensions if the startpoint @xmath104 of @xmath56 has an inneighbour in @xmath816 outside @xmath56 . we repeat this ` extension ' procedure as long as we can . let @xmath56 denote the path obtained in this way , say @xmath56 joins @xmath1069 to @xmath1070 . note that @xmath57 must be the endpoint of a white edge in @xmath1018 and @xmath58 the startpoint of a white edge in @xmath1018 . we will now apply a ` rotation ' procedure to close @xmath56 into a cycle . by ( @xmath1053 ) @xmath57 has at least @xmath1071 inneighbours in @xmath816 and @xmath58 has at least @xmath1071 outneighbours in @xmath816 and all these in- and outneighbours lie on @xmath56 since we could not extend @xmath56 any further . let @xmath1072 and @xmath1073 . so @xmath1074 and @xmath1075 and @xmath1076 by @xmath1045 . moreover , whenever @xmath1077 and @xmath1078 is the successor of @xmath1079 on @xmath56 , then either @xmath1080 was a white edge in @xmath1018 or @xmath1081 . thus in both cases @xmath1082 . so the set @xmath587 of successors in @xmath56 of all the vertices in @xmath131 lies in @xmath896 and no vertex in @xmath131 sends out a red edge in @xmath56 . similarly one can show that the set @xmath588 of predecessors in @xmath56 of all the vertices in @xmath132 lies in @xmath403 and no vertex in @xmath132 receives a red edge in @xmath56 . together with ( @xmath1054 ) this shows that we can apply lemma [ rotationlemma ] with @xmath1083 playing the role of @xmath1 and @xmath896 playing the role of @xmath403 and @xmath403 playing the role of @xmath62 to obtain a cycle @xmath1084 containing precisely the vertices of @xmath56 such that @xmath1085 , @xmath1086 and such that @xmath1087 consists of edges from @xmath131 to @xmath587 and edges from @xmath588 to @xmath132 . thus @xmath1087 contains no red edges . replacing @xmath56 with @xmath1084 gives us a @xmath48-factor ( which we still call @xmath94 ) with fewer cycles . also note that if the number of cycles is reduced by @xmath39 , then we use at most @xmath1088 edges in @xmath97 to achieve this . so @xmath94 still satisfies all requirements with the possible exception of ( @xmath1035 ) . if it still does not satisfy ( @xmath1035 ) , we will repeatedly apply this ` rotation - extension ' procedure until the current @xmath48-factor @xmath1018 also satisfies ( @xmath1035 ) . however , we need to be careful since we do not want to use edges of @xmath97 several times in this process . simply deleting the edges we use may not work as ( @xmath1089 ) might fail later on ( when we will repeat the above process for @xmath94 with @xmath1090 ) . so each time we modify @xmath1018 , we also modify @xmath97 as follows . all the edges from @xmath97 which are used in @xmath1018 are removed from @xmath97 . all the edges which are removed from @xmath1018 in the rotation - extension procedure are added to @xmath97 . ( note that by @xmath1091 we never add red edges to @xmath97 . ) when we refer to @xmath97 , we always mean the ` current ' version of @xmath97 , not the original one . furthermore , at every step we still refer to an edge of @xmath97 as green , even if initially the edge did not lie in @xmath97 . similarly at every step we refer to the non - red edges of our current @xmath48-factor as white , even if initially they belonged to @xmath97 . note that if we added a green edge @xmath1025 into @xmath1018 , then @xmath103 lost an outneighbour in @xmath97 , namely @xmath1042 . however , @xmath1009 implies that we also moved some ( white ) edge @xmath105 of @xmath1018 to @xmath97 , where @xmath104 lies in the same cluster @xmath1092 as @xmath1042 ( here @xmath992 ) . so we still have that @xmath1093 . similarly , at any stage @xmath1094 . when @xmath97 is modified , then @xmath816 is modified accordingly . this will occur if we add some white edges to @xmath97 whose start or endpoint lies on a red edge in @xmath1018 . however , claim [ nice1factors](iv ) implies that at any stage we still have @xmath1095 also note that by ( @xmath1096 ) , the modified version of @xmath97 still satisfies @xmath1097 so @xmath97 and @xmath816 will satisfy ( @xmath1098)(@xmath1099 ) throughout and thus the above argument still works . so after at most @xmath532 steps @xmath1018 will also satisfy ( @xmath1035 ) . suppose that for some @xmath1100 we have found @xmath48-factors @xmath1101 satisfying @xmath1008@xmath1009 . we can now carry out the rotation - extension procedure for @xmath94 in the same way as for @xmath1018 until @xmath94 also satisfies ( @xmath1035 ) . in the construction of @xmath94 , we do not use the original @xmath97 , but the modified version obtained in the construction of @xmath1102 . we then introduce the oriented spanning subgraph @xmath816 of @xmath97 similarly as before ( but with respect to the current @xmath48-factor @xmath94 ) . then all the above bounds on these graphs still hold , except that in the middle expression of ( [ h3i ] ) we multiply the term @xmath1103 by @xmath350 to account for the total number of edges removed from @xmath97 so far . but this does not affect the next inequality . so eventually , all the @xmath94 will satisfy ( @xmath740)(@xmath995 ) . our final aim is to piece together the cycles in @xmath94 , for each @xmath82 and @xmath350 , to obtain edge - disjoint hamilton cycles of @xmath1 . since we have @xmath1104 @xmath48-factors @xmath1007 for each @xmath86 , in total we will find @xmath1105 edge - disjoint hamilton cycles of @xmath1 , as desired . recall that @xmath303 was defined in section [ applydrl ] . given any @xmath82 , apply lemma [ shiftedwalk ] to obtain a closed shifted walk @xmath1106 in @xmath303 with respect to @xmath70 such that each cycle in @xmath70 is traversed at most @xmath662 times . so @xmath1107 is the set of all cycles in @xmath70 , @xmath1108 is the successor of @xmath1109 on @xmath1110 and @xmath1111 for each @xmath1112 ( where @xmath1113 ) . moreover , @xmath1114 for each @xmath48-factor @xmath94 we will now use the edges of @xmath98 and @xmath99 to obtain a hamilton cycle @xmath1115 with the following properties : * if @xmath1116 or @xmath989 then @xmath1115 and @xmath1117 are edge - disjoint . * @xmath1118 consists of edges from @xmath94 , @xmath98 and @xmath99 only . * there are at most @xmath1119 edges from @xmath98 lying in @xmath1115 . * there are at most @xmath1120 edges from @xmath99 lying in @xmath1115 . for each @xmath350 , we will use @xmath1121 to ` guide ' us how to merge the cycles in @xmath94 into the hamilton cycle @xmath1115 . suppose that we have already defined @xmath1122 of the hamilton cycles @xmath1117 satisfying ( i)(iv ) , but have yet to define @xmath1115 . we remove all those edges which have been used in these @xmath39 hamilton cycles from both @xmath98 and @xmath99 . for each @xmath797 , we denote by @xmath1123 the subcluster of @xmath403 containing all those vertices which do not lie on a red edge in @xmath94 . we refer to @xmath1123 as the _ white subcluster of @xmath403_. thus @xmath1124 by property @xmath1014 in section [ 4.6 ] . note that the outneighbours of the vertices in @xmath1123 on @xmath94 all lie in @xmath896 while their inneighbours lie in @xmath852 . for each @xmath1125 we will denote the white subcluster of a cluster @xmath1109 by @xmath1126 . we use similar notation for @xmath1108 and @xmath1127 . consider any @xmath1128 . recall that @xmath62 and @xmath403 are viewed as clusters of size @xmath80 in @xmath303 , but when considering @xmath70 we are in fact considering subclusters of @xmath62 and @xmath403 of size @xmath274 . when viewed as clusters in @xmath303 , @xmath1129 initially corresponded to an @xmath162-regular pair of density at least @xmath1130 in @xmath98 . thus when viewed as clusters in @xmath70 , @xmath1129 initially corresponded to a @xmath1131-regular pair of density at least @xmath1132 in @xmath98 . moreover , initially the edges from @xmath1133 to @xmath1123 in @xmath98 induce a @xmath1134-regular pair of density at least @xmath1135 . however , we have removed all the edges lying in the @xmath39 hamilton cycles @xmath1117 which we have defined already . property ( iii ) implies that we have removed at most @xmath1136 edges from @xmath98 . thus we have the following property : * given any @xmath1137 , let @xmath1138 , @xmath1139 be such that @xmath1140 . then @xmath1141 . when constructing @xmath1115 we will remove at most @xmath1119 more edges from @xmath98 . but since @xmath1142 is far from being tight , it will hold throughout the argument below . similarly , the initial definition of @xmath99 ( c.f . section [ randomsplit ] ) and ( iv ) together imply the following property : * consider any edge @xmath1143 . let @xmath1031 and @xmath1032 be such that @xmath1144 . then @xmath1145 . we now construct @xmath1115 from @xmath94 . condition @xmath1146 in section [ 4.6 ] implies that , for each @xmath1125 , every vertex in @xmath1147 lies on the same cycle , @xmath1148 say , in @xmath94 . let @xmath1149 be such that @xmath1150 has at least @xmath1151 outneighbours in @xmath98 which lie in @xmath1152 . by @xmath1142 all but at most @xmath1153 vertices in @xmath1154 have this property . note that the outneighbour in @xmath94 of any such vertex lies in @xmath1155 . however , by @xmath1156 all but at most @xmath1157 vertices in @xmath1155 have at least @xmath1158 inneighbours in @xmath99 which lie in @xmath1154 . thus we can choose @xmath1150 with the additional property that its outneighbour @xmath1159 in @xmath94 has at least @xmath1160 inneighbours in @xmath99 which lie in @xmath1154 . let @xmath56 denote the directed path @xmath1161 from @xmath1162 to @xmath1150 . we now have two cases to consider . * case 1 . * @xmath1163 . note that @xmath1150 has at least @xmath1164 outneighbours @xmath1165 in @xmath1166 such that the inneighbour of @xmath1167 in @xmath94 lies in @xmath1168 . however , by @xmath1169 all but at most @xmath1153 vertices in @xmath1168 have at least @xmath1170 outneighbours in @xmath98 which lie in @xmath1171 . thus we can choose an outneighbour @xmath1165 of @xmath1150 in @xmath98 such that the inneighbour @xmath1172 of @xmath1167 in @xmath94 lies in @xmath1168 and @xmath1172 has at least @xmath1170 outneighbours in @xmath98 which lie in @xmath1171 . we extend @xmath56 by replacing it with @xmath1173 . * case 2 . * @xmath1174 . in this case the vertices in @xmath1175 already lie on @xmath56 . we will use the following claim to modify @xmath56 . [ rotateclaim ] there is a vertex @xmath1176 such that : * @xmath1177 . * the predecessor @xmath1178 of @xmath1179 on @xmath56 lies in @xmath1168 . * there is an edge @xmath1180 in @xmath99 such that @xmath1181 and @xmath1179 precedes @xmath1167 on @xmath56 ( but need not be its immediate predecessor ) . * the predecessor @xmath1172 of @xmath1167 on @xmath56 lies in @xmath1168 . * @xmath1172 has at least @xmath1170 outneighbours in @xmath98 which lie in @xmath1171 . [ ] [ ] @xmath1162 [ ] [ ] @xmath1182 [ ] [ ] @xmath1183 [ ] [ ] @xmath1184 [ ] [ ] @xmath1185 [ ] [ ] @xmath1186 in case 2,title="fig : " ] since @xmath1187 has at least @xmath1170 outneighbours in @xmath98 which lie in @xmath1188 , at least @xmath1189 of these outneighbours @xmath104 are such that the predecessor @xmath103 of @xmath104 on @xmath56 lies in @xmath1168 and at least @xmath1190 outneighbours of @xmath103 in @xmath98 lie in @xmath1171 . this follows since all such vertices @xmath104 have their predecessor on @xmath56 lying in @xmath1191 ( since @xmath1192 ) , since @xmath1193 and since by @xmath1169 all but at most @xmath1153 vertices in @xmath1168 have at least @xmath1190 outneighbours in @xmath1171 . let @xmath595 denote the set of all such vertices @xmath104 , and let @xmath592 denote the set of all such vertices @xmath103 . so @xmath1194 , @xmath1195 , @xmath1196 . let @xmath1197 denote the set of the first @xmath1198 vertices in @xmath592 on @xmath56 and @xmath1199 the set of the last @xmath1200 vertices in @xmath595 on @xmath56 . then @xmath1201 implies the existence of an edge @xmath1202 from @xmath1203 to @xmath1204 in @xmath99 . then the successor @xmath1179 of @xmath1178 on @xmath56 satisfies the claim . let @xmath1205 and @xmath1167 be as in claim [ rotateclaim ] . we modify @xmath56 by replacing @xmath56 with @xmath1206 ( see figure 2 ) . in either of the above cases we obtain a path @xmath56 from @xmath1162 to some vertex @xmath1207 which has at least @xmath1170 outneighbours in @xmath98 lying in @xmath1171 . we can repeat the above process : if @xmath1208 then we extend @xmath56 as in case 1 . if @xmath1209 or @xmath1210 then we modify @xmath56 as in case 2 . in both cases we obtain a new path @xmath56 which starts in @xmath1162 and ends in some @xmath1211 that has at least @xmath1170 outneighbours in @xmath98 lying in @xmath1212 . we can continue this process , for each @xmath1148 in turn , until we obtain a path @xmath56 which contains all the vertices in @xmath1213 ( and thus all the vertices in @xmath1 ) , starts in @xmath1162 and ends in some @xmath1214 having at least @xmath1170 outneighbours in @xmath98 which lie in @xmath1215 . [ rotateclaim2 ] there is a vertex @xmath1216 such that : * @xmath1217 . * the predecessor @xmath1218 of @xmath1219 on @xmath56 lies in @xmath1154 . * there is an edge @xmath1220 in @xmath99 such that @xmath1221 and @xmath1219 precedes @xmath1222 on @xmath56 . * the predecessor @xmath1223 of @xmath1222 on @xmath56 lies in @xmath1154 . * @xmath1223 has at least @xmath1224 outneighbours in @xmath99 which lie in @xmath1215 . the proof is almost identical to that of claim [ rotateclaim ] except that we apply @xmath1201 to ensure that @xmath1223 has at least @xmath1224 outneighbours in @xmath99 which lie in @xmath1215 . let @xmath1225 and @xmath1222 be as in claim [ rotateclaim2 ] . we modify @xmath56 by replacing it with the path @xmath1226 from @xmath1162 to @xmath1223 . so @xmath56 is a hamilton path in @xmath1 which is edge - disjoint from the @xmath39 hamilton cycles @xmath1117 already defined . in each of the @xmath364 steps in our construction of @xmath56 we have added at most one edge from each of @xmath98 and @xmath99 . so by ( [ s ] ) @xmath56 contains at most @xmath1119 edges from @xmath98 and at most @xmath1119 edges from @xmath99 . all other edges of @xmath56 lie in @xmath94 . recall that @xmath1162 has at least @xmath1160 inneighbours in @xmath99 which lie in @xmath1154 and @xmath1223 has at least @xmath1224 outneighbours in @xmath99 which lie in @xmath1215 . thus we can apply lemma [ rotationlemma ] to @xmath1227 with @xmath1228 playing the role of @xmath403 and @xmath1229 playing the role of @xmath62 to obtain a hamilton cycle @xmath1115 in @xmath1 where @xmath1230 . by construction , @xmath1115 satisfies ( i)(iv ) . thus we can indeed find @xmath1231 hamilton cycles in @xmath1 , as desired . in this section , we describe how theorem [ main ] can be extended to ` almost regular ' oriented graphs whose minimum semidegree is larger than @xmath21 . more precisely , we say that an oriented graph @xmath1 on @xmath2 vertices is _ @xmath1232-regular _ if @xmath1233 and @xmath1234 . [ main38 ] for every @xmath1235 there exist @xmath1236 and @xmath1237 such that the following holds . suppose that @xmath1 is an @xmath1238-regular oriented graph on @xmath13 vertices where @xmath1239 . then @xmath1 contains at least @xmath1240 edge - disjoint hamilton cycles . theorem [ main38 ] is best possible in the sense that there are almost regular oriented graphs whose semidegrees are all close to @xmath21 but which do not contain a hamilton cycle . these were first found by hggkvist @xcite . however , we believe that if one requires @xmath1 to be completely regular , then one can actually obtain a hamilton decomposition of @xmath1 . note this would be a significant generalization of kelly s conjecture . for every @xmath1235 there exists @xmath1236 such that for all @xmath1241 and all @xmath1242 each @xmath1243-regular oriented graph on @xmath2 vertices has a decomposition into hamilton cycles . at present we do not even have any examples to rule out the possibility that one can reduce the constant @xmath1244 in the above conjecture : is there a constant @xmath1245 such that for every sufficiently large @xmath2 every @xmath1246-regular oriented graph @xmath1 on @xmath2 vertices has a hamilton decomposition or at least a set of edge - disjoint hamilton cycles covering almost all edges of @xmath1 ? it is clear that we can not take @xmath1247 since there are non - hamiltonian @xmath32-regular oriented graphs on @xmath2 vertices with @xmath1248 ( consider a union of 2 regular tournaments ) . * sketch proof of theorem [ main38 ] . * the proof of theorem [ main38 ] is similar to that of theorem [ main ] . a detailed proof of theorem [ main38 ] can be found in @xcite . the main use of the assumption of high minimum semidegree in our proof of theorem [ main ] was that for any pair @xmath34 , @xmath130 of large sets of vertices , we could assume the existence of many edges between @xmath34 and @xmath130 ( see lemma [ keevashmult ] ) . this enabled us to prove the existence of very short paths , shifted walks and skeleton walks between arbitrary pairs of vertices . lemma [ keevashmult ] does not hold under the weaker degree conditions of theorem [ main38 ] . however , ( e.g. by lemma 4.1 in @xcite ) these degree conditions are strong enough to imply the following ` expansion property ' : for any set @xmath853 of vertices , we have that @xmath1249 ( provided @xmath1250 is not too close to @xmath2 ) . lemma 3.2 in @xcite implies that this expansion property is also inherited by the reduced graph . so in the proof of lemma [ multifactor1 ] , this expansion property can be used to find paths of length @xmath1251 which join up given pairs of vertices . similarly , in lemma [ shiftedwalk ] we find closed shifted walks so that each cycle @xmath55 in @xmath531 is traversed @xmath1251 times instead of just @xmath873 times ( such a result is proved explicitly in corollary 4.3 of @xcite ) . finally , in the proof of claim [ shadow ] we now find shadow skeleton walks whose length is @xmath1251 instead of 5 . in each of these cases , the increase in length does not affect the remainder of the proof . we would like to thank demetres christofides for helpful discussions . 10 n. alon and a. shapira , testing subgraphs in directed graphs , _ journal of computer and system sciences _ * 69 * ( 2004 ) , 354382 . b. alspach , j .- c . bermond and d. sotteau , decompositions into cycles . i. hamilton decompositions , _ cycles and rays ( montreal , pq , 1987 ) _ , kluwer acad . publ . , dordrecht , 1990 , 918 . d. khn and d. osthus , embedding large subgraphs into dense graphs , in _ surveys in combinatorics _ ( s. huczynska , j.d . mitchell and c.m . roney - dougal eds . ) , _ london math . lecture notes _ * 365 * , 137167 , cambridge university press , 2009 . c.st.j.a . nash - williams , edge - disjoint hamiltonian circuits in graphs with vertices of large valency , in _ 1971 studies in pure mathematics ( presented to richard rado ) _ , academic press , london , 1971 , 157183 . a. treglown , _ phd thesis , university of birmingham _ , in preparation . zhang , every regular tournament has two arc - disjoint hamilton cycles , j. qufu normal college , special issue oper . research ( 1980 ) , 7081 .

we show that every sufficiently large regular tournament can almost completely be decomposed into edge - disjoint hamilton cycles . more precisely , for each @xmath0 every regular tournament @xmath1 of sufficiently large order @xmath2 contains at least @xmath3 edge - disjoint hamilton cycles . this gives an approximate solution to a conjecture of kelly from 1968 . our result also extends to almost regular tournaments . msc2000 : 5c20 , 5c35 , 5c45 .

hydrodynamics describes the evolution of a fluid perturbed away from thermal equilibrium by long wave length fluctuations . the long wave length physics ( long compared with the mean field path of particle collisions ) can be systematically described by an expansion of space - time derivatives on classical fields with prefactors called transport coefficients . these transport coefficients encode the physics of short ( compared with the mean free path ) distance and are inputs to hydrodynamics . but they can be computed , in principle , once the microscopic theory of the system is known . we are interested in computing the transport coefficients in quantum chromodynamics ( qcd ) with @xmath0 flavors of massless quarks at finite temperature ( @xmath1 ) and chemical potentials ( @xmath2 , @xmath3 ) . the leading transport coefficients at the first derivative order include the shear viscosity ( @xmath4 ) , bulk viscosity ( @xmath5 ) , and the conductivity matrix ( @xmath6 ) . the shear viscosity of qcd has attracted a lot of attention recently . its ratio with the entropy density ( @xmath7 ) extracted from the hot and dense matter created at relativistic heavy ion collider ( rhic ) arsene:2004fa , adcox:2004mh , back:2004je , adams:2005dq just above the phase transition temperature ( @xmath8 ) yields @xmath9 at @xmath10 @xcite , which is close to a conjectured universal lower bound of @xmath11 @xcite inspired by the gauge / gravity duality maldacena:1997re , gubser:1998bc , witten:1998qj . this value of @xmath12 can not be explained by extrapolating perturbative qcd result arnold:2000dr , arnold:2003zc , chen:2010xk , chen:2011 km . the smallest @xmath12 is likely to exist near @xmath8 @xcite ( see , e.g. , ref . @xcite for a compilation and more references ) . alsofinite @xmath13 results suggests that @xmath12 is smaller at smaller @xmath13 . this is based on results of perturbative qcd at @xmath14 @xmath8 chen:2012jc and of a hadronic gas at @xmath15 @xmath8 and small @xmath13 chen:2007xe . it is speculated that the same pattern will persist at @xmath8 such that the smallest @xmath12 might exist near @xmath8 with @xmath16 chen:2012jc . for the bulk viscosity , the sum rule study kharzeev:2007wb , karsch:2007jc shows that @xmath5 increases rapidly near @xmath8 when @xmath1 approaches @xmath8 from above . this is consistent with the lattice gluon plasma result near @xmath8 @xcite and perturbative qcd result @xcite at much higher @xmath1 . this , when combined with pion gas results below @xmath8 chen:2007kx , fernandezfraile:2008vu , lu:2011df , dobado:2011qu , chakraborty:2010fr , suggests that @xmath17 has a local maximum near @xmath8 ( see , e.g. , chen:2011 km for a compilation ) . unlike @xmath12 , perturbative qcd result shows very small @xmath13 dependence in @xmath17 @xcite . note that at high @xmath13 , there are also bulk viscosities governed by the weak interaction such as the urca processes which have consequences in neutron star physics dong:2007mb , alford:2006gy , alford:2008pb , sad:2006qv , sad:2007ud , wang:2010ydb . these are quite different from the transport coefficients from the strong interaction mentioned above . the perturbative qcd calculations of @xmath4 and @xmath5 with finite @xmath13 were performed at the leading - log ( ll ) order of the strong coupling constant ( @xmath18 ) expansion in ref . either @xmath1 or @xmath13 in the calculation is much larger than @xmath19 which is the scale where qcd becomes non - perturbative . but the calculation is not applicable to the color superconducting phase at @xmath20 , since the vacuum in the calculation has no symmetry breaking . in this work , we apply the same perturbative qcd approach to compute the conductivity matrix @xmath6 at the ll order . the conductivity is an important transport coefficient which plays an essential role in the evolution of electromagnetic fields in heavy ion collisions huang:2013iia , mclerran:2013hla . the conductivity in strongly coupled quark gluon plasma was calculated with lattice qcd @xcite and dyson - schwinger equation @xcite . we first review the constraints from the second law of thermal dynamics ( i.e. the entropy production should be non - negative ) which show that the particle diffusion , heat conductivities , and electric conductivity are all unified into one single conductivity in this system . when @xmath21 , the conductivity becomes a @xmath22 matrix . we then show through the boltzmann equation that the conductivity matrix @xmath6 at the ll order is symmetric and positive definite ( @xmath23 for any real , non - vanishing vector @xmath24 ) . the former is a manifestation of the onsager relation while the latter is a manifestation of the second law of thermal dynamics . for simplicity , we show the numerical results of @xmath6 with all fermion chemical potential to be identical . in this limit , there are only two independent entries in @xmath6 . all the diagonal matrix elements are degenerate and positive since @xmath6 is positive definite . however , the off - diagonal matrix elements are degenerate but negative at finite @xmath13 . this means a gradient @xmath25 can drive a current of flavor @xmath26 alone the gradient direction , but it will also drive currents of different flavors in the opposite direction . this backward current phenomenon might seem counter intuitive , but we find that it is generic and it has a simple explanation . we speculate that this phenomenon might be most easily measured in cold atom experiments . let us start from the hydrodynamical system with only one flavor of quark of electric charge @xmath27 . the energy - momentum conservation and current conservation yield @xmath28where @xmath29 is the energy - momentum tensor , @xmath30 is the quark current and @xmath31 is the electromagnetic field strength tensor . the long wave length physics can be systematically described by the expansion of space - time derivatives@xmath32where we have used the parameter @xmath33 to keep track of the expansion and we will set @xmath34 at the end . @xmath31 is counted as @xmath35 . we will then assume the system is isotropic and homogeneous in thermal equilibrium so there is no special directions or intrinsic length scales macroscopically . we also assume the underlying microscopic theory satisfies parity , charge conjugation and time reversal symmetries such that the antisymmetric tensor @xmath36 does not contribute to @xmath29 and @xmath30 . also , we assume the system is fluid - like , describable by one ( and only one ) velocity field ( the conserved charged is assumed to be not broken spontaneously , otherwise the superfluid velocity needs to be introduced as well ) . also , at @xmath37 , the system is in local thermal equilibrium , i.e. the system is in equilibrium in the comoving frame where the fluid velocity is zero . with these assumptions , we can parametrize @xmath38where @xmath39diag(@xmath40 ) and @xmath41 , @xmath42 and @xmath43 are the energy density , pressure and number density , respectively . the fluid velocity @xmath44 and @xmath45 @xmath46 , @xmath47 , @xmath48 and @xmath49 are the bulk viscous pressure , shear viscous tensor , heat flow vector and diffusion current . they satisfy the orthogonal relations , @xmath50 . the covariant entropy flow is given by @xcite @xmath51where @xmath52 and @xmath53 is the entropy density . taking the space - time derivative of @xmath54 , then using the gibbs - duhem relation @xmath55 and the conservation equations ( eq : conservation_01 ) , we obtain the equation for entropy production : @xmath56 + h^{\mu } \left ( \partial _ { \mu } \beta + \beta u^{\nu } \partial _ { \nu } u_{\mu } \right ) \notag \\ & + \beta \pi ^{\mu \nu } \partial _ { \langle\mu } u_{\nu \rangle}-\beta \pi \partial \cdot u , \label{div - s-1}\end{aligned}\]]where the symmetric traceless tensor @xmath57 is defined by , @xmath58 \partial ^{\alpha } u^{\beta } , \]]and where@xmath59 and @xmath60 is the electric field in the comoving frame . at @xmath35 , this equation yields @xmath62 , \label{eq : relation_01}\]]where we have used the thermodynamic equation @xmath63 . this identity simplifies eq . ( [ div - s-1 ] ) to @xmath64 \notag \\ & + \beta \pi ^{\mu \nu } \partial _ { \left\langle \mu \right . } \nu \right\rangle } -\beta \pi \partial \cdot u. \label{eq : h_nu_02 - 1}\end{aligned}\ ] ] the second law of thermodynamics requires @xmath65 . it can be satisfied if , up to terms orthogonal to @xmath66 , @xmath67 and @xmath68 $ ] , @xmath46 , @xmath47 , @xmath48 and @xmath49 have the following forms at @xmath35 : @xmath69 , \label{eq : definition_01}\end{aligned}\]]where @xmath70 is inserted because @xmath71 . the coefficients @xmath4 , @xmath5 and @xmath6 are transport coefficients with names of shear viscosity , bulk viscosity and conductivity , respectively . the second law of thermodynamics requires these transport coefficients to be non - negative . on the right hand side of eq . ( [ eq : definition_01 ] ) , the three vectors @xmath72 , @xmath73 and @xmath74 form a unique combination and share the same transport coefficient @xmath6 israel:1979wp . it is obtained by assuming @xmath75 and @xmath76 has the ideal fluid form described in eq.([t ] ) . in general , we do not expect this to be true in all systems ( e.g. a solid might not have the ideal fluid description ) and hence there could be more transport coefficients . conventionally , the transport coefficients corresponding to @xmath72 , @xmath73 and @xmath74 are called particle diffusion , heat conductivity , and electric conductivity , respectively . in hydrodynamics , the choice of the velocity field is not unique . one could choose @xmath77 to align with the momentum density @xmath78 or the current @xmath79 , or their combinations . however , the system should be invariant under the transformation @xmath80 as long as @xmath81 is maintained ( or @xmath82 at @xmath35 ) . under this transformation , @xmath83 @xmath84 and @xmath85 @xmath86 . however , the entropy production equation ( [ eq : h_nu_02 - 1 ] ) remains invariant under this transformation . in this paper , we will be working at the landau frame with @xmath77 proportional to the momentum density @xmath78 such that @xmath87 in the comoving frame . then @xmath88 \label{aa}\ ] ] from eq . ( [ eq : definition_01 ] ) . @xmath6 is positive , the sign makes sense for particle diffusion and electric conduction because the diffusion is from high to low density and positively charged particles move along the @xmath89 direction . however , heat conduction induces a flow from low to high temperature ! this result is counter intuitive . this is because @xmath90 induces a momentum flow @xmath91 . if we choose to boost the system to the landau frame where @xmath92 , then the physics is less transparent . for particle diffusion and electric conduction this is not a problem , because one could have particles and antiparticles moving in opposite directions and still keep the net momentum flow zero . the physics of heat conduction becomes clear in the eckart frame where @xmath77 is proportional to the current @xmath79 and we have @xmath93.\ ] ] in this frame , the direction of heat conduction is correct ( while the physics of particle diffusion and electric conduction become less transparent ) . as expected , @xmath91 stays finite when @xmath94 but @xmath95 . when the flavor of massless quarks is increased to @xmath0 , then there are @xmath0 conserved currents @xmath96 ( the conserved electric current is just a combination of them ) . the hydrodynamical equations becomes @xmath97then the entropy production yields @xmath98 + \beta \pi ^{\mu \nu } \partial _ { \left\langle \mu \right . } u_{\left . \nu \right\rangle } -\beta \pi \partial \cdot u \notag \\ & \geq & 0\end{aligned}\]]working in the landau frame , we have@xmath99 . \label{d1}\]]our task is to compute the @xmath6 matrix which can be achieved by setting @xmath100 but @xmath101 . the second law of thermodynamics dictates @xmath6 being a positive definite matrix . we will use the boltzmann equation to compute our ll result of @xmath6 . it has been shown that boltzmann equation gives the same leading order result as the kubo formula in the coupling constant expansion in a weakly coupled @xmath102 theory @xcite and in hot qed @xcite , provided the leading @xmath1 and @xmath13 dependence in particle masses and scattering amplitudes are included . this conclusion is expected to hold in perturbative qcd as well @xcite . the boltzmann equation of a quark gluon plasma describes the evolution of the color and spin averaged distribution function @xmath103 for particle @xmath104 ( @xmath105 with @xmath106 for gluon , @xmath0 quarks and @xmath0 anti - quarks ) : @xmath107where @xmath103 is a function of space - time @xmath108 and momentum @xmath109 . for the ll calculation , we only need to consider two - particle scattering processes denoted as @xmath110 . the collision term has the form @xmath111 . \label{definition of c ab - cd}\]]where @xmath112 and @xmath113 and @xmath114where @xmath115 is the matrix element squared with all colors and helicities of the initial and final states summed over . the scattering amplitudes can be regularized by hard thermal loop propagators and in this paper we use the same scattering amplitudes as in ref . @xcite ( see also table i of ref . then the collision term for a quark of flavor @xmath116 is @xmath117where @xmath118 is the quark helicity and color degeneracy factor and the factor @xmath119 is included when the initial state is formed by two identical particles . similarly , @xmath120where @xmath121 is the gluon helicity and color degeneracy factor . in equilibrium , the distributions are denoted as @xmath122 and @xmath123 , with @xmath124where @xmath1 is the temperature , @xmath125 is the fluid four velocity and @xmath2 is the chemical potential for the quark of flavor @xmath116 . they are all space time dependent . the thermal masses of gluon and quark / anti - quark for external states ( the asymptotic masses ) can be computed via arnold:2002zm , mrowczynski:2000ed @xmath126where @xmath127 , @xmath128 , and @xmath129 . this yields @xmath130where we have set @xmath131 in the integrals on the right hand sides of eqs . ( [ mg ] ) and ( [ thermal mass g ] ) . the difference from non - vanishing masses is of higher order . in this work , we only need the fact that the thermal masses are proportional to @xmath132 for the ll results . matching to the derivative expansion in hydrodynamics , we expand the distribution function of particle @xmath104 as a local equilibrium distribution plus a correction @xmath133where the upper / lower sign corresponds to the femion / boson distribution . inserting eq . ( [ eq : expansion_01 ] ) into eq . ( [ eq : be_01 ] ) , we can solve the linearized boltzmann equation by keeping linear terms in space - time derivatives . here we neglect the viscous terms related to @xmath134 in @xmath135 and consider only the @xmath136 terms . at the zeroth order , @xmath37 , the system is in local thermal equilibrium and the boltzmann equation ( [ eq : be_01 ] ) is satisfied , @xmath137=0 $ ] . at @xmath35 , the left hand side of the boltzmann equation yields @xmath138 , \label{b1}\]]and @xmath139to derive this result , we have used @xmath140 in the local fluid rest frame where @xmath141 and @xmath142 and @xmath143 which yields@xmath144and@xmath145and then by applying thermodynamic relations , we can replace the time derivatives of @xmath1 , @xmath13 and @xmath77 with spatial derivatives : @xmath146 \bm{\nabla } \mathbf{\cdot u } , \label{dt / dt } \\ \frac{\partial \mathbf{u}}{\partial t } & = -\beta \bm{\nabla } t-\sum_{a=1}^{n_{f}}\frac{n_{a}t}{\epsilon + p}\bm{\nabla } \left ( \frac{\mu _ { a}}{t}\right ) . \notag\end{aligned}\]]those relations lead to eqs.([b1],[b2 ] ) . to get the right hand side of the boltzmann equation at @xmath35 , we parametrize @xmath135 of eq . ( [ eq : expansion_01 ] ) as @xmath147the matrix @xmath148 is @xmath149 . we will see there are @xmath149 equations to constrain them . for each boltzmann equation , we have a linear combination of @xmath0 terms of @xmath136 . since each @xmath136 is linearly independent to each other , thus there are @xmath0 equations for each boltzmann equation . totally we have @xmath150 boltzmann equations , thus we have @xmath149 equations to solve for @xmath148 . these equations are @xmath151 , \label{eq : constr - g}\]]@xmath152 , \label{eq : constr - q}\end{aligned}\]]and @xmath153 , \label{eq : constr - qb}\end{aligned}\]]where @xmath154 .\end{aligned}\]]formally we can rewrite these linearized boltzmann equations in a compact form , @xmath155where @xmath156 and @xmath157 are both vectors of @xmath158 components and @xmath159 is a @xmath160 matrix . in the kinetic theory , the quark current of flavor @xmath116 is @xmath161expanding this expression to @xmath35 and matching it to eq.([d1 ] ) , we have@xmath162since we are working in the landau frame , we should impose the landau - lifshitz condition @xmath163this implies @xmath164we can use these constraints to rewrite eq.([e1 ] ) as@xmath165this form can be schematically written as @xmath166where we have used eq.([linearized equation ] ) for the second equality . more explicitly , @xmath167where @xmath168 \notag \\ & \cdot \left [ \mathbf{a}^{c_{1}b}(k_{1})+\mathbf{a}^{c_{2}b}(k_{2})-\mathbf{a}^{c_{3}b}(k_{3})-\mathbf{a}^{c_{4}b}(k_{4})\right ] . \label{f2}\end{aligned}\ ] ] from eq.([linearized equation ] ) , it is clear that if @xmath169then from momentum conservation this implies @xmath170those modes are called zero modes ( denoted by the subscribe @xmath171 in eq.(0 ) ) . they would have been a problem for eq.([f3 ] ) unless @xmath172 , but this is guaranteed from the total momentum conservation at @xmath35,@xmath173and eqs.([b1],[b2 ] ) . thus , we can just solve for @xmath157 in eq.([f3 ] ) by discarding the zero modes . from eqs.([f1 ] ) and ( [ f2 ] ) , we can see easily that @xmath174 . this is a manifestation of the onsager relation which appears when particle scattering is symmetric under the time - reversal transformation . we can also see that @xmath6 is positive definite . now we are ready to solve the conductivity matrix @xmath6 . our strategy to solve for @xmath175 is to make use of eq.([f3 ] ) to solve for @xmath176 from @xmath177 ( no summation over @xmath116 ) . once all the @xmath176 are obtained , @xmath175 can be computed . also , in solving for @xmath177 , one can use the standard algorithm to systematically approach the answer from below chen:2011 km . the dependence on the strong coupling constant is similar to that in shear viscosity it is inversely proportional to the scattering rate which scales as @xmath178 with the @xmath179 dependence coming from regularizing the collinear infrared singularity by the thermal masses of quarks or gluons . @xmath6 is of mass dimension two , thus we will present our result in the normalized conductivity @xmath180such that @xmath181 is dimensionless and coupling constant independent . for simplicity , we will concentrate on the linear response of a thermal equilibrium system with all fermion chemical potentials to be identical , i.e. @xmath182 for all @xmath116 s but each @xmath183 could be varied independently . this symmetry makes all the diagonal matrix elements ( denoted as @xmath184 ) identical and all the off - diagonal ones ( denoted as @xmath185 ) identical . @xmath186 and @xmath185 are even in @xmath13 ( and so are @xmath187 and @xmath188 ) because our microscopic interaction ( in vacuum ) is invariant under charge conjugation , thus @xmath189 should be invariant under @xmath190 . it is easy to diagonalize @xmath6 . one eigenvalue is@xmath191corresponding to the conductivity of the flavor singlet total quark current ( @xmath192 is the total quark current conductivity)@xmath193the other @xmath194 eigenvalues are degenerate with the value @xmath195they are the conductivities of the flavor non - singlet currents @xmath196,\ ] ] with @xmath197 . ( upper panel ) and off - diagonal conductivity @xmath198 ( lower panel ) as functions of @xmath199 for different @xmath0 . ] ( upper panel ) and off - diagonal conductivity @xmath198 ( lower panel ) as functions of @xmath199 for different @xmath0 . ] ( upper panel ) and @xmath200 ( lower panel ) as functions of @xmath199 for different @xmath0 . ] ( upper panel ) and @xmath200 ( lower panel ) as functions of @xmath199 for different @xmath0 . ] @xmath187 and @xmath188 are shown as functions of @xmath201 in fig . 1 ] for various @xmath0 with @xmath202 such that the system is asymptotically free , while @xmath203 and @xmath204 are shown in fig . 2 ] ( note that there is no @xmath188 or @xmath204 for @xmath205 ) . the fact that the matrix @xmath6 is positive definite makes @xmath187 , @xmath203 and @xmath204 positive , but it imposes no constraint on the sign of @xmath188 . when @xmath206 , we can expand @xmath207 , and @xmath208 . we find @xmath209 for all @xmath0 while the values of @xmath210 and @xmath211 for different @xmath0 are tabulated in table . [ power expansion coefficients lambdaqq ] . our result for @xmath212 agrees within @xmath213 to that of arnold , moore and yaffe ( amy ) calculated up to @xmath214 listed in table iii of ref . @xcite . the @xmath215 property is due to a bigger symmetry enjoyed by the ll results : if we just change all the quarks of flavor @xmath116 into anti - quarks while the rest of the system stays the same , then as far as collision is concerned , the other quarks and the gluons will not feel any difference . this is because the ll result only depends on two - particle scattering , and although this action could change the sign of certain amplitudes , it does not change the collision rate . for example , the amplitudes of @xmath216 @xmath217 and @xmath218 @xmath219 ( @xmath220 ) have different signs because one of the couplings changes sign when we change the color into its anti - color , but the amplitude squared is of the same . this makes the diagonal terms even in all the chemical potentials @xmath221while the off - diagonal term @xmath175 is odd in @xmath2 and @xmath222 but even in other chemical potentials @xmath223thus , at the ll order , @xmath6 becomes diagonal when all the chemical potentials vanish . to understand the other features of @xmath187 and @xmath188 , we first turn to @xmath203 and @xmath204 in the @xmath224 limit . in this large chemical potential limit , the quark contribution dominates over those of anti - quark and gluon . the fermi - dirac distribution function @xmath225 of quark @xmath226 multiplied by its pauli blocking factor @xmath227 can be well approximated by a @xmath228function , @xmath229 . we then first set @xmath230 for all @xmath116 so all the currents @xmath231 becomes identical . @xmath232 can be rewritten as @xmath233 , and eq . ( [ 01 ] ) yields@xmath234the summation gives @xmath235 and @xmath236 . on the other hand , eq.([f1 ] ) gives @xmath237 where @xmath238 comes from summing the @xmath116 , @xmath239 , @xmath240 , @xmath241 indices of @xmath242 and we have used @xmath243 in eq.([f2 ] ) . these two conditions yield @xmath244 . this is indeed what happens in fig . 2 ] at large @xmath13 ( although the @xmath245 dependence is not so obvious in this plot but we have checked this at much larger @xmath246 ) . we can perform the similar counting to the scaling of @xmath247 . from eq . ( [ 01 ] ) , @xmath248 and from eq.([f1 ] ) @xmath249 . thus , @xmath250 which is also observed in fig . [ fig . 2 ] . the main difference in @xmath251 and @xmath247 is the @xmath252 dependence@xmath247 has no cancellation factor of @xmath253 in large @xmath13 . the different @xmath13 scaling between @xmath192 and @xmath247 at large @xmath13 is due to collisions , which change the direction of the current and reduce the conductivity . while both flavor singlet and non - singlet fermions can collide among themselves , they do not collide with each other ( the scattering amplitude vanishes ) . thus , when @xmath13 , the flavor singlet chemical potential , is increased , the flavor singlet current experiences more collisions . therefore the flavor singlet conductivity @xmath192 is reduced . for the flavor non - singlet current , the increase of @xmath13 does not affect the collision . however , it will increase the averaged fermi momentum such that the induced current and the flavor non - singlet conductivity @xmath247 will be increased . given the large @xmath13 behavior of @xmath203 and @xmath204 , the large @xmath13 behavior of @xmath187 and @xmath188 is now easily reconstructed : @xmath254 ( @xmath255 ) and @xmath256 . the sign of @xmath188 can be best understood from the flavor non - singlet current effect such that a gradient of @xmath2 induces anti-@xmath239 currents ( @xmath257 ) and yields @xmath258 . we can then interpolate @xmath188 to @xmath259 at zero @xmath13 . there is no non - trivial structure at intermediate @xmath13 . for @xmath187 , the @xmath205 curve seems to be at odd with other @xmath0 curves , but this anomaly disappears when viewed in the @xmath203 plot . the fact that @xmath260 while @xmath261 at finite @xmath13 is intriguing . it means a gradient @xmath25 can drive a current @xmath231 along the @xmath262 direction , but it will also drive currents of different flavors in the opposite direction . this backward current phenomenon seems counter intuitive at the first sight . but the physics behind is just that the flavor singlet current experiences more collisions in a flavor singlet medium than the flavor non - singlet ones . if the medium is flavor non - singlet , e.g. @xmath263 while the other chemical potentials all vanish , then the flavor non - singlet current @xmath264 will experience more collisions than the flavor singlet current . therefore , we will have @xmath265 . this is consistent with eq.([lab ] ) derived from the symmetry of the ll order along . thus the simple explanation based on collisions that we presented above seems quite generic . it might happen in other systems such as cold atoms as well . in that case , cold atom experiments might be the most promising ones to observe this backward current phenomenon . we have calculated the conductivity matrix of a weakly coupled quark - gluon plasma at the leading - log order . by setting all quark chemical potentials to be identical , the diagonal conductivities become degenerate and positive , while the off - diagonal ones become degenerate but negative ( or zero when the chemical potential vanishes ) . this means a potential gradient of a certain fermion flavor can drive backward currents of other flavors . a simple explanation is provided for this seemingly counter intuitive phenomenon . it is speculated that this phenomenon is generic and most easily measured in cold atom experiments . acknowledgement : sp thanks tomoi koide and xu - guang huang for helpful discussions on the onsager relation . jwc thanks jan m. pawlowski for useful discussions and the u. of heidelberg for hospitality . jwc , yfl and sp are supported by the cts and casts of ntu and the nsc ( 102 - 2112-m-002 - 013-my3 ) of roc . yks is supported in part by the ccnu - qlpl innovation fund under grant no . . this work is also supported by the national natural science foundation of china under grant no . 11125524 and 11205150 , and in part by the china postdoctoral science foundation under the grant no . 2011m501046 . 99 i. arsene _ et al . _ [ brahms collaboration ] , nucl . phys . a * 757 * , 1 ( 2005 ) [ nucl - ex/0410020 ] . k. adcox _ et al . _ [ phenix collaboration ] , nucl . a * 757 * , 184 ( 2005 ) [ nucl - ex/0410003 ] . b. b. back , m. d. baker , m. ballintijn , d. s. barton , b. becker , r. r. betts , a. a. bickley and r. bindel _ et al . _ , nucl . phys . a * 757 * , 28 ( 2005 ) [ nucl - ex/0410022 ] . j. adams _ et al . _ [ star collaboration ] , nucl . phys . a * 757 * , 102 ( 2005 ) [ nucl - ex/0501009 ] . h. song , s. a. bass , u. heinz , t. hirano and c. shen , phys . lett . * 106 * , 192301 ( 2011 ) [ erratum - ibid . * 109 * , 139904 ( 2012 ) ] . p. kovtun , d. t. son and a. o. starinets , phys.rev . lett . * 94 * , 111601 ( 2005 ) . j. m. maldacena , adv . * 2 * , 231 ( 1998 ) [ int . j. theor . phys . * 38 * , 1113 ( 1999 ) ] . s. s. gubser , i. r. klebanov and a. m. polyakov , phys . b * 428 * , 105 ( 1998 ) e. witten , adv . * 2 * , 253 ( 1998 ) p. b. arnold , g. d. moore and l. g. yaffe , jhep * 0011 * , 001 ( 2000 ) [ hep - ph/0010177 ] . b. arnold , g. d. moore and l. g. yaffe , jhep * 0305 * , 051 ( 2003 ) [ hep - ph/0302165 ] . chen , j. deng , h. dong and q. wang , phys . d * 83 * , 034031 ( 2011 ) [ erratum - ibid . d * 84 * , 039902 ( 2011 ) ] [ arxiv:1011.4123 [ hep - ph ] ] . j .- w . chen and e. nakano , b * 647 * , 371 ( 2007 ) [ hep - ph/0604138 ] . chen , y. -f . liu , y .- k . song and q. wang , phys . d * 87 * , 036002 ( 2013 ) [ arxiv:1212.5308 [ hep - ph ] ] . chen , y. -h . li , y. -f . liu and e. nakano , phys . d * 76 * , 114011 ( 2007 ) [ hep - ph/0703230 ] . d. kharzeev , k. tuchin , jhep * 0809 * , 093 ( 2008 ) . h. b. meyer , jhep * 1004 * , 099 ( 2010 ) . b. arnold , c. dogan and g. d. moore , phys . d * 74 * , 085021 ( 2006 ) [ hep - ph/0608012 ] . j .- w . chen and j. wang , c * 79 * , 044913 ( 2009 ) [ arxiv:0711.4824 [ hep - ph ] ] . d. fernandez - fraile and a. gomez nicola , phys . rev . lett . * 102 * , 121601 ( 2009 ) . e. lu , g. d. moore , phys . * c83 * , 044901 ( 2011 ) . a. dobado , f. j. llanes - estrada and j. m. torres - rincon , phys . b * 702 * , 43 ( 2011 ) . p. chakraborty and j. i. kapusta , phys . rev . c * 83 * , 014906 ( 2011 ) . h. dong , n. su and q. wang , phys . d * 75 * , 074016 ( 2007 ) [ astro - ph/0702104 ] . m. g. alford and a. schmitt , j. phys . g * 34 * , 67 ( 2007 ) [ nucl - th/0608019 ] . m. g. alford , m. braby and a. schmitt , j. phys . g * 35 * , 115007 ( 2008 ) [ arxiv:0806.0285 [ nucl - th ] ] . b. a. sad , i. a. shovkovy and d. h. rischke , phys . d * 75 * , 065016 ( 2007 ) [ astro - ph/0607643 ] . b. a. sad , i. a. shovkovy and d. h. rischke , phys . d * 75 * , 125004 ( 2007 ) [ astro - ph/0703016 ] . x. wang and i. a. shovkovy , phys . d * 82 * , 085007 ( 2010 ) [ arxiv:1006.1293 [ hep - ph ] ] . x. -g . huang and j. liao , phys . * 110 * , 232302 ( 2013 ) [ arxiv:1303.7192 [ nucl - th ] ] . l. mclerran and v. skokov , arxiv:1305.0774 [ hep - ph ] . h. -t . ding , a. francis , o. kaczmarek , f. karsch , e. laermann and w. soeldner , phys . d * 83 * , 034504 ( 2011 ) [ arxiv:1012.4963 [ hep - lat ] ] . a. amato , g. aarts , c. allton , p. giudice , s. hands and j. -i . skullerud , arxiv:1307.6763 [ hep - lat ] . s. -x . qin , arxiv:1307.4587 [ nucl - th ] . w. israel and j. m. stewart , annals phys . * 118 * , 341 ( 1979 ) . s. pu , arxiv:1108.5828 [ hep - ph ] . s. jeon , phys . d * 52 * , 3591 ( 1995 ) . y. hidaka and t. kunihiro , phys . d * 83 * , 076004 ( 2011 ) .

we calculate the conductivity matrix of a weakly coupled quark - gluon plasma at the leading - log order . by setting all quark chemical potentials to be identical , the diagonal conductivities become degenerate and positive , while the off - diagonal ones become degenerate but negative ( or zero when the chemical potential vanishes ) . this means a potential gradient of a certain fermion flavor can drive backward currents of other flavors . a simple explanation is provided for this seemingly counter intuitive phenomenon . it is speculated that this phenomenon is generic and most easily measured in cold atom experiments .

the theme of these lectures at the 46th course of the zakopane school is multi - particle production in hadronic collisions at high energies in the color glass condensate ( cgc ) effective field theory . there has been tremendous progress in our theoretical understanding in the seven years since one of the authors last lectured here . at that time , the author s lectures covered the state of the art ( in the cgc framework ) in both deeply inelastic scattering ( dis ) studies and in hadronic multi - particle production @xcite . a sign of rapid progress in the field is that there were several talks and lectures at this school covering various aspects of this physics in dis alone . we will restrict ourselves here to developments in our understanding of multi - particle production in hadronic collisions . another significant development in the last seven years has been the large amount of data from the relativistic heavy ion collider ( rhic ) at bnl , key features of which were nicely summarized in the white - papers of the experimental collaborations @xcite culminating in the announcement of the discovery of a perfect fluid " at rhic . the exciting experimental observations were discussed here in the lectures of jacak @xcite . the rhic data have had a tremendous impact on the cgc studies of multi - particle production . while we will discuss rhic phenomenology , and indeed specific applications of theory to data , our primary focus will be on attempts to develop a systematic theoretical framework in qcd to compute multi - particle production in hadronic collisions . some applications of the cgc approach to rhic phenomenology were also covered at this school by kharzeev as part of his lectures @xcite . for recent comprehensive reviews , see ref . @xcite . the collider era in high energy physics has made possible investigations of qcd structure at a deep level in studies of multi - particle final states . much attention has been focused on the nature of multi - particle production in jets ; for a nice review , see refs . the problem is however very general . theoretical developments in the last couple of decades suggest that semi - hard particle production in high energy hadronic interactions is dominated by interactions between partons having a small fraction @xmath0 of the longitudinal momentum of the incoming nucleons . in the regge limit of small @xmath0 and fixed momentum transfer squared @xmath1 ( corresponding to very large center of mass energies @xmath2 ) the balitsky fadin kuraev lipatov ( bfkl ) evolution equation @xcite predicts that parton densities grow very rapidly with decreasing @xmath0 . a rapid growth with @xmath0 , in the gluon distribution , for fixed @xmath3 , was observed in the hera experiments @xcite . ( it is not clear however that the observed growth of the gluon distribution is a consequence of bfkl dynamics @xcite . ) because the rapid growth in the regge limit corresponds to very large phase space densities of partons in hadronic wave functions , saturation effects may play an important role in hadronic collisions at very high energies @xcite . these slow down the growth of parton densities relative to that of bfkl evolution and may provide the mechanism for the unitarization of cross - sections at high energies . the large parton phase space density suggests that small @xmath0 partons can be described by a classical color field rather than as particles @xcite . light cone kinematics ( more simply , time dilation ) further indicates that there is a natural separation in time scales , whereby the small @xmath0 partons are the dynamical degrees of freedom and the large @xmath0 partons act as static color sources for the classical field . in the mclerran venugopalan ( mv ) model , the large color charge density of sources is given by the density of large @xmath0 partons in a big nucleus which contains @xmath4 valence quarks ( where @xmath5 is the atomic number of the nucleus ) . in this limit of strong color sources , one has to solve the non linear classical yang - mills equations to obtain the classical field corresponding to the small @xmath0 parton modes . this procedure properly incorporates , at tree level , the recombination interactions that are responsible for gluon saturation . in the mv model , the distribution of large @xmath0 color sources is described by a gaussian statistical distribution @xcite . a more general form of this statistical distribution , for @xmath6 gauge theories , valid for large @xmath5 and moderate @xmath0 , is given in refs . @xcite . the separation between large @xmath0 and small @xmath0 , albeit natural , is somewhat arbitrary in the mv model ; the physics should in fact be independent of this separation of scales . this property was exploited to derive a functional renormalization group ( rg ) equation , the jimwlk equation , describing the evolution of the gauge invariant source distributions to small @xmath0 @xcite . the jimwlk functional rg equation is equivalent to an infinite hierarchy of evolution equations describing the behavior of multi - parton correlations at high energies first derived by balitsky @xcite . a useful ( and tremendously simpler ) large @xmath7 and large @xmath5 mean - field approximation independently derived by kovchegov @xcite , is commonly known as the balitsky - kovchegov equation . the general effective field theory framework describing the non - trivial behavior of multi - parton correlations at high energies is often referred to as the color glass condensate ( cgc ) @xcite . several lecturers at the school discussed the state of the art in our understanding of the small @xmath0 wavefunction @xcite . for previous discussions at recent schools , see refs . @xcite . to compute particle production in the cgc framework , in addition to knowing the distribution of sources in the small @xmath0 nuclear wavefunction , one must calculate the properties of multi - particle production for any particular configuration of sources . in this paper , we will assume that the former is known . all we require is that these sources ( as the rg equations tell us ) are strong sources , parametrically of the order of the inverse coupling constant , and are strongly time dependent . we will describe a formalism to compute multi - particle production for an arbitrary distribution of such sources . these lectures are organized as follows . in the first lecture , we shall describe the formalism for computing particle production in a field theory coupled to strong time - dependent external classical sources . we will consider as a toy model a @xmath8 scalar theory , where @xmath9 is coupled to a strong external source . although the complications of qcd such as gauge invariance are very important , many of the lessons gained from this simpler scalar theory apply to studies of particle production in qcd . we will demonstrate that there is no simple power counting in the coupling constant @xmath10 for the probability @xmath11 to produce @xmath12 particles . a simple power counting however exists for moments of @xmath11 . we will discuss how one computes the average multiplicity and ( briefly ) the variance . with regard to the former , we will show how it can be computed to next - to - leading order in the multiplicity . we will also discuss what it takes to compute the generating function for the multiplicity distribution to leading order in the coupling . in lecture ii , we will relate the formal considerations developed in lecture i , to the results of real time numerical simulations of the average multiplicity of gluons and quarks produced in heavy ion collisions . inclusive gluon production , to lowest order in the loop expansion discussed in lecture i , is obtained by solving the classical yang - mills equations for two color sources moving at the speed of light in opposite directions @xcite . this problem has been solved numerically in @xcite for the boost - invariant case . the multiplicity of quark - pairs is computed from the quark propagator in the background field of @xcite it has been studied numerically in @xcite . a first computation for the boost non - invariant case has also been performed recently @xcite . it was shown there that rapidity dependent fluctuations of the classical fields lead to the non abelian analog of the weibel instability @xcite , first studied in the context of electromagnetic plasmas . such instabilities may be responsible for the early thermalization required by phenomenological studies of heavy ion collisions . we will discuss how a better understanding of the small quantum fluctuations discussed in lecture i may provide insight into early thermalization . in the third and final lecture , we will discuss how the formalism outlined in lecture i simplifies in the case of proton - nucleus collisions . ( a similar simplification occurs in hadron - hadron collisions at forward / backward rapidities where large @xmath0 s in one hadron ( small color charge density ) and small @xmath0 in the other ( large color charge density ) are probed . ) at leading order in the coupling , lowest order in the proton charge density , and all orders in the nuclear color charge density , analytical results are available for both inclusive gluon and quark production factorization breaks down even at leading order in pa collisions @xcite . ] . in the former case , the analytical formula can be written , in @xmath13 factorized form , as the product of unintegrated distributions in both the proton and the nucleus convoluted with the matrix element for the interactions squared . this formula is used extensively in the literature for phenomenological applications . we will discuss one such application , that of limiting fragmentation . as outlined in the introduction , the glasma is formed when two sheets of colored glass collide , producing a large number of partons . a cartoon depicting this collision is shown in fig . [ fig : figi-1 ] . in the cgc framework , it is expected that observables can be expressed as @xmath14\,[d\rho_2]\ , w_{_{y_{\rm beam}-y}}[\rho_1]\,w_{_{y_{\rm beam}+y } } [ \rho_2 ] \ , { \cal o}[\rho_1,\rho_2 ] \ ; , \label{eq : li-0}\ ] ] where they are first computed as a functional of the color charge densities @xmath15 and @xmath16 of the two nuclei and then averaged over all possible configurations of these sources , with the likelihood of a particular configuration at a given rapidity @xmath17 (= @xmath18 ) specified by the weight functionals @xmath19 $ ] and @xmath20 $ ] respectively . here @xmath21 is the beam rapidity in a hadronic collision ( @xmath22 denoting the proton mass ) with the center of mass energy @xmath2 . the evolution of the @xmath23 $ ] s with rapidity is described by the jimwlk equation @xcite . for small @xmath0 ( large @xmath17 ) and/or large nuclei , the rapid growth of parton densities corresponds to light cone source densities @xmath24 in other words , the sources are strong . thus understanding how two sheets of colored glass shatter to produce the glasma requires that we understand the nature of particle production in a field theory with strong , time dependent sources . in this lecture , we will outline the tools to systematically compute particle production in such theories . more details can be found in refs . @xcite . field theories with strong time dependent sources are different from field theories in the vacuum in one key respect . the `` vacuum '' in the former , even in weak coupling , is non - trivial because it can produce particles . specifically , the amplitude from the vacuum state @xmath25 to a populated state @xmath26 is @xmath27 unitarity requires that the sum over all out " states satisfies the identity @xmath28 we therefore conclude that @xmath29 in other words , the probability that the vacuum stays empty is strictly smaller than unity . following the conventions of @xcite , we can write the vacuum - to - vacuum transition amplitude as @xmath30 } \ ; , \label{eq : li-2}\ ] ] where @xmath31 $ ] compactly represents the sum of the connected vacuum - vacuum diagrams in the presence of the external ( in our case , strong , time dependent , colored ) source @xmath32 . therefore , the inequality ( [ eq : li-1 ] ) means that vacuum - vacuum diagrams have a non - zero imaginary part , since @xmath33)$ ] . in stark contrast , for a field theory without external sources , eq . ( [ eq : li-1 ] ) would be an equality , and the vacuum - vacuum diagrams would be purely real , thereby only giving a pure phase for the vacuum - to - vacuum amplitude in eq . ( [ eq : li-2 ] ) . they therefore do not contribute to the probabilities for producing particles . ( [ eq : li-1 ] ) tells us that one has to be more careful in field theories with external sources . to illustrate how particle production works in such theories , we shall , for simplicity , consider a real scalar field with cubic self - interactions , coupled to an external source . ( the lessons we draw carry over straightforwardly to qcd albeit their implementation is in practice significantly more complex . ) the lagrangian density is @xmath34 note that the coupling @xmath10 in this theory has dimensions of the mass ; and that the theory is super - renormalizable in @xmath35 dimensions . the source densities @xmath36 can be envisioned as the scalar analogue of the sum of two source terms @xmath37 corresponding respectively in the cgc framework to the recoil - less color currents of the two hadronic projectiles . let us now consider how the perturbative expansion for such a theory looks like in weak coupling . the power of a generic simply connected diagram is given simply by @xmath38 where @xmath39 are the number of external lines , @xmath40 the number of loops and @xmath41 the number of sources . for vacuum vacuum graphs , @xmath42 . as @xmath43 , the power counting for a theory with strong sources is given entirely by an expansion in the number of loops . in particular , at tree level ( @xmath44 ) , the vacuum vacuum graphs are all of order @xmath45 . the tree graphs contributing to the connected vacuum - vacuum amplitude in eq . ( [ eq : li-1 ] ) can be represented as 1=to 7 cm @xmath46\equiv i\sum_{\rm conn}v = \quad\;\raise -3.5mm\box1 \label{eq : li-4}\ ] ] there are also loop contributions in this expression which we have not represented here . to proceed with the perturbative computation , we need to consider the analog of the well known cutkosky rules for this case . for each diagram in the computation , begin by assigning for each vertex and source , two kinds of vertices denoted by @xmath47 or @xmath48 . a vertex of type @xmath47 is the ordinary vertex and appears with a factor @xmath49 in feynman diagrams . a vertex of type @xmath48 is the opposite is real in an unitary theory , the vertex of type @xmath48 is also the complex conjugate of the vertex of type @xmath47 . ] of a @xmath47 vertex , and its feynman rule is @xmath50 . likewise , for insertions of the source @xmath32 , insertions of type @xmath47 appear with the factor @xmath51 while insertions of type @xmath48 appear instead with @xmath52 . thus for each feynman diagram @xmath53 in eq . ( [ eq : li-3 ] ) , containing only @xmath47 vertices and sources ( denoted henceforth as @xmath54 ) contributing to the sum of connected vacuum - vacuum diagrams , we obtain a corresponding set of diagrams @xmath55 by assigning the symbol @xmath56 to the vertex @xmath57 of the original diagram ( and connecting a vertex of type @xmath58 to a vertex of type @xmath59 with a propagator @xmath60 to be discussed further shortly ) . the generalized set of diagrams therefore includes @xmath61 such diagrams if the original diagram had @xmath12 vertices and sources . using recursively the so - called `` largest time equation '' @xcite , one obtains the identity , @xmath62 where the prime in the sum means that we sum over all the combinations of @xmath63 s , except the two terms where the vertices and sources are all of type @xmath47 or all of type @xmath48 . ( there are therefore @xmath64 terms in this sum . ) we now need to specify the propagators connecting the @xmath65 vertices and sources . the usual feynman ( time - ordered ) free propagator is the propagator connecting two vertices of type @xmath47 , i.e. @xmath66 . it can be decomposed as @xmath67 which defines the propagators @xmath68 and @xmath69 . likewise , the anti time - ordered free propagator @xmath70 is defined as . ] @xmath71 the fourier transforms of the free propagators @xmath60 for our scalar theory are @xmath72 for a given term in the right hand side of eq . ( [ eq : li-5 ] ) , one can divide the diagram in several regions , each containing only @xmath47 or only @xmath48 vertices and sources . ( there is at least one external source in each of these regions because of energy conservation constraints . ) the @xmath47 regions and @xmath48 regions of the diagram are separated by a `` cut '' , and one thus obtains a cut vacuum - vacuum diagram " . at tree level , the first terms generated by these cutting rules ( applied to compute the imaginary part of the sum of connected vacuum - vacuum diagrams in eq . ( [ eq : li-5 ] ) ) are 1=to 8 cm @xmath73 the @xmath47 and @xmath48 signs adjacent to the grey line in each diagram here indicate the side on which the set of @xmath47 and @xmath48 vertices is located . as one can see , there are cuts intercepting more than one propagator . the sum of the diagrams with @xmath74 cut propagators is denoted by @xmath75 the identity in eq . ( [ eq : li-7 ] ) ( and eq . ( [ eq : li-5 ] ) ) is a statement of unitarity . these @xmath76 are sometimes called combinants " in the literature @xcite . it is important to note that cut connected vacuum - vacuum diagrams would be zero in the vacuum because energy can not flow from one side of the cut to the other in the absence of external sources . this is of course consistent with a pure phase in eq . ( [ eq : li-5 ] ) . this constraint on the energy flow is removed if the fields are coupled to * time - dependent * external sources . cut vacuum - vacuum diagrams , and therefore the imaginary part of vacuum - vacuum diagrams , differ from zero in this case . we now turn to the probabilities for producing @xmath12 particles . the probability to produce one particle from the vacuum can be parameterized as @xmath77 where @xmath78 , a series in @xmath79 ( @xmath80 ) is obtained by summing the 1-particle cuts through connected vacuum - vacuum diagrams . the exponential prefactor is the square of the sum of all the vacuum - vacuum diagrams , which arises in any transition probability . the probability @xmath81 for producing two particles from the vacuum contains two pieces . one is obtained by squaring the @xmath82 piece of the probability for producing one particle ( dividing by @xmath83 for identical particles ) in this case , the two particles are produced independently from one another . the other @xmath84 is a correlated " contribution from a 2-particle cut through connected vacuum - vacuum diagrams . we therefore obtain @xmath85\ ; . \label{eq : li-8.2}\ ] ] in a similar vein , the probability @xmath86 can be shown to consist of three pieces . one ( uncorrelated " ) term is the cube of @xmath82 ( preceded by a symmetry factor @xmath87 ) . another is the combination @xmath88 , corresponding to the production of two particles in the same subdiagram with the third produced independently from the first two . finally , there is the correlated " three particle production probability @xmath89 corresponding to the production of three particles from the same diagram . the sum of these three pieces is thus @xmath90\ ; . \label{eq : li-8.3}\ ] ] some of the graphs contributing to @xmath78 , @xmath91 and @xmath92 are shown in fig . [ fig : b123 ] . 1=to 8 cm 2=to 8 cm 3=to 5.3 cm @xmath93 following this line of inductive reasoning , one obtains a general formula for the production of @xmath12 particles @xmath94 for any @xmath12 . in this formula , @xmath95 is the number of disconnected subdiagrams producing the @xmath12 particles , and @xmath75 denotes the sum of all @xmath74-particle cuts through the connected vacuum - vacuum diagrams . this formula gives the probability of producing @xmath12 particles to all orders in the coupling @xmath10 in a field theory with strong external sources . it is important to realize that all the details of the dynamics of the theory under consideration are hidden in the numbers @xmath76 , and that eq . ( [ eq : li-8.f ] ) is a generic form for transition probabilities when many disconnected graphs as well as vacuum - vacuum graphs contribute . this formula is therefore equally valid for qcd . it is useful to introduce a generating function for these probabilities , @xmath96\ , . \label{eq : li-8.g}\ ] ] one can use this object in order to compute moments of the distribution of probabilities @xmath97 where each prime " denotes a derivative with respect to @xmath98 . note that @xmath99 . this demonstrates explicitly that the exponential prefactor in eq . ( [ eq : li-8.f ] ) is essential for the theory to be unitary . though we derived eqs . ( [ eq : li-8.f ] ) and ( [ eq : li-8.g ] ) independently , we were alerted by dremin @xcite that an earlier version of the formulas in eqs . ( [ eq : li-8.f ] ) and ( [ eq : li-8.g ] ) was derived by gyulassy and kauffmann @xcite nearly 30 years ago also using general combinatoric arguments that did not rely on specific dynamical assumptions . these combinatoric rules for computing probabilities ( and moments thereof ) in field theory with strong external sources can be mapped on to the agk cutting rules derived in the context of reggeon field theory @xcite by writing eq . ( [ eq : li-8.f ] ) as @xmath100 where @xmath101 denotes the probability of producing @xmath12 particles in @xmath95 cut sub - diagrams . one can ask directly what the probability of @xmath95 cut sub - diagrams is by summing over @xmath12 to obtain @xmath102 this is a poisson distribution , which is unsurprising in our framework , because disconnected vacuum - vacuum graphs are uncorrelated by definition . the average number of such cut diagrams is simply @xmath103 an exact identification with ref . @xcite is obtained by expanding the exponential in eq . ( [ eq : li-9 ] ) to order @xmath104 , and defining @xmath105 where @xmath106 is the probability of having @xmath95 cut sub - diagrams out of @xmath107 sub - diagrams ( with @xmath104 being the number of uncut diagrams ) . this distribution of probabilities can be checked to satisfy the relations @xmath108 this set of identities is strictly equivalent to the eqs . ( 24 ) of ref . @xcite where @xmath104 and @xmath95 there are identified as the numbers of uncut and cut reggeons respectively . the first relation means that diagrams with two or more subdiagrams cancel in the calculation of the multiplicity . these relations are therefore a straightforward consequence of the fact that the distribution of the numbers of cut subdiagrams is a poisson distribution . they do not depend at all on whether these subdiagrams are `` reggeons '' or not . in the agk approach , the first identity in eq . ( [ eq : li-11 ] ) suggests that the average number of cut reggeons @xmath109 can be computed from diagrams with one cut reggeon alone . the average multiplicity satisfies the relation @xmath110 where @xmath111 is the average number of particles in one cut reggeon . computing this last quantity of course requires a microscopic model of what a reggeon is . before going on , it is useful to summarize what we have learnt at this stage about field theories with strong external sources . we derived a general formula in eq . ( [ eq : li-8.f ] ) for the probability to produce @xmath12 particles in terms of cut connected vacuum - vacuum diagrams , where @xmath76 is the sum of the terms with @xmath74 cuts . this formula is a purely combinatoric expression ; it does not rely on the microscopic dynamics generating the @xmath76 . nevertheless , it tells us several things that were not obvious . firstly , the probability distribution in eq . ( [ eq : li-8.f ] ) is not a poisson distribution , _ even at tree level _ , if any @xmath112 for @xmath113 . it is often assumed that classical dynamics is poissonian . we see here that the non - trivial correlations ( symbolized by non - zero @xmath76 terms with @xmath114 ) in theories with self - interacting fields can produce significant modifications of the poisson distribution . another immediate observation is that even the probability to produce one particle ( eq . ( [ eq : li-8.1 ] ) ) is completely non - perturbative in the coupling constant @xmath10 for arbitrarily small coupling . in other words , @xmath115 can not be expressed as an analytic expansion in powers of @xmath10 . therefore , while weak coupling techniques are certainly valid , such theories ( the cgc for instance ) are always non - perturbative . interestingly , we will see shortly that a simple expansion in powers of the coupling exists for moments of the probability distribution . finally , we saw that there was a simple mapping between the cutting rules first discussed in ref . @xcite and those for cut connected vacuum - vacuum graphs in field theories with strong sources . in the rest of this lecture , we shall sketch the derivation of explicit expressions for the @xmath12-particle probabilities and for the first moment of the multiplicity distribution . specifically , we will outline an algorithm to compute the average multiplicity up to next - to - leading order in the coupling constant . the probability for producing @xmath12 particles is given by the expression @xmath116 \left|\big<\p_1\cdots \p_n{}_{\rm out}\big|0_{\rm in}\big>\right|^2 \ ; , \label{eq : li-12}\ ] ] where @xmath117 . the well known lehman symanzik zimmerman ( lsz ) reduction formula @xcite relates the transition amplitude for producing @xmath12 particles from the vacuum to the residue of the multiple poles of green s functions of the interacting theory . it can be expressed as @xmath118 \ ; e^{i{\cal v}[\rho]}\ ; , \label{eq : li-13}\ ] ] where the factors of @xmath119 correspond to self - energy corrections on the cut propagators of the vacuum - vacuum diagrams . substituting the r.h.s . of this equation into eq . ( [ eq : li-12 ] ) , and noting that @xmath120 is the fourier transform of the propagator given in eq . ( [ eq : li-6 ] ) , we can write the probability @xmath11 directly as @xmath121 \ ; \left . e^{i{\cal v}[\rho_+]}\;e^{-i{\cal v}^*[\rho_- ] } \right|_{\rho_+=\rho_-=\rho}\ ; , \label{eq : li-14}\end{aligned}\ ] ] where @xmath122 $ ] is the operator @xmath123 the sources in the amplitude and the complex conjugate amplitude are labeled as @xmath124 and @xmath125 respectively to ensure that the functional derivatives act only on one of the two factors . a useful and interesting identity is @xmath126 } \ ; e^{i{\cal v}[\rho_+]}\;e^{-i{\cal v}^*[\rho_- ] } = e^{i{\cal v}_{_{sk}}[\rho_+,\rho_-]}\ ; , \label{eq : li-16}\ ] ] where @xmath127 $ ] is the sum of all connected vacuum - vacuum diagrams in the schwinger - keldysh formalism @xcite , with the source @xmath124 on the upper branch of the contour and likewise , @xmath125 on the lower branch . when @xmath128 , it is well known that this sum of all such connected vacuum - vacuum diagrams is zero . the generating function @xmath129 , from eqs . ( [ eq : li-8.g ] ) and ( [ eq : li-14 ] ) is simply @xmath130}\;e^{-i{\cal v}^*[\rho_- ] } \right|_{\rho_+=\rho_-=\rho}\ ; . \label{eq : li-17}\ ] ] from the expression for the operator @xmath131 in eq . ( [ eq : li-15 ] ) , it is clear that @xmath129 can be formally obtained by substituting the off - diagonal propagators @xmath132 in the usual cut vacuum - vacuum diagrams . we shall now proceed to discuss how one computes the average multiplicity ( @xmath133 ) of produced particles . from eqs . ( [ eq : li-17 ] ) and ( [ eq : li-15 ] ) , we obtain @xmath134_{\rho_+=\rho_-=\rho}\ ; , \label{eq : li-18}\ ] ] where @xmath135 and @xmath136 are the 1- and 2-point amputated green s functions in the schwinger - keldysh formalism : @xmath137}{\delta \rho_\pm(x)}\ ; , \nonumber\\ & & \gamma^{(+-)}(x , y ) \equiv \frac{\square_x+m^2}{z}\;\frac{\square_y+m^2}{z}\ ; \frac{\delta^2 i{\cal v}_{_{sk}}[\rho_+,\rho_-]}{\delta \rho_+(x)\delta \rho_-(y)}\ ; . \label{eq : li-19}\end{aligned}\ ] ] diagrammatically , @xmath138 can be represented as 1to 4 cm @xmath139 unlike the probabilities , there is a well defined power counting for the moments of the multiplicity distribution . this is simply because the overall absorption factor " @xmath140 present in the computation each probability , cancels when one computes averaged quantities . this is a crucial simplification , because it means that the moments of the distribution have a sensible perturbative expansion as a series in powers of @xmath141 . at leading order in the coupling constant , @xmath142 , only the left diagram in eq . ( [ eq : li-20 ] ) contributes . the right diagram , that contains the connected 2-point function , is a one loop diagram ; in our power counting ( see eq . ( [ eq : li-3.1 ] ) ) it starts at order @xmath143 . the lowest order , where we need only tree level diagrams , can therefore be expressed as 1=to 1.8 cm @xmath144 where the sum is over all the tree diagrams on the left and on the right of the propagator @xmath145 ( represented in boldface ) as well as a sum over the labels @xmath146 of the vertices whose type is not written explicitly . at this order , the mass in @xmath145 is simply the bare mass , and @xmath147 . the diagrams in eq . ( [ eq : li-21 ] ) can be computed using the cutkosky rules we discussed previously . beginning with one of the `` leaves '' of the tree ( attached to the rest of the diagram by a @xmath47 vertex for instance ) , one has two contributions @xmath148 and @xmath149 for the propagators connecting it to the vertex just below . ( the source can be factored out , because we set @xmath128 . ) this difference in the propagators gives @xmath150 where @xmath151 is the free _ retarded propagator_. ( likewise , @xmath152 . ) repeating this procedure recursively , propagators from all the leaves " down to the root are converted into retarded propagators . it is well known that the retarded solution @xmath153 of the classical equations of motion with the initial conditions @xmath154 and @xmath155 can be expressed as a sum of tree diagrams built with retarded propagators . the sum over all the trees on each side of the cut in eq . ( [ eq : li-21 ] ) can therefore be identified as 1=to 1.55 cm 2=to 1.55 cm @xmath156 from this discussion and eq . ( [ eq : li-18 ] ) , the leading order inclusive multiplicity can be expressed as @xmath157 using the identity @xmath158\phi_c(x)\right)$ ] and the boundary conditions obeyed by the retarded classical field @xmath153 , one obtains @xmath159\phi_c(x)\right|^2\ ; . \label{eq : li-25}\ ] ] the corresponding formula for gluon production in heavy ion collisions in the color glass condensate framework is @xmath160 where @xmath161 is the polarization vector for the produced gluon . this is precisely the expression that was computed in previous real time numerical simulations of yang mills equations _ for each configuration of color sources in each of the nuclei_. to compute the distribution of gluons , we need to average over the distribution over all possible color sources as specified in eq . ( [ eq : li-0 ] ) . we will discuss results from these simulations further in lecture ii . the leading order result in eqs . ( [ eq : li-25 ] ) and ( [ eq : li-26 ] ) is well known . we shall now discuss the computation to next - to - leading order in the coupling to order @xmath162 . at this order , both terms in eq . ( [ eq : li-18 ] ) contribute to the multiplicity . the right diagram in eq . ( [ eq : li-20 ] ) contributes with the blob evaluated at tree level , 1=to 2 cm @xmath163 this contribution to the inclusive multiplicity is analogous to that of quark - anti - quark pair production or gluon pair production to the respective average multiplicities for these quantities . the left diagram in eq . ( [ eq : li-20 ] ) , at this order , contains 1-loop corrections to diagrams of the kind displayed in eq . ( [ eq : li-21 ] ) . a blob on one side of the cut in eq . ( [ eq : li-20 ] ) is evaluated at the 1 loop level ( corresponding to the contribution from one loop correction to the classical field ) while the other blob is evaluated at tree level ( corresponding to the contribution from the classical field itself ) . this can be represented as 1=to 1.8 cm @xmath164 the inclusive multiplicity at nlo includes contributions from both eqs . ( [ eq : li-27 ] ) and ( [ eq : li-28 ] ) . to evaluate the diagram in eq . ( [ eq : li-27 ] ) , one needs to compute the propagator @xmath165 in the presence of the background field @xmath166 . this can be done by solving a lippmann - schwinger equation for @xmath167 @xcite . in practice , numerical solutions of this equation can be obtained only for _ retarded _ or _ advanced _ green s functions in the background field . it turns out that one can express @xmath167 in terms of these as @xmath168 where @xmath169 here @xmath170 ( @xmath171 ) is the free retarded ( advanced ) propagator and @xmath172 ( @xmath173 ) is the retarded ( advanced ) scattering @xmath174-matrix . substituting eq . ( [ eq : li-30 ] ) in eq . ( [ eq : li-29 ] ) and using the resulting expression in the second term of eq . ( [ eq : li-18 ] ) , the contribution of this term to the nlo multiplicity can be expressed as @xmath175 one can then show that @xcite @xmath176 \eta_q(x)\ ; , \label{eq : li-32}\ ] ] where @xmath177 is a small fluctuation field about @xmath153 and is the _ retarded _ solution of the partial differential equation @xmath178 with the initial condition @xmath179 when @xmath180 . note here that @xmath10 has the dimension of a mass . note also that , despite being similar , the equation for @xmath181 is not the classical equation of motion but is instead the equation of motion of a small fluctuation . this nlo contribution to the inclusive multiplicity can be computed by solving an initial value problem with boundary conditions set at @xmath182 . the other contribution of order @xmath162 to the average multiplicity is from the diagram in eq . ( [ eq : li-28 ] ) . this contribution can be written as @xmath183\phi_c(x ) \big]\nonumber\\ & & \qquad\qquad\times \big [ \lim_{x_0\to+\infty}\int d^3\x \ ; e^{ip\cdot x } \big[\partial_0-ie_p\big]\phi_{c,1}(x ) \big]^*+\mbox{c.c.}\nonumber\\ & & \label{eq : li-34}\end{aligned}\ ] ] the one loop contribution to the classical field 1=to 4 cm @xmath184 includes arbitrary insertions of the background field @xmath153 . following the discussion before eq . ( [ eq : li-22 ] ) of the cutkosky rules in this case , it can be written as @xcite @xmath185 we have used here the identity @xmath186 . in practice , @xmath187 can also be obtained as the _ retarded _ solution to the equation @xmath188 with an initial condition such that @xmath189 and its derivatives vanish at @xmath190 . the source term in this equation can be rewritten as @xmath191 , where @xmath192 . after a little algebra @xcite , one can show that @xmath193 here @xmath194 and @xmath195 are solutions of eq . ( [ eq : li-33 ] ) with plane wave initial conditions at @xmath196 of @xmath197 and @xmath198 respectively . we observe that @xmath199 contains ultraviolet divergences that arise from the integration over the momentum @xmath200 in eq . ( [ eq : li-38 ] ) . they can be identified with the usual 1-loop ultraviolet divergences of the @xmath8 field theory in the vacuum and must be subtracted systematically in order to obtain a finite result . to summarize , the two nlo contributions to the inclusive multiplicity , eqs . ( [ eq : li-31 ] ) and ( [ eq : li-34 ] ) can be computed systematically as follows . one first computes the lowest order classical field @xmath153 by solving the classical equations of motion , as a function of time , with the retarded boundary condition @xmath201 at @xmath202 . this computation was performed previously in the cgc framework @xcite . the small fluctuation equation of motion in eq . ( [ eq : li-33 ] ) is then solved in the background of @xmath153 , also with retarded boundary conditions at @xmath202 for the small fluctuation field @xmath177 . this is then sufficient , from eqs . ( [ eq : li-32 ] ) and ( [ eq : li-31 ] ) , to compute one contribution to the nlo multiplicity . to compute the other , solutions of the small fluctuation equations of motion can also be used , following eq . ( [ eq : li-38 ] ) , to determine @xmath199 . subsequent to this determination , the temporal evolution of the one loop classical field can be computed by solving eq . ( [ eq : li-37 ] ) , again with retarded boundary conditions at @xmath202 . finally , this result can be substituted in eq . ( [ eq : li-34 ] ) in order to compute the second contribution to the nlo multiplicity . albeit involved and technically challenging , the algorithm we have outlined is straightforward . the extension to the qcd case can be done . indeed , this computation is similar to a numerical computation ( performed by gelis , kajantie and lappi @xcite ) of the number of produced quark pairs in the classical background field of two nuclei . an interesting question we shall briefly consider now is whether we can directly compute the generating function itself to some order in the coupling ; even a leading order computation would contain a large amount of information . from eqs . ( [ eq : li-17 ] ) and ( [ eq : li-15 ] ) , we obtain has a well defined perturbative expansion in powers of @xmath141 ( that starts at the order @xmath203 ) , while this is not the case for @xmath129 itself . ] @xmath204\ ; , \label{eq : li-39}\ ] ] where @xmath205 and @xmath206 are defined as in eq . ( [ eq : li-19 ] ) , but must be evaluated with the substitution @xmath132 of the off - diagonal propagators . unsurprisingly , this equation involves the same topologies as that for the average multiplicity in eq . ( [ eq : li-20 ] ) . if we can compute the expression in eq . ( [ eq : li-39 ] ) even to leading order , the generating function can be determined directly by integration over @xmath98 , since we know that @xmath207 . at leading order , as we have seen , only the first term in eq . ( [ eq : li-39 ] ) contributes and ( using the same trick as in eq . ( [ eq : li-25 ] ) ) eq . ( [ eq : li-39 ] ) can be written as @xmath208_{x_0=-\infty}^{x_0=+\infty } \nonumber\\ & & \qquad\qquad\qquad\times \big [ \int d^3\y\ ; e^{-ip\cdot y}\;(\partial_y^0+ie_p)\,\phi_-(z|y ) \big]_{y_0=-\infty}^{y_0=+\infty}\ , , \nonumber\\ & & \label{eq : li-41}\end{aligned}\ ] ] where @xmath209 correspond to the tree diagrams 1=to 1.55 cm 2=to 1.55 cm @xmath210 evaluated with cutkosky s rules where the off - diagonal propagators @xmath145 are multiplied by a factor @xmath98 . we will now see why computing the generating function at leading order is significantly more complicated than computing the average multiplicity . the fields in eq . ( [ eq : li-42 ] ) can equivalently be expressed as the integral equation @xmath211\nonumber\\ & & \qquad\qquad\qquad -zg^0_{+-}(x , y)\left[j(y)-\frac{g}{2}\phi_-^2(z|y)\right ] \big\ } \nonumber\\ \phi_-(z|x)&=&i\int d^4y\ ; \big\{zg^0_{-+}(x , y)\left[j(y)-\frac{g}{2}\phi_+^2(z|y)\right]\nonumber\\ & & \qquad\qquad\qquad -g^0_{--}(x , y)\left[j(y)-\frac{g}{2}\phi_-^2(z|y)\right ] \big\}\ ; . \label{eq : li-43}\end{aligned}\ ] ] note now that when @xmath212 , @xmath213 ( defined in eq . ( [ eq : li-23 ] ) ) and the propagators in eqs . ( [ eq : li-43 ] ) , from eq . ( [ eq : li-22 ] ) , can be rearranged to involve only the retarded free propagator @xmath214 . it is precisely for this reason that the computation of the inclusive multiplicity simplifies ; the field @xmath166 can be determined by solving an initial value problem with boundary conditions at @xmath182 . this simplification clearly does not occur when @xmath215 . in order to understand the boundary conditions for @xmath209 in eqs . ( [ eq : li-42 ] ) and ( [ eq : li-43 ] ) , we begin by expressing them as a sum of plane waves , @xmath216 note here that @xmath217 is positive . since @xmath209 does not obey the free klein - gordon equation , the coefficients functions must themselves depend on time . however , assuming that both the source @xmath36 and the coupling constant @xmath10 are switched off adiabatically at large negative and positive times , the coefficient functions @xmath218 become constants in the limit of infinite time ( @xmath219 ) . the technique we use for determining the boundary conditions for the coefficients @xmath218 is reminiscent of the derivation of green s theorem in electrostatics . we will not go into the derivation here ( see ref . @xcite for the detailed derivation ) . the boundary conditions at @xmath220 are @xmath221 using eqs . ( [ eq : li-44 ] ) and ( [ eq : li-45 ] ) , we can write eq . ( [ eq : li-41 ] ) as @xmath222 therefore evaluating the generating function at leading order requires that we know the coefficient functions at @xmath223 . unlike the case of partial differential equations with retarded boundary conditions , there are no straightforward algorithms for finding the solution with the boundary conditions listed in eq . ( [ eq : li-45 ] ) . methods for solving these sorts of problems are known as `` relaxation processes '' . a fictitious `` relaxation time '' variable @xmath224 is introduced and the simulation is begun at @xmath225 with functions @xmath226 that satisfy all the boundary conditions but not the equation of motion . these fields evolve in @xmath224 with the equation ( preserving the boundary conditions for each @xmath224 ) @xmath227 which admits solutions of the eom as fixed points . the r.h.s . can in principle be replaced by any function that vanishes when @xmath226 is a solution of the classical eom . this function should be chosen to ensure that the fixed point is attractive . a similar algorithm has been developed recently to study the real time non - equilibrium properties of quantum fields @xcite . higher moments of the multiplicity distribution can also be computed following the techniques described here . interestingly , the variance ( at leading order ) can be computed once one obtains the solutions of the small fluctuation equations of motion . the computation is outlined in ref . thus both the leading order variance and the nlo inclusive multiplicity can be determined simultaneously . the variance contains useful information that can convey information about the earliest stages of a heavy ion collision . in particular , correlations between particles in a range of rapidity windows can provide insight into the early stages of a heavy ion collision @xcite . this provides a segue for the topic of the second lecture on the properties of the glasma . in the previous lecture , we outlined a formalism to compute particle production in field theories with strong time dependent sources . as argued previously , the color glass condensate is an example of such a field theory . in the cgc framework , the high energy factorization suggested by eq . ( [ eq : li-0 ] ) is assumed to compute final states . in this lecture , we will address the question of how one computes in practice the initial glasma fields after a heavy ion collision , what the properties of these fields are and outline theoretical approaches to understanding how these fields may thermalize to form a quark gluon plasma . a cartoon depicting the various stages of the spacetime evolution of matter in a heavy ion collision is shown in fig . [ fig : figii-00 ] . in the cgc effective field theory , hard ( large @xmath0 ) parton modes in each of the nuclei are lorentz contracted , static sources of color charge for the soft ( small @xmath0 ) wee parton , weizscker williams modes in the nuclei . here @xmath0 is the longitudinal momentum fraction of partons in the colliding nuclei . wee modes with @xmath228 and @xmath229 are coherent across the longitudinal extent of the nucleus and therefore couple to a large density of color sources . with increasing energy , the scale separating soft and hard modes shifts towards smaller values of @xmath0 ; how this happens can be quantified by a wilsonian rg @xcite . in a heavy ion collision , the color current corresponding to the large @xmath0 modes can be expressed as @xmath230 where the color charge densities @xmath231 of the two nuclei are independent sources of color charge on the light cone . let us recall that @xmath232 . the @xmath233 functions represent the fact that lorentz contraction has squeezed the nuclei to infinitesimally thin sheets . the absence of a longitudinal size scale ensures that the gauge fields generated by these currents will be boost - invariant they are independent of the space time rapidity @xmath234 . the gauge fields before the collision are obtained by solving the yang - mills equations @xmath235 where @xmath236 $ ] and @xmath237 $ ] are the gauge covariant derivative and field strength tensor , respectively , in the fundamental representation and @xmath238 $ ] denotes a commutator . before the nuclei collide ( @xmath239 ) , a solution of the equations of motion is @xcite @xmath240 where , here and in the following , the transverse coordinates @xmath241 are labeled by the latin index @xmath242 . the subscript @xmath58 on the @xmath243-functions denote that they are smeared by an amount @xmath58 in the respective @xmath244 light cone directions . we require that the functions @xmath245 ( @xmath246 denote the labels of the colliding nuclei ) are such that @xmath247 they are pure gauge solutions of the equations of motion . the gauge fields , just as the weizscker williams fields in qed , are therefore plane polarized sheets of radiation before the collision . the functions @xmath248 satisfy @xmath249 this equation has an analytical solution given by @xcite @xmath250 to obtain this result one has to assume path ordering in @xmath244 respectively for nucleus 1 and 2 ; we assume that the limit @xmath251 is taken at the end of the calculation . we now introduce the proper time @xmath252 the initial conditions for the evolution of the gauge field in the collision are formulated on the proper time surface @xmath253 . they are obtained @xcite by generalizing the previous ansatz for the gauge field to @xmath254 where we adopt the fock schwinger gauge condition @xmath255 . this gauge is an interpolation between the two light cone gauges @xmath256 on the @xmath257 surfaces respectively . the gauge fields @xmath258 in the forward light cone can be determined from the known gauge fields @xmath259 of the respective nuclei before the collision by invoking a physical `` matching condition '' which requires that the yang - mills equations @xmath260 be regular at @xmath261 . the @xmath233-functions of the current in the yang mills equations therefore have to be compensated by identical terms in spatial derivatives of the field strengths . interestingly , it leads to the unique solution @xcite @xmath262\ ; . \label{eq : lii-6}\end{aligned}\ ] ] further , the only condition on the derivatives of the fields that would lead to regular solutions are @xmath263 . for the purpose of solving the yang - mills equations for a heavy - ion collision on a lattice , we shall work with the @xmath264 co - ordinates and re - express the initial conditions for the fields and their derivatives in terms of the fields and their conjugate momenta in these co - ordinates . our gauge condition is @xmath265 , and the initial conditions in eq . ( [ eq : lii-6 ] ) for the functions @xmath266 and @xmath267 at @xmath261 can be expressed in terms of the fields @xmath268 where we have made manifest the fact that these fields are boost - invariant i.e. independent of @xmath181 . this is a direct consequence of the assumption in eq . ( [ eq : lii-1 ] ) that the currents are @xmath233-function sources on the light cone . the light cone hamiltonian in @xmath265 gauge , in this case of boost invariant fields , can be written as @xcite @xmath269 \ , . \label{eq : lii-8}\ ] ] here the conjugate momenta to the fields are the chromo - electric fields @xmath270 note that the contribution of the hard valence current does not appear explicitly in the @xmath265 hamiltonian expressed in ( @xmath271 , @xmath181 ) co - ordinates . the dependence on the color source densities is entirely contained in the dependence of the initial conditions on the source densities . boost invariance simplifies the problem tremendously because the qcd hamiltonian in this case is dimensionally reduced " to a @xmath272-d ( qcd + adjoint scalar field ) hamiltonian . in terms of these glasma fields and their conjugate momenta , the initial conditions in eq . ( [ eq : lii-6 ] ) at @xmath261 can be rewritten as @xmath273\;. \label{eq : lii-10}\end{aligned}\ ] ] the magnetic fields being defined as @xmath274 , these initial conditions suggest that @xmath275 and @xmath276 . note that the latter condition follows from the constraint on the derivatives of the gauge field that ensure regular solutions at @xmath261 . thus one obtains the interesting results that the initial glasma fields correspond to large initial longitudinal electric and magnetic fields ( @xmath277 ) and zero transverse electric and magnetic fields ( @xmath278 ) . this is in sharp contrast to the electric and magnetic fields of the nuclei before the collision ( the weizscker williams fields ) which are purely transverse ! their importance was emphasized recently by lappi and mclerran @xcite who also coined the term glasma " to describe the properties of these fields prior to equilibration . an immediate consequence of these initial conditions , as noted by kharzeev , krasnitz and venugopalan @xcite , is that non - zero chern - simons charge can be generated in these collisions . the dynamics of the chern - simons number in nuclear collisions however differs from the standard discussion in two ways . firstly , the time translational invariance of the fields is broken by the singularity corresponding to the collision . secondly , due to the boost invariance of the solutions , there can be no non - trivial boost invariant gauge transformations . this can be seen as follows . in ref . @xcite , it was shown that the chern - simons charge per unit rapidity could be expressed as @xmath279 because this density is manifestly invariant under rapidity dependent transformations , such transformations ( which correspond to sphaleron transitions ) can not change the chern - simons charge . thus sphaleron transitions are disallowed for boost - invariant field configurations . ( [ eq : lii-10 ] ) tell us that @xmath280 ; therefore the chern - simons charge generated in a given window in rapidity at a time @xmath271 is simply , by definition , @xmath281 . since @xmath181 s of either sign are equally likely , the ensemble average @xmath282 is zero . however , @xmath283 is non zero . its value was computed in ref . the topological charge squared per unit rapidity generated for rhic and lhc collisions is about 1 - 2 units . in contrast , estimates of the same quantity in a thermal plasma are one to two orders of magnitude larger . if boost invariance is violated ( as we shall soon discuss ) , sphaleron transitions can go , and can potentially be large . this possibility , in a different formulation , was discussed previously by shuryak and collaborators @xcite . we shall now discuss the particle distributions that correspond to the gauge fields and their conjugate momenta in the forward light cone . from the hamilton equations @xmath284 the yang - mills equations are @xmath285 they also satisfy the gauss law constraint @xmath286 these equations are non - linear and have to be solved numerically . a lattice discretization is convenient because it preserves gauge invariance explicitly . one can write down the analogue of the well known kogut susskind hamiltonian in this case and solve eq . ( [ eq : lii-13 ] ) numerically on a discretized spatial is treated as a continuous variable , that can have increments as small as required to reach the desired accuracy in the solution of the equations of motion . ] lattice with the initial conditions in eq . ( [ eq : lii-10 ] ) . we shall not describe the numerical procedure here but instead refer the reader to refs . @xcite . solving hamilton s equations , the average gluon multiplicity can be computed using precisely the formula we discussed previously in eq . ( [ eq : li-26 ] ) . the result in eq . ( [ eq : li-26 ] ) is the average multiplicity for _ a _ configuration of color charge densities in each of the nuclei . it is an average in the sense of being the first moment of the multiplicity distribution . this multiplicity has to be further averaged over the distribution of sources @xmath19 $ ] and @xmath287 $ ] , as specified in eq . ( [ eq : li-0 ] ) . these weight functionals have to be specified at an initial scale @xmath288 in rapidity , and are then evolved to higher rapidities by the jimwlk renormalization group equation @xcite . for the purposes of computing the average multiplicity in _ central _ au - au collisions at rhic , i.e. for rapidities where evolution effects _ a la jimwlk _ are not yet important , the weight functionals @xmath23 $ ] are gaussian distributions specified in the mv model ( discussed briefly in the introduction to these lectures ) : @xmath289 = \exp\left ( - \int d^2 \x_\perp \frac{\rho_{1,2}^a\rho_{1,2}^a}{2\ , \lambda_s^2 } \right)\ ; , \label{eq : lii-15}\ ] ] here @xmath290 , where @xmath291 is the color charge squared of the sources per unit area . the nuclei , for simplicity , are assumed to be identical nuclei . @xmath292 is the only dimensionful scale ( besides the nuclear radius @xmath293 ) in the problem . it is simply related , in leading order , to the nuclear saturation scale @xmath294 by the expression @xmath295 , where @xmath296 is an infrared scale of order @xmath297 . the nuclear saturation scale , performing a simple extrapolation of the hera data on the gluon distribution of the proton to au nuclei , is of the order @xmath298 gev at rhic energies in several estimates @xcite . for @xmath299 gev , this corresponds to a value @xmath300 gev . clearly , there are logarithmic uncertainties in this estimate at least of order 10% . for the rest of this lecture , we will assume the gaussian form in eq . ( [ eq : lii-15 ] ) for the averaging over sources ; modifications to account for the ( very likely ) significant effects of small @xmath0 quantum evolution will have to be considered at lhc energies . in order to compute gluon number distributions , we impose the transverse coulomb gauge @xmath301 to fix the gauge freedom completely . the result for the number distributions , averaged over the sources in eq . ( [ eq : lii-15 ] ) , is computed at a time @xmath302 to be @xmath303 where @xmath304 is a function of the form @xmath305^{-1 } & ( k_\perp/\lambda_s \leq 1.5 ) \\ \\ a_2\,\lambda_s^4\;\ln(4\pi k_\perp/\lambda_s)\;k_\perp^{-4 } & ( k_\perp/\lambda_s > 1.5 ) \\ \end{array } \right.\ ; , \label{eq : lii-17}\end{aligned}\ ] ] with @xmath306 , @xmath307 , @xmath308 , and @xmath309 . these results are plotted in fig . [ fig : dndkt ] they are compared to those computed independently by lappi @xcite . the different lines in the figure correspond to different lattice discretizations ; the differences at large @xmath13 therefore indicate the onset of lattice artifacts , which can be eliminated by going closer to the continuum limit ( larger lattices ) . comparison of gluon transverse momentum distributions per unit area as a function of @xmath310 . knv i ( circles ) : the number defined with @xmath311 taken to mean the lattice wave number along one of the principal directions . knv ii ( squares ) and lappi ( solid line ) : the number defined by averaging over the entire brillouin zone and with @xmath311 taken to mean the frequency @xmath312 . , width=259 ] from eq . ( [ eq : lii-17 ] ) , the number distribution at large @xmath13 has the power law dependence one expects in perturbative qcd at leading order . for small @xmath13 , the result is best fit by a massive 2-d bose - einstein distribution even though one is solving classical equations of motion ! there is an interesting discussion in the statistical mechanics literature that suggests that such a distribution may be generic for classical glassy " systems far from equilibrium @xcite . another interesting observation is that the non - perturbative real time dynamics of the gauge fields generates a mass scale @xmath107 which makes the number distributions infrared safe for finite times . such a `` plasmon mass '' can be extracted from the single particle dispersion relation ; it behaves dynamically as a function of time precisely as a screening mass does @xcite . this can be seen in fig . [ fig : plasmon_tau ] . as we shall discuss shortly , this plasmon mass can be related to the growth rate of instabilities in the glasma . the total transverse energy and number can be obtained independently , from the hamiltonian density and from a gauge invariant relaxation ( cooling ) technique respectively . these agree with those obtained by integrating eq . ( [ eq : lii-16 ] ) over @xmath13 and can be expressed as @xmath313 where @xmath314 and @xmath315 for the wide range @xmath316 respectively . for larger values of @xmath317 , the functions @xmath318 and @xmath319 have a weak logarithmic dependence on @xmath320 . if we assume parton - hadron duality and directly compare the number of gluons from eq . ( [ eq : lii-18 ] ) to the number of hadrons measured at @xmath321 in @xmath322 gev / nucleon au - au collisions at rhic , one obtains a good agreement for @xmath323 gev . this value is a little larger than the values we extracted from extrapolations of the hera data ; one should however keep in mind that additional contributions to the multiplicity of hadrons will accrue from quark and gluon production at next - to - leading order @xcite . if we include these contributions , as we hope to eventually , @xmath324 will be lower than this value . the `` formation time '' @xmath325 , defined as the time when the energy density @xmath326 behaves as @xmath327 , is defined as @xmath328 , where @xmath329 in the range of interest . the initial energy density for times @xmath330 ( @xmath331 fm for @xmath332 gev)is then @xmath333 this energy density , again for @xmath332 gev ( and @xmath334 ) , is @xmath335 gev/@xmath336 at @xmath337 . because the energy density is ultraviolet sensitive , this number is probably an overestimate because the spectrum at large @xmath338 in practice falls much faster than the lowest order estimate in eq . ( [ eq : lii-17 ] ) . in a recent paper @xcite , lappi has shown that the energy density computed in this framework , at early times has the form @xmath339 ; it is finite for any @xmath340 but is not well defined strictly at @xmath261 . in the discussion up to this point , we have assumed that the color charge squared per unit area of the source , @xmath341 , is constant . however , for finite nuclei , this is not true and one can define an impact parameter dependent @xmath324 , i.e. @xmath342 . this generalization , in the classical yang - mills framework described here was discussed previously in ref . @xcite and is given by @xmath343 where @xmath344 is the nuclear thickness profile , @xmath345 is the transverse coordinate vector ( the reference frame here being the center of the nucleus ) , @xmath346 is the woods - saxon nuclear density profile , and @xmath347 is the color charge squared per unit area in the center of the nucleus . one can use this expression to compute the multiplicity as a function of impact parameter in the collision . then , by using a glauber model to relate the average impact parameter to the average number of participants @xcite , one can obtain the dependence of the multiplicity on the number of participants . previous computations of the centrality dependence of the multiplicity and of rapidity distributions were performed in the kln approach @xcite . there however , unlike eq . ( [ eq : lii-20 ] ) , the saturation scale depends on the number of participant nucleons : @xmath348 with @xmath349\ ; . \label{eq : lii-22}\ ] ] in this formula , @xmath350 is the nucleon - nucleon cross - section , and @xmath351 the impact parameter between the two nuclei . note that as this form involves the thickness functions of both nuclei @xmath5 and @xmath352 , it is manifestly not universal in contrast to the definition in eq . [ eq : lii-20 ] . for the centrality dependence of the multiplicity distributions , the saturation scales defined through eqs . ( [ eq : lii-20 ] ) or ( [ eq : lii-21 ] ) lead to very similar results . this is because the multiplicity , at any particular @xmath345 , depends on the lesser of the two saturation scales , say @xmath353 . the dependence on the `` non - universal '' factor in eq . ( [ eq : lii-22 ] ) is then weak because , by definition , @xmath354 is large . however , the two prescriptions can be distinguished by examining a quantity of phenomenological importance , the eccentricity @xmath58 defined as @xmath355 this quantity is a measure of the asymmetry of the overlap region between the two nuclei in collisions at non zero impact parameter . in an ideal hydrodynamical description of heavy ion collisions , a larger initial eccentricity may lead to larger elliptic flow @xcite than observed , thereby necessitating significant viscous effects . comparisons of model predictions , with different initial eccentricities , to data may therefore help constrain the viscosity of the quark gluon plasma . in fig . [ fig : ecc ] , we show results for the eccentricity from the @xmath13 factorized kln approach with the saturation scale defined as in eq . ( [ eq : lii-21 ] ) compared to the classical yang - mills ( cym ) result computed with the definition in eq . ( [ eq : lii-20 ] ) . is denoted by cym @xcite . the traditional initial eccentricity used in hydrodynamics is a linear combination of mostly `` glauber @xmath356 '' and a small amount of `` glauber @xmath357 '' . the `` kln '' curve is the eccentricity obtained from the cgc calculation in refs . , width=259 ] the kln @xmath356 definition of @xmath294 leads to the largest eccentricity . the universal cym definition gives smaller values of @xmath58 albeit larger than the traditional parameterization ( used in hydrodynamical model computations ) where the energy density is taken to be proportional to the number of participating nucleons . this result is also shown to be insensitive to two different choices of the infrared scale @xmath107 which regulates the spatial extent of the coulomb tails of the gluon distribution . a qualitative explanation of the differences in the eccentricity computed in the two approaches is given in ref . @xcite we refer the reader to the discussion there . also , a further elaboration of the discussion of refs . @xcite was very recently presented in ref . @xcite the revised curves our closer to the result in ref . @xcite . our discussion thus far of the glasma has assumed strictly boost invariant initial conditions on the light cone , of the form specified in eq . ( [ eq : lii-10 ] ) . however , this is clearly an idealization because it requires strict @xmath233-function sources as in eq . ( [ eq : lii-1 ] ) , _ and _ that one completely disregards quantum fluctuations . because the collision energy is ultra - relativistic and because quantum fluctuations are suppressed by one power of @xmath358 , this was believed to be a good approximation . in particular , it was not realized that violations of boost invariance lead to a non - abelian version of the weibel instability @xcite well known in electromagnetic plasmas . to understand the potential ramifications of this instability for thermalization , let us first consider where the boost invariant results lead us . from eq . ( [ eq : lii-19 ] ) , it is clear that the energy density is far from thermal in which event , it would decrease as @xmath359 . the momentum distributions become increasingly anisotropic : @xmath360 and @xmath361 . once the particle - like modes of the classical field ( @xmath338 ) begin to scatter , the occupation number of the field modes begins to decrease . how this occurs through scattering was outlined in an elegant scenario dubbed `` bottom up '' by baier et al @xcite . at very early times , small elastic scattering of gluons with @xmath362 dominates and is responsible for lowering the gluon occupation number . the debye mass @xmath363 ( see fig . [ fig : plasmon_tau ] ) sets the scale for these scattering , and the typical @xmath364 is enhanced by collisions . one obtains @xmath365 . from this dependence , one can estimate that the occupation number of gluons is @xmath366 for @xmath367 . for proper times greater than these , the classical field description becomes less reliable . in the bottom up scenario , soft gluon radiation from @xmath368 scattering processes becomes important at @xmath369 . the system thermalizes shortly thereafter at @xmath370 with a temperature @xmath371 . the thermalization time scale in this scenario is parametrically faster than that obtained by solving the boltzmann equation for @xmath372 processes , which gives @xmath373 @xcite . the debye mass scale is key to the power counting in the bottom up scenario . however , as pointed out recently @xcite this power counting is affected by an instability that arises from a change in sign of the debye mass squared for anisotropic momentum distributions @xcite . the instability is the non - abelian analog of the weibel instability @xcite in electromagnetic plasmas and was discussed previously in the context of qcd plasmas by mrowczynski @xcite . one can view the instability , in the configuration space of the relevant fields , as the development of specific modes for which the effective potential is unbound from below @xcite . detailed simulations in the hard - loop effective theory in @xmath374-dimensions @xcite and in @xmath375-dimensions @xcite have confirmed the existence of this non - abelian weibel instability . particle field simulations of the effects of the instability on thermalization have also been performed recently @xcite . all of these simulations consider the effect of instabilities in systems at rest . however , as discussed previously , the glasma expands into the vacuum at nearly the speed of light . are they seen in the glasma ? no such instabilities were seen in the boost invariant @xmath272-d numerical simulations . in the rest of this lecture , we will discuss the consequences of relaxing boost invariance in ( now ) @xmath375-d numerical simulations of the glasma fields -d numerical simulations is based on work in refs . @xcite ] ; as may be anticipated , non - abelian weibel instabilities also arise in the glasma . in heavy - ion collisions , the initial conditions on the light cone are never exactly boost invariant . besides the simple kinematic effect of lorentz contraction at high energies , one also has to take into account quantum fluctuations at high energies . for instance , as we discussed in the last lecture , we will have small quantum fluctuations at nlo , for each configuration of the color sources , which are not boost invariant . parametrically , from the power counting discussed there , quantum fluctuations may be of order unity , compared to the leading classical fields which are of order @xmath376 . in the following , we will discuss two simple models of initial conditions containing rapidity dependent fluctuations . a more complete theory should specify , from first principles , the initial conditions in the boost non - invariant case . we will discuss later some recent work in that direction . the only condition we impose is that these initial conditions satisfy gauss law . we construct these by modifying the boost - invariant initial conditions in eq . ( [ eq : lii-10 ] ) to @xmath377 + \delta e_\eta(\eta,\x_\perp,)\ ; , \label{eq : lii-24}\end{aligned}\ ] ] while keeping @xmath378 unchanged . the rapidity dependent perturbations @xmath379 are in principle arbitrary , except for the requirement that they satisfy the gauss law . for these initial conditions , it takes the form @xmath380 the boost invariance violating perturbations are constructed as follows . * we first generate random configurations @xmath381 with @xmath382 * next , for our first model of rapidity perturbations , we generate a gaussian random function @xmath383 with amplitude @xmath384 @xmath385 for the second model , we also generate a gaussian random function , but subsequently remove high - frequency components of @xmath383 @xmath386 where @xmath387 acts as a `` band filter '' suppressing the high frequency modes . this model is introduced because the white noise gaussian fluctuations of the previous model leads to identical amplitudes for all modes . as a consequence , the high momentum modes dominate bulk observables such as the pressure . the unstable modes we wish to focus on are sensitive to infrared modes at early times but their effects are obscured by the higher momentum modes from the white noise spectrum . this is particularly acute for large violations of boost invariance . therefore , damping these high frequency modes allows us to also study the effect of instabilities for larger values of @xmath384 , or `` large seeds '' that violate boost - invariance . * for both models , once @xmath383 is generated , we obtain for the fluctuation fields @xmath388 these fluctuations , by construction , satisfy gauss law . to implement rapidity fluctuations in the above manner , one requires @xmath389 . this is a consequence of the @xmath264 coordinates , as can be seen from the fact that the jacobian for the transformation from cartesian coordinates vanishes in this limit . ] and does not have a physical origin . we therefore implement these initial conditions for @xmath390 with @xmath391 . our results are only weakly dependent on the specific choice of @xmath392 . the primary gauge invariant observables in simulations of the classical yang - mills equations are the components of the energy - momentum tensor @xcite . we will discuss specifically @xmath393\ ; , \nonumber\\ & & \tau^2 t^{\eta \eta}=\tau^{-2}\ { \rm tr}\left[f_{\eta i}^2+e_i^2\right ] -{\rm tr}\left[f_{xy}^2+e_\eta^2\right]\;. \label{eq : lii-30}\end{aligned}\ ] ] note that the hamiltonian density is @xmath394 . these components can be expressed as @xmath395 which correspond to @xmath271 times the mean transverse and longitudinal pressure , respectively . when studying the time evolution of rapidity - fluctuations , it is useful to introduce fourier transforms of observables with respect to the rapidity . for example , @xmath396 where @xmath397 denotes averaging over the transverse coordinates @xmath398 . apart from @xmath399 , this quantity would be strictly zero in the boost - invariant ( @xmath400 ) case , while for non - vanishing @xmath384 and @xmath401 , @xmath402 has a maximum amplitude for some specific momentum @xmath401 . using a very small but finite value of @xmath384 , this maximum amplitude is very much smaller than the corresponding amplitude of a typical transverse momentum mode . the physical parameters in this study are @xmath403 ( = @xmath324 ; see the discussion after eq . ( [ eq : lii-15 ] ) ) , @xmath404 , where @xmath293 is the nuclear radius , @xmath384 , the initial size of the rapidity dependent fluctuations and finally , the band filter @xmath387 , which as discussed previously , we employ only for large values of @xmath384 . physical results are expressed in terms of the dimensionless combinations @xmath405 and @xmath406 . for rhic collisions of gold nuclei , one has @xmath407 ; for collisions of lead nuclei at lhc energies , this will be twice larger . the physical properties of the spectrum of fluctuations ( specified in our simple model here by @xmath384 and @xmath387 ) will presumably be further specified in a complete theory . for our present purposes , they will be treated as arbitrary parameters , and results presented for a large range in their values . briefly , the lattice parameters in this study , in dimensionless units , are ( i ) @xmath408 and @xmath409 , the number of lattice sites in the @xmath345 and @xmath181 directions respectively ; ( ii ) @xmath410 and @xmath411 , the respective lattice spacings ; ( iii ) @xmath412 and @xmath413 , the time at which the simulations are initiated and the stepping size respectively . the continuum limit is obtained by holding the physical combinations @xmath414 and @xmath415 fixed , while sending @xmath416 , @xmath410 and @xmath411 to zero . for this study , we pick @xmath417 units of rapidity . the magnitude of violations of boost invariance , as represented by @xmath384 , is physical and deserves much study . the initial time is chosen to ensure that for @xmath400 , we recover earlier @xmath272-d results ; we set @xmath418 . our results are insensitive to variations that are a factor of 2 larger or smaller than this choice . for further details on the numerical procedure we refer the reader to refs . @xcite . in fig . [ fig : maxfm ] , we plot the maximal value and the related discussion there . ] of @xmath419 at each time step , as a function of @xmath405 . the data are for a @xmath420 lattice and correspond to @xmath421 and @xmath417 . the maximal value remains nearly constant until @xmath422 , beyond which it grows rapidly . a best fit to the functional form @xmath423 gives @xmath424 for @xmath425 ; the coefficients @xmath426 , @xmath427 are small numbers proportional to the initial seed . it is clear from fig . [ fig : maxfm ] that the form @xmath428 is preferred to a fit with an exponential growth in @xmath271 . this @xmath428 growth of the unstable soft modes is closely related to the mass generated by the highly non - linear dynamics of soft modes in the glasma . as we discussed previously , and showed in fig . [ fig : plasmon_tau ] , a plasmon mass @xmath429 , is generated . after an initial transient behavior , it is of the form @xmath430 with @xmath431 ( this parameterization is robust as one approaches the continuum limit ) . the dependence on @xmath406 is weak . in the finite temperature hard thermal loop ( htl ) formalism for anisotropic plasmas , the maximal unstable modes of the stress - energy tensor grow as @xmath432 ) , where the growth rate @xmath433 satisfies the relation @xmath434 for maximally anisotropic particle distributions @xcite . here @xmath435 where @xmath436 is the anisotropic single particle distribution of the hard modes . it was shown in ref . @xcite that @xmath437 for both isotropic and anisotropic plasmas . one therefore obtains @xmath438 for @xmath421 , @xmath439 gives the coefficient @xmath440 , which is quite close to the value obtained by a fit to the numerical data @xmath441 . however , this agreement is misleading because a proper treatment would give in the exponent @xmath442 , with @xmath443 . the observed growth rate is approximately half of that predicted by directly applying the htl formalism to the glasma . despite obvious similarities , it is not clear that the equivalence can be expected to hold at this level of accuracy . nevertheless , the similarities in the two frameworks is noteworthy as we will discuss now . @xmath444 as a function of momentum @xmath401 , averaged over 160 initial conditions on a @xmath445 lattice with @xmath446 and @xmath447 , @xmath448 . four different simulation times show how the softest modes start growing with an distribution reminiscent of results from hard - loop calculations @xcite . also indicated are the respective values of @xmath449 for three values of @xmath450 ( see text for details).,width=259 ] in fig . [ fig : earlytimes ] we show the ensemble - averaged @xmath451 for four different simulation times . the earliest time ( @xmath452 ) shows the configuration before the instability sets in . at the next time , one sees a bump above the background , corresponding to the distribution of unstable modes . the unstable mode with the biggest growth rate ( the cusp of the `` bumps '' in fig.[fig : earlytimes ] ) was precisely what was used to determine the maximal growth rate @xmath453 by fitting the time dependence of this mode to the form @xmath454 . the two later time snapshots shown in fig . [ fig : earlytimes ] ( for @xmath455 and @xmath456 ) indicate that the growth rate of the unstable modes closely resembles the analytic prediction from hard - loop calculations @xcite . in fig . [ fig : earlytimes ] , @xmath449 is the largest mode number that is sensitive to the instability . its behavior is shown in fig . [ fig : numax ] . , on a lattices with @xmath457 , @xmath458 and various violations of boost - invariance @xmath384 . the dashed line represents the linear scaling behavior . right : time evolution of the maximum amplitude @xmath451 . when this amplitude reaches a certain size ( denoted by the dashed horizontal line ) , @xmath449 starts to grow fast . ] , on a lattices with @xmath457 , @xmath458 and various violations of boost - invariance @xmath384 . the dashed line represents the linear scaling behavior . right : time evolution of the maximum amplitude @xmath451 . when this amplitude reaches a certain size ( denoted by the dashed horizontal line ) , @xmath449 starts to grow fast . ] from this figure , one observes an underlying trend indicating a linear increase of @xmath449 with approximately @xmath459 . for sufficiently small violations of boost - invariance , this seems to be fairly independent of the transverse or longitudinal lattice spacing we have tested . for much larger violations of boost - invariance or sufficiently late times one observes that @xmath449 deviates strongly from this `` linear law '' . in fig . [ fig : numax ] we show that this deviation seems to occur when the maximum amplitude of @xmath460 reaches a critical size , independent of other simulation parameters . this critical value is denoted by a dashed horizontal line and has the magnitude @xmath461 in the dimensionless units plotted there . a possible explanation for this behavior is that once the transverse magnetic field modes in the glasma ( with small @xmath13 ) reach a critical size , the corresponding lorenz force in the longitudinal direction is sufficient to bend `` particle '' ( hard gauge mode ) trajectories out of the transverse plane into the longitudinal direction . this is essentially what happens in electromagnetic plasmas . note however that in electromagnetic plasmas the particle modes are the charged fermions.in contrast , the particle modes here are the hard ultraviolet transverse modes of the field itself . we will comment shortly on how this phenomenon may impact thermalization . , for @xmath462 , @xmath458 , @xmath463 , @xmath464 and @xmath409 ranging from @xmath465 to @xmath466 . larger lattices correspond to smaller @xmath384 . this explains why the early - time behavior is not universal for the simulations shown here.,width=259 ] the saturation seen in the right of fig . [ fig : numax ] is shown clearly in fig . [ fig : lssat ] where we plot , in order to suppress the ultraviolet modes in the initial fluctuation . note further that for larger seeds the instability systematically saturates at earlier times , as is clear from the right of fig . [ fig : numax ] . ] the temporal evolution of the maximum amplitude of the ensemble averaged @xmath467 , for lattices with different @xmath411 . early times in this figure ( @xmath468 ) correspond to the stage when the weibel instability is operative . interestingly , the simulations show saturation of the growth at approximately the same amplitude . these preliminary results are similar to the phenomenon of `` non - abelian saturation '' , found in the context of simulations of plasma instabilities in the hard loop framework @xcite . in the small seed case , the longitudinal fluctuations carry a tiny fraction of the total system energy . in the large seed case , in contrast , for the simulations shown here , the initial energy contained in the longitudinal modes is @xmath469% of the total system energy . in reality , we expect this fraction to be significantly larger . however , this would require us to study the dynamics on even larger longitudinal lattices than those included in this study to ensure that the contributions to the pressure from ultraviolet modes are not contaminated by lattice artifacts . in the left fig . [ fig : tetaeta ] , we plot @xmath470 as a function of @xmath271 for different lattice spacings @xmath411 . for large @xmath411 ( low lattice uv cutoff ) , the longitudinal pressure is consistent with zero ; it is clearly finite when the lattice uv cutoff is raised . however , the rise saturates as there is no notable difference between the simulations for the three smallest values of the lattice spacing . at face value , this result suggests that the rise in the longitudinal pressure is physical and not a discretization artifact . clearly , further studies on larger transverse lattices are needed to strengthen this claim . , for lattices with @xmath462 , @xmath458 , @xmath463 , @xmath471 and @xmath409 ranging from @xmath465 to @xmath466 . note : reduced statistical ensemble of 2 runs for @xmath472 . right : hamiltonian density @xmath473 , @xmath474 and @xmath475 , for @xmath476 and @xmath445 lattices . the energy density is fit to @xmath477 at late times . all curves are calculated on lattices with @xmath446 , @xmath458 and @xmath471 . ] , for lattices with @xmath462 , @xmath458 , @xmath463 , @xmath471 and @xmath409 ranging from @xmath465 to @xmath466 . note : reduced statistical ensemble of 2 runs for @xmath472 . right : hamiltonian density @xmath473 , @xmath474 and @xmath475 , for @xmath476 and @xmath445 lattices . the energy density is fit to @xmath477 at late times . all curves are calculated on lattices with @xmath446 , @xmath458 and @xmath471 . ] in the right of fig . [ fig : tetaeta ] , we investigate the time evolution of the transverse pressure and the energy density for ( i ) a simulation with a low uv cutoff ( @xmath476 lattice ) and ( ii ) a simulation with a high uv cutoff ( @xmath445 lattice ) . we observe that the rise in the mean longitudinal pressure accompanies a drop both in the mean transverse pressure and energy density . this result is consistent with the previously advocated physical mechanism of the lorenz force bending transverse uv modes ( thereby decreasing the transverse pressure ) into longitudinal uv modes ( simultaneously raising the longitudinal pressure ) , thereby pushing the system closer to an isotropic state . the energy density depends on the proper time as @xmath478 , which , while not the free streaming result of @xmath479 , is also distinct from the @xmath480 required for a locally isotropic system undergoing one dimensional expansion . furthermore , the time scales ( noting that for rhic energies @xmath481 gev ) are much larger than the time scales of interest for early thermalization of the glasma into a qgp . similar results were obtained in an analytical model of the late time behavior of expanding anisotropic fields in the hard loop formalism @xcite . nevertheless , these simulations are proof in principle that non - trivial dynamics can take place in the glasma driving the system towards equilibrium . a mode analysis along the lines of that performed recently by bdeker and rummukainen @xcite is required to understand this dynamics in greater detail . in particular , it would be useful to understand whether the rapid shift of unstable modes to the ultraviolet ( as seen in fig . [ fig : numax ] ) is due to a turbulent kolmogorov cascade as discussed in refs . the most important task however is to understand from first principles the spectrum of initial fluctuations that break boost invariance . a first step in this direction was taken in ref . this issue is closely related to the nlo computation of small fluctuations outlined in lecture i. at nlo , some of these quantum fluctuations are accompanied by large logs in @xmath0 . thus for @xmath482 , these effects are large . to completely understand which contributions from the small fluctuations can be absorbed in the evolution of the initial wavefunctions . ] , and to isolate the remainder that contributes to the spectrum of initial fluctuations , requires that we demonstrate factorization for inclusive multiplicities . this work is in progress @xcite . finally , another interesting problem is whether one can match the temporal evolution of glasma fields into kinetic equations at late times . such a matching was considered previously in ref . the early time strong field dynamics may however modify the power counting assumed in these studies this possibility is also under active investigation @xcite . in conclusion , understanding the early classical field dynamics of the glasma and its subsequent thermalization is crucial to understand how and when the system thermalizes to form a qgp . the phenomenological implications of these studies are significant because they influence the initial conditions for hydrodynamic models . one such example that we discussed is the initial eccentricity of the qcd matter ; its magnitude may be relevant for our understanding of just how `` perfect '' , the perfect fluid created at rhic is . in the first two lectures , we discussed the problem of multi - particle production for hadronic collisions where the large @xmath0 modes are strong sources @xmath483 . this is a good model of the dynamics in proton - proton collisions at extremely high energies or in heavy ion collisions already at somewhat lower energies . in eq . ( [ eq : li-0 ] ) , one has @xmath484 , where @xmath13 is the typical momentum of the partons in the nuclei . as we then discussed in lectures i and ii , there is no small parameter in the expansion in powers of these sources and one has to solve classical equations of motion numerically to compute the average inclusive multiplicities for gluon and quark production . however , for asymmetric collisions , the most extreme example of which are collisions of protons with heavy nuclei , one has a situation where @xmath485 and @xmath486 . the other situation where a similar power counting is applicable is when one probes forward ( or backward ) rapidities in proton - proton or nucleus - nucleus collisions . in these cases , one is probing large @xmath0 parton distributions in one hadron ( small color charge density @xmath487 ) and small @xmath0 parton distributions in the other ( large color charge density @xmath488 ) . in these situations , analytical computations are feasible in the cgc framework . in this lecture , we will discuss the phenomenon of limiting fragmentation in this framework . the hypothesis of limiting fragmentation @xcite in high energy hadron - hadron collisions states that the pseudo - rapidity distribution @xmath489 ( where @xmath490 is the pseudo - rapidity shifted by the beam rapidity @xmath491 ) becomes independent of the center - of - mass energy @xmath2 in the region around @xmath492 , i.e. @xmath493 where @xmath387 is the impact parameter . limiting fragmentation appears to have a wide regime of validity . it was confirmed experimentally in @xmath494 and @xmath495 collisions at high energies @xcite . more recently , the brahms and phobos experiments at the relativistic heavy ion collider ( rhic ) at brookhaven national laboratory ( bnl ) performed detailed studies of the pseudo - rapidity distribution of the produced charged particles @xmath496 for a wide range ( @xmath497 ) of pseudo - rapidities , and for several center - of - mass energies ( @xmath498 ) in nucleus - nucleus ( au - au and cu - cu ) and deuteron - nucleus ( d - au ) collisions . results for pseudo - rapidity distributions have also been obtained over a limited kinematic range in pseudo - rapidity by the star experiment at rhic @xcite . these measurements have opened a new and precise window on the limiting fragmentation phenomenon . it is worth noting that this scaling is in strong disagreement with boost invariant scenarios which predict a fixed fragmentation region and a broad central plateau extending with energy . it would therefore be desirable to understand the nature of hadronic interactions that lead to limiting fragmentation , and the deviations away from it . in a recent article , bialas and jeabek @xcite , argued that some qualitative features of limiting fragmentation can be explained in a two - step model involving multiple gluon exchange between partons of the colliding hadrons and the subsequent radiation of hadronic clusters by the interacting hadrons . here we will discuss how the limiting fragmentation phenomenon arises naturally within the cgc approach we shall address its relation to the bialas - jeabek model briefly later . inclusive gluon production in proton - nucleus collisions was first computed in refs . @xcite , and shown to be @xmath13 factorizable in ref . @xcite . in ref . @xcite , the gluon field produced in pa collisions was computed explicitly in lorentz gauge @xmath499 . more recently , the gluon field was also determined explicitly in the @xmath500 light - cone gauge @xcite . the inclusive multiplicity distribution of produced gluons factorization is explicitly violated @xcite . ] can be expressed in the @xmath13-factorized form as @xcite , @xmath501 the formula , as written here , is only valid at zero impact parameter and assumes that the nuclei have a uniform density in the transverse plane ; the functions @xmath502 are defined for the entire nucleus . @xmath503 denotes the transverse area of the overlap region between the two nuclei , while @xmath504 are the total transverse area of the nuclei , and @xmath505 is the casimir in the fundamental representation . the longitudinal momentum fractions @xmath506 and @xmath507 are defined as @xmath508 where @xmath509 is the beam rapidity , @xmath22 is the proton mass , and @xmath510 is the transverse momentum of the produced gluon . the kinematics here is the @xmath511 eikonal kinematics , which provides the leading contribution to gluon production in the cgc picture . the functions @xmath512 and @xmath513 are obtained from the dipole - nucleus cross - sections for nuclei @xmath5 and @xmath352 respectively , @xmath514 where @xmath515 and where the matrices @xmath516 are adjoint wilson lines evaluated in the classical color field created by a given partonic configuration of the nuclei @xmath5 or @xmath352 in the infinite momentum frame . for a nucleus moving in the @xmath517 direction , they are defined to be @xmath518 \ ; . \label{eq : liii-5}\end{aligned}\ ] ] here the @xmath519 are the generators of the adjoint representation of @xmath6 and @xmath520 denotes the `` time ordering '' along the @xmath521 axis . @xmath522 is a certain configuration of the density of color charges in the nucleus under consideration , and the expectation value @xmath523 corresponds to the average over these color sources @xmath524 . as discussed previously , in the mclerran - venugopalan ( mv ) model @xcite , where no quantum evolution effects are included , the @xmath32 s have a gaussian distribution , with a 2-point correlator given by @xmath525 where @xmath526 is the color charge squared per unit area . this determines @xmath502 completely @xcite , since the 2-point correlator is all we need to know for a gaussian distribution . we will shortly discuss the small @xmath0 quantum evolution of the correlator on the r.h.s . ( [ eq : liii-4 ] ) . these distributions @xmath502 , albeit very similar to the canonical unintegrated gluon distributions in the hadrons , should not be confused with the latter @xcite . however , at large @xmath13 ( @xmath527 ) , they coincide with the usual unintegrated gluon distribution . note that the unintegrated gluon distribution here is defined such that the proton gluon distribution , to leading order satisfies @xmath528 from eq . ( [ eq : liii-2 ] ) , it is easy to see how limiting fragmentation emerges in the limit where @xmath529 . in this situation , the typical transverse momentum @xmath13 in the projectile at large @xmath506 is much smaller than the typical transverse momentum @xmath530 in the other projectile , because these are controlled by saturation scales evaluated respectively at @xmath506 and at @xmath507 respectively . therefore , at sufficiently high energies , it is legitimate to approximate @xmath531 by @xmath532 . integrating the gluon distribution over @xmath510 , we obtain @xmath533 this expression is nearly independent of @xmath507 and therefore depends only weakly on on @xmath534 . to obtain the second line in the above expression , we have used eq . ( [ eq : liii-4 ] ) and the fact that the wilson line @xmath516 is a unitary matrix . therefore , details of the evolution are unimportant for limiting fragmentation , only the requirement that the evolution equation preserves unitarity . the residual dependence on @xmath507 comes from the the upper limit @xmath535 of the integral in the second line . this ensures the applicability of the approximation that led to the expression in the second line above . the integral over @xmath536 gives the integrated gluon distribution in the projectile , evaluated at a resolution scale of the order of the saturation scale of the target . therefore , the residual dependence on @xmath537 arises only via the scale dependence of the gluon distribution of the projectile . this residual dependence on @xmath537 is very weak at large @xmath506 because it is the regime where bjorken scaling is observed . the formula in eq . ( [ eq : liii-6 ] ) was used previously in ref . the nuclear gluon distribution here is determined by global fits to deeply inelastic scattering and drell - yan data . we note that the glue at large @xmath0 is very poorly constrained at present @xcite . the approach of bialas and jezabek @xcite also amounts to using a similar formula , although convoluted with a fragmentation function ( see eqs . ( 1 ) , ( 4 ) and ( 5 ) of @xcite in addition , both the parton distribution and the fragmentation function are assumed to be scale independent in this approach ) . we will discuss the effect of fragmentation functions later in our discussion . though limiting fragmentation can be simply understood as a consequence of unitarity in the high energy limit , what may be more compelling are observed deviations from limiting fragmentation and how these vary with energy . we will now see whether deviations from limiting fragmentation can be understood from the renormalization group ( rg ) evolution of the unintegrated gluon distributions in eq . [ eq : liii-2 ] . in particular , we study the rg evolution of these distributions given by the balitsky - kovchegov ( bk ) equation @xcite . the bk equation is a non - linear evolution equation and large @xmath5 ) approximation where higher order dipole correlators are neglected . ] in rapidity @xmath538 for the forward scattering amplitude @xmath539 of a _ quark - antiquark dipole _ of size @xmath540 scattering off a target in the limit of very high center - of - mass energy @xmath2 where @xmath174 is defined as : @xmath541 here @xmath542 is the corresponding wilson line for the scattering of a quark - anti - quark dipole in the _ fundamental _ representation . the correlators @xmath516 in eq . ( [ eq : liii-4 ] ) , which are in the _ adjoint _ representation are wilson lines for the scattering of a gluon dipole on the same target instead . the bk equation captures essential features of high energy scattering . when @xmath543 , one has color transparency ; for @xmath544 , the amplitude @xmath545 , and one obtains gluon saturation is defined in terms of the requirement that @xmath546 for @xmath547 . ] . it is therefore an excellent model to study both limiting fragmentation as well as deviations from it . it is convenient to express the bk equation in momentum space in terms of the bessel - fourier transform of the amplitude @xmath548 one obtains @xmath549 where we denote @xmath550 . the operator @xmath551 is the well known bfkl kernel in momentum space @xcite . in the large @xmath7 and large @xmath5 limit , the correlators of wilson lines in the fundamental and adjoint representations are simply related : @xmath552 ^ 2\ ; .\ ] ] one can therefore express the unintegrated gluon distribution in eq . ( [ eq : liii-4 ] ) in terms of @xmath174 as @xmath553 ^ 2 \ ; . \label{eq : liii-10}\ ] ] in ref . @xcite we solved the bk equation numerically , in both fixed and running coupling cases , in order to investigate limiting fragmentation in hadronic collisions . the results are shown in fig . [ fig:1a ] . ) ( dashed lines ) and ( ii ) correlators in the fundamental representation see text ( solid lines).,width=259 ] the solid line is the result obtained for the unintegrated distribution corresponding to correlators in the fundamental representation , i.e. proportional to the fourier transform of @xmath554 instead of that of @xmath555 in eq . ( [ eq : liii-10 ] ) . our results for limiting fragmentation are obtained through the following procedure : * one first solves the bk equation in eq . ( [ eq : liii-9 ] ) to obtain ( via eq . ( [ eq : liii-8 ] ) ) eq . ( [ eq : liii-10 ] ) for the unintegrated distributions @xmath556 . the solution is performed for @xmath557 with the initial condition @xmath558 , given by the mclerran - venugopalan model @xcite with a fixed initial value of the saturation scale @xmath559 . for a gold nucleus , extrapolations from hera and estimates from fits to rhic data suggest that @xmath560 . the saturation scale in the proton is taken to be @xmath561 . for comparison , we also considered initial conditions from the golec - biernat and wusthoff ( gbw ) model @xcite . the values of @xmath562 were varied in this study to obtain best fits to the data . * we used the ansatz @xmath563 in order to extrapolate our results to larger values of @xmath564 , where the parameter @xmath565 is fixed by qcd counting rules . * the resulting expressions are substituted in eq . ( [ eq : liii-2 ] ) to determine rapidity distribution of the produced gluons . the pseudo - rapidity distributions are determined by multiplying eq . ( [ eq : liii-2 ] ) with the jacobian for the transformation from @xmath566 to @xmath181 . this transformation requires one to specify an infrared mass , which is also the mass chosen to regulate the ( logarithmic ) infrared sensitivity of the rapidity distributions . for further details on how the results are obtained , we refer the reader to ref . @xcite . for charged particles from nucleon - nucleon collisions at ua5 energies @xcite @xmath567 and phobos data @xcite at @xmath568 . upper plots : initial distribution from the mv model , lower plots : initial distribution from the gbw model . left panels : @xmath569 , right panels @xmath570 . , width=384 ] in figure [ fig:2 ] , we plot the pseudo - rapidity distributions of the charged particles produced in nucleon - nucleon collisions for center of mass energies ranging from @xmath571 to @xmath572 . the computations were performed for input distributions ( for bk evolution ) at @xmath573 from the the gbw and mv models . the normalization is a free parameter which is fitted at one energy . plots on the left of figure [ fig:2 ] are obtained for @xmath569 ( the free parameter in the large @xmath0 extrapolation ) whereas the right plots are for @xmath570 . the different values of @xmath574 are obtained for different values of @xmath358 as inputs to the bk equation . while these values of @xmath358 might appear small , they can be motivated as follows . the amplitude has the growth rate @xmath575 . thus @xmath576 , which gives reasonable fits ( more on this in the next paragraph ) to the pp data for the mv initial conditions , corresponds to @xmath577 . thus a small value of @xmath358 in fixed coupling computations `` mimics '' the value for the energy dependence of the amplitude in next - to - leading order resummed bk computations @xcite and in empirical dipole model comparisons to the hera data @xcite . our computations are extremely sensitive to the extrapolation prescription to large @xmath0 . this is not a surprise as the wave - function of the projectile is probed at fairly large values of @xmath506 . from our analysis , we see that the data naively favors a non - zero value for @xmath578 . the value @xmath579 results in distributions which , in both the mv and gbw cases , give reasonable fits ( albeit with different normalizations ) at lower energies but systematically become harder relative to the data as the energy is increased . to fit the data in the mv model up to the highest ua5 energies , a lower value of @xmath580 than that in the gbw model is required . this is related to the fact that mv model has tails which extend to larger values in @xmath13 than in the gbw model . as the energy is increased , the typical @xmath581 does as well . we will return to this point shortly . but plotted versus @xmath582 to illustrate the region of limiting fragmentation . , width=384 ] in figure [ fig:3 ] the same distributions are shown as a function of the @xmath582 . the calculations for @xmath569 are consistent with scaling in the limiting fragmentation region . there is a slight discrepancy between the calculations and the data in the mid - rapidity region . this discrepancy may be a hint that one is seeing violations of @xmath13 factorization in this regime because @xmath13 factorization becomes less reliable the further one is from the dilute - dense kinematics of the fragmentation regions @xcite . this discrepancy should grow with increasing energy . however , our parameters are not sufficiently constrained that a conclusive statement can be made . for instance , as we mentioned previously , there is a sensitivity to the infrared mass chosen in the jacobian of the transformation from @xmath566 to @xmath181 . this is discussed further in ref . @xcite . in figure [ fig:4 ] we show the extrapolation to higher energies , in particular the lhc range of energies for the calculation with the gbw input . we observed previously that the mv initial distribution , when evolved to these higher energies , gives a rapidity distribution which is very flat in the range @xmath583 . we noted that this is because the average transverse momentum grows with the energy giving a significant contribution from the high @xmath13 tail of the distribution in the mv input at @xmath584 . the effect of fragmentation functions on softening the spectra in the limiting fragmentation region can be simply understood by the following qualitative argument . the inclusive hadron distribution can be expressed as @xmath585 where @xmath586 is the fragmentation function denoting the probability , at the scale @xmath587 , that a gluon fragments into a hadron carrying a fraction @xmath98 of its transverse momentum . for simplicity , we only consider here the probability for gluons fragmenting into the hadron . the lower limit of the integral can be determined from the kinematic requirement that @xmath588 we obtain , @xmath589 if @xmath590 were zero , the effect of including fragmentation effects would simply be to multiply eq . ( [ eq : liii-11 ] ) by an overall constant factor . at lower energies , the typical value of @xmath591 is small for a fixed @xmath534 ; the value of @xmath590 is quite low . however , as the center of mass energy is increased , the typical @xmath591 value grows slowly with the energy . this has the effect of raising @xmath590 for a fixed @xmath534 , thereby lowering the value of the multiplicity in eq.([eq : liii-11 ] ) for that @xmath534 . note further that eq . ( [ eq : liii-12 ] ) suggests that there is a kinematic bound on @xmath591 as a function of @xmath534 only very soft gluons can contribute to the inclusive multiplicity . for gbw input model . the parameter in the large @xmath0 extrapolation was set to @xmath569.,width=259 ] in figure [ fig : pt_pp ] we display the @xmath592 distributions obtained from the mv input compare to the ua1 data @xcite . we compare the calculation with and without the fragmentation function . the fragmentation function has been taken from @xcite . clearly the `` bare '' mv model does not describe the data at large @xmath13 because it does not include fragmentation function effects which , as discussed , make the spectrum steeper . in contrast , because the @xmath13 spectrum of the gbw model dies exponentially at large @xmath13 , this `` unphysical '' @xmath13 behavior mimics the effect of fragmentation functions see figure [ fig : pt_pp ] . hence extrapolations of this model , as shown in figure [ fig:5 ] give a more reasonable looking result . similar conclusions were reached previously in ref . @xcite . distribution from eq . ( [ eq : liii-2 ] ) with mv ( full squares ) and gbw ( full triangles ) initial conditions . the mv initial condition with the fragmentation function included is denoted by the open squares . the distribution is averaged over the rapidity region @xmath593 , to compare with data ( in 200 gev / nucleon proton - antiproton collisions in the same pseudo - rapidity range ) on charged hadron @xmath592 distributions from the ua1 collaboration : full circles.,width=259 ] we next compute the pseudo - rapidity distribution in deuteron - gold collisions . in figure [ fig:6 ] we show the result for the calculation compared with the da data @xcite . the unintegrated gluons were extracted from the pp and aa data . the overall shape of the distribution matches well on the deuteron side with the minimum - bias data . the disagreement on the nuclear fragmentation side is easy to understand since , as mentioned earlier , it requires a better implementation of nuclear geometry effects . similar conclusions were reached in ref . @xcite in their comparisons to the rhic deuteron - gold data . we now turn to to nucleus nucleus collisions . in figure [ fig:5 ] we present fits to data on the pseudo - rapidity distributions in gold gold collisions from the phobos , brahms and star collaborations . the data @xcite are for @xmath594 and the brahms data @xcite are for @xmath595 . a reasonable description of limiting fragmentation is achieved in this case as well . one again has discrepancies in the central rapidity region as in the pp case . we find that values of @xmath596 gev for the saturation scale give the best fits . this value is consistent with the other estimates discussed previously @xcite . apparently the gold - gold data are better described by the calculations which have @xmath570 . this might be related to the difference in the large @xmath0 distributions in the proton and nucleus . further , slightly higher values of @xmath580 are preferred to the pp case . this variation of parameters from aa to pp case might be also connected with the fact that in our approach the impact parameter is integrated out thereby averaging over details of the nuclear geometry . in fig . [ fig : auauext ] we show the extrapolation of two calculations to higher energy @xmath597 . we note that the calculation within the mv model gives results which would violate the scaling in the limiting fragmentation region by approximately @xmath598 at larger @xmath534 . this violation is due partly to the effect of fragmentation functions discussed previously and partly to the fact that the integrated parton distributions from the mv model do not obey bjorken scaling at large values of @xmath0 . in the latter case , the violations are proportional to @xmath599 as discussed previously . the effects of the former are simulated by the gbw model the extrapolation of which , to higher energies , is shown by the dashed line . the band separating the two therefore suggests the systematic uncertainity in such an extrapolation coming from ( i ) the choice of initial conditions and ( ii ) the effects of fragmentation functions which are also uncertain at lower transverse momenta . ( filled triangles , squares and circles ) , brahms collaboration at energies @xmath600 ( open squares and circles ) . the data from the star collaboration at energy @xmath601 ( open triangles ) are not visible on this plot but can be seen more clearly in fig . [ fig : auauext ] . upper solid line : initial distributions from the mv model ; lower solid line : initial distributions from the gbw model . , width=384 ] to the lhc energy @xmath597/ nucleon . for comparison , the same data at lower energies are shown . ( see fig . [ fig:5 ] . ) dashed line - gbw input @xmath602 , solid line - mv input with @xmath602 . , width=259 ] to summarize the discussion in this lecture , we studied the phenomenon of limiting fragmentation in the color glass condensate framework . in the dilute - dense ( projectile - target ) kinematics of the fragmentation regions , one can derive ( in this framework ) an expression for inclusive gluon distributions which is @xmath13 factorizable into the product of `` unintegrated '' gluon distributions in the projectile and target . from the general formula for gluon production ( eq . ( [ eq : liii-2 ] ) ) , limiting fragmentation is a consequence of two factors : * unitarity of the @xmath516 matrices which appear in the definition of the unintegrated gluon distribution in eq . ( [ eq : liii-4 ] ) . * bjorken scaling at large @xmath506 , namely , the fact that the integrated gluon distribution at large @xmath0 , depends only on @xmath506 and not on the scale @xmath603 . ( the residual scale dependence consequently leads to the dependence on the total center - of - mass energy . ) deviations from the limiting fragmentation curve at experimentally accessible energies are very interesting because they can potentially teach us about how parton distributions evolve at high energies . in the cgc framework , the balitsky - kovchegov equation determines the evolution of the unintegrated parton distributions with energy from an initial scale in @xmath0 chosen here to be @xmath573 . this choice of scale is inspired by model comparisons to the hera data . we compared our results to data on limiting fragmentation from pp collisions at various experimental facilities over a wide range of collider energies , and to collider data from rhic for deuteron - gold and gold - gold collisions . we obtained results for two different models of initial conditions at @xmath604 ; the mclerran - venugopalan model ( mv ) and the golec - biernat wusthoff ( gbw ) model . we found reasonable agreement for this wide range of collider data for the limited set of parameters and made predictions which can be tested in proton - proton and nucleus - nucleus collisions at the lhc . clearly these results can be fine tuned by introducing further details about nuclear geometry . more parameters are introduced , however there is more data for different centrality cuts we leave these detailed comparisons for future studies . in addition , an important effect , which improves agreement with data , is to account for the fragmentation of gluons in hadrons . in particular , the mv model , which has the right leading order large @xmath13 behavior at the partonic level , but no fragmentation effects , is much harder than the data . the latter falls as a much higher power of @xmath13 . as rapidity distributions at higher energies are more sensitive to larger @xmath13 , we expect this discrepancy to show up in our studies of limiting fragmentation and indeed it does . taking this into account leads to more plausible extrapolations of fits of existing data to lhc energies . these lectures were delivered by one of us ( rv ) at the 46th zakopane school in theoretical physics . this school held a special significance because it coincided with the 70th birthday of andrzej bialas who is a founding member of the school . rv would like to thank michal praszalowicz for his excellent organization of the school . this work has drawn on recent results obtained by one or both of us in collaboration with h. fujii , k. fukushima , s. jeon , k. kajantie , t. lappi , l. mclerran , p. romatschke and a. stasto . we thank them all . rv was supported by doe contract no . de - ac02 - 98ch10886 . k. adcox et al . , [ phenix collaboration ] , nucl . a 757 * , 184 ( 2005 ) ; j. adams et al . , [ star collaboration ] , _ ibid . _ , 102 ( 2005 ) ; b.b . back et al . , [ phobos collaboration ] , _ ibid . _ , 28 ( 2005 ) ; i. arsene et al . , [ brahms collaboration ] , _ ibid . _ , 1 ( 2005 ) . , phys . * b 214 * , 587 ( 1988 ) ; phys . * b 314 * , 118 ( 1993 ) ; phys . lett . * b 363 * , 26 ( 1997 ) ; j. randrup , s. mrwczyski , phys . rev . * c 68 * , 034909 ( 2003 ) ; s. mrowczynski , acta phys . polon . * b 37 * , 427 ( 2006 ) . m. hirai , s. kumano , m. miyama , phys . rev . * d 64 * , 034003 ( 2001 ) ; m. hirai , s. kumano , t .- h . nagai , phys . rev . * c 70 * , 044905 ( 2004 ) ; k.j . eskola , v. kolhinen , c. salgado , eur . j. * c 9 * , 61 ( 1999 ) ; d. de florian , r. sassot , phys . rev . * d 69 * , 074028 ( 2004 ) . e. gotsman , e.m . levin , m. lublinsky , u. maor , nucl . phys . * a 696 * , 851 ( 2001 ) ; eur . j. * c 27 * , 411 ( 2003 ) ; e. levin , m. lublinsky , nucl . phys . * a 696 * , 833 ( 2001 ) ; m. lublinsky , eur . j. * c 21 * , 513 ( 2001 ) .

in the color glass condensate ( cgc ) effective field theory , when two large sheets of colored glass collide , as in a central nucleus - nucleus collision , they form a strongly interacting , non - equilibrium state of matter called the glasma . how colored glass shatters to form the glasma , the properties of the glasma , and the complex dynamics transforming the glasma to a thermalized quark gluon plasma ( qgp ) are questions of central interest in understanding the properties of the strongly interacting matter produced in heavy ion collisions . in the first of these lectures , we shall discuss how these questions may be addressed in the framework of particle production in a field theory with strong time dependent external sources . albeit such field theories are non - perturbative even for arbitrarily weak coupling , moments of the multiplicity distribution can in principle be computed systematically in powers of the coupling constant . we will demonstrate that the average multiplicity can be ( straightforwardly ) computed to leading order in the coupling and ( remarkably ) to next - to - leading order as well . the latter are obtained from solutions of small fluctuation equations of motion with _ retarded boundary conditions_. in the second lecture , we relate our formalism to results from previous 2 + 1 and 3 + 1 dimensional numerical simulations of the glasma fields . the latter show clearly that the expanding glasma is unstable ; small fluctuations in the initial conditions grow exponentially with the square root of the proper time . whether this explosive growth of small fluctuations leads to early thermalization in heavy ion collisions requires at present a better understanding of these fluctuations on the light cone . in the third and final lecture , motivated by recent work of biaas and jeabek @xcite , we will discuss how the widely observed phenomenon of limiting fragmentation is realized in the cgc framework . = by -1 1 . cea , service de physique thorique ( ura 2306 du cnrs ) , + 91191 , gif - sur - yvette cedex , france 2 . department of physics , bldg . 510 a , + brookhaven national laboratory , upton , ny-11973 , usa

it has become fashionable to invoke feedback from accreting black holes ( bhs ) as an influential element of galaxy evolution ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? regulatory mechanisms are sorely needed to keep massive galaxies from forming too many stars and becoming overly massive or blue at late times ( e.g. , * ? ? ? * ; * ? ? ? feedback from an accreting bh provides a tidy solution . for one thing , the gravitational binding energy of a supermassive bh is completely adequate to unbind leftover gas in the surrounding galaxy . furthermore , using simple prescriptions for black hole feedback leads to a natural explanation for the observed scaling relations between the bh mass and properties of the surrounding galaxy , including stellar velocity dispersion and bulge luminosity and mass ( e.g. , * ? ? ? * ; * ? ? ? the problem remains to find concrete evidence of bh self - regulation , and to determine whether or not accretion energy has a direct impact on the surrounding galaxy . there are some special circumstances in which accretion energy clearly has had an impact on its environment . for instance , jet activity in massive elliptical galaxies and brightest cluster galaxies deposits energy into the hot gas envelope ( see review in * ? ? ? * ) , although the efficiency of coupling the accretion energy to the gas remains uncertain ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , as does the relative importance of heating by the active nucleus as opposed to other possible sources ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? likewise , there is clear evidence that powerful radio jets entrain warm gas and carry significant amounts of material out of their host galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? however , as only a minority ( @xmath7 ) of active bhs are radio - loud , invoking this mechanism as the primary mode of bh feedback would require all galaxies to have undergone a radio - loud phase a conjecture which lacks direct evidence and contradicts a theoretical paradigm in which radio - loudness is determined by the spin of the black hole ( e.g. , * ? ? ? thus , it is not clear whether bh activity in radio galaxies accounts for more than a small fraction of the bh growth ( e.g. , * ? ? ? * ; * ? ? ? * ) and therefore whether this mode of feedback is in fact the dominant one . nuclear activity is known to drive outflows on small scales . broad absorption - line troughs are seen in @xmath8 of luminous quasars ( e.g , * ? ? ? * ) , and there is good reason to believe that the outflows are ubiquitous but have a covering fraction of @xmath9 , at least for high-@xmath10 systems ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? the velocities in broad absorption lines are high ( @xmath11 km s@xmath3 ) , and they most likely emerge from a wind blown off of the accretion disk ( e.g. , * ? ? ? in a few rare objects the outflow appears to extend out to large distances from the nucleus @xcite , but it is unclear whether most of these outflows have any impact beyond hundreds of schwarzschild radii . narrow associated absorption - line systems are signposts of outflows extending to larger distances , but determining their physical radii ( and thus the mass outflow rate ) is notoriously challenging . in cases where it is possible , the estimated outflow rates are thought to be significant fractions of the accretion onto the bh ( see review in * ? ? ? it is clear that some quasars affect their environment some of the time . the extent and the dominant mode of these interactions remain open to interpretation . in particular , it is not clear whether quasars are effectively removing the interstellar medium ( ism ) of their host galaxies during the high accretion rate episodes those that account for the majority of the bh growth . such feedback has been postulated by numerical simulations ( e.g. , * ? ? ? * ) , but direct observational evidence for this process is lacking . in this work , we look for direct evidence of extended warm gas in emission , using the narrow - line region ( nlr ) and specifically the strong and ubiquitous [ ] @xmath12 line . the nlr is in some respects the ideal tracer of the interface between the galaxy and the active galactic nucleus ( agn ) , as the gas is excited by the agn but extended on galaxy - wide scales . for a long time , following the seminal work of @xcite , it was thought that truly extended emission - line regions ( so - called eelrs with radii of 10 - 50 kpc ) were only found in radio - loud objects . using narrow - band imaging , these authors examined known luminous , @xmath13 agns and found that @xmath14 of the radio - loud objects had luminous extended [ ] nebulosities , while none of the radio - quiet objects did . it is not clear if the extended gas has an internal or external origin nor whether it is only present in radio - loud systems or is only well - illuminated in the presence of radio jets ( e.g. , * ? ? ? * ; * ? ? ? emission - line regions around radio - quiet systems @xcite are not usually as extended nor as luminous as those seen in the presence of powerful radio jets . this statement depends on the flux limit . at very low surface - brightness levels ( @xmath15erg s@xmath3 cm@xmath3 arcsec@xmath17 ) , diverse morphologies are observed in emission line gas ( e.g. , * ? ? ? * ; * ? ? ? an interesting exception may be the broad - line active galaxy mrk 231 . this galaxy shows outflowing neutral and ionized gas that is extended on @xmath6 kpc scales and moving at thousands of km s@xmath3@xcite . there is a jet in this galaxy ( as well as a starburst ) but the jet is not likely the source of acceleration of the neutral outflow @xcite . rather than focus on unobscured ( broad - line ) quasars , where detailed study of the nlr extent and kinematics is hampered by the presence of a luminous nucleus , we look instead at obscured quasars . the experiment is worth revisiting in light of the discovery of a large sample of obscured quasars with the sloan digital sky survey ( sdss ; * ? ? ? * ) . the sample , with @xmath18 , was selected based on the [ ] line luminosity @xcite and now comprises nearly 1000 objects @xcite . extensive follow - up with the _ hubble space telescope _ @xcite , _ chandra _ and _ xmm - newton_@xcite , _ spitzer_@xcite , spectropolarimetry @xcite , gemini @xcite , and the vla @xcite yield a broad view of the properties of the optically selected obscured quasar population . we target the low - redshift end of the sample , to maximize our spatial resolution of the nlr . in our first paper , we examined the host galaxies of our targets ( * ? ? ? * paper i hereafter ) . here we study the spatial distribution and kinematics of the ionized gas . after describing the sample and observations ( 2 ) , we turn to the nlr sizes ( 3 ) and then the spatially resolved kinematics of the sample as a whole ( 4 ) . we present two candidate dual obscured agns ( 5 ) and then summarize and conclude ( 6 ) . -0 mm the sample and data reduction were introduced in detail in paper i ( table 1 ) . the sample was selected from @xcite . we focused on targets with @xmath19 to ensure that [ ] @xmath12 was accessible in the observing window and imposed a luminosity cut on the [ ] line of @xmath20}}}}\geq 10^{42}$ ] erg s@xmath3 to pre - select luminous quasars ( estimated intrinsic luminosity @xmath21 mag ) . radio flux densities at 1.4 ghz were obtained from the faint images of the radio sky at twenty cm survey ( first ; @xcite ) and the nrao vla sky survey ( nvss ; @xcite ) . with one exception , all objects are radio - quiet , as determined by their position on the @xmath20}}}}- \nu l_{\nu}$ ] ( 1.4 ghz ) diagram @xcite , and they are at least an order of magnitude below the nominal radio - loud vs. radio - quiet separation line in this plane . the single radio - loud object in the sample , sdss j1124 + 0456 , is a double - lobed radio galaxy ( alternate name 4c+05.50 ) with @xmath22 ( 1.4 ghz)@xmath23 erg s@xmath3 which was observed with a slit nearly perpendicular to the orientation of its large - scale radio lobes . we observed 15 objects over two observing runs using the low - dispersion survey spectrograph ( ldss3 ; * ? ? ? * ) with a @xmath24 slit at the magellan / clay telescope on las campanas . the seeing was typically @xmath25 over the two runs . we integrated for at least one hour per target and covered one or two slit positions ( table 1 ) . lower-@xmath26 targets were observed with the vph - blue grism in the reddest setting , for a wavelength coverage of @xmath27 , while the higher-@xmath26 targets were observed with the bluest setting of the vph - red grism ( @xmath28 ) . the velocity resolution in each setting is @xmath29 km s@xmath3 . in addition to the primary science targets , at least two flux calibrator stars were observed per night and a library of velocity template stars consisting of f m giants was observed over the course of the run . since we have only long - slit observations , we do not sample the full velocity field of the gas or stars in the galaxy . with a few exceptions , the galaxy images were only marginally resolved in the sdss images . thus in selecting position angles to observe we were mainly guided by visual inspection of the color composite images . since these galaxies typically have very high equivalent width [ ] lines , we attempted to identify [ ] structures based on color - gradients in the images . as a result , the slit is not necessarily oriented along the major or minor axis of a given galaxy . in particular , it is important to keep in mind when judging the radial velocity curves of the spiral galaxies ( sdss j1106 + 0357 , sdss j1222@xmath300007 , sdss j1253@xmath300341 , sdss j2126 + 0035 and likely sdss j1124 + 0456 ) . of these , sdss j1106 + 0357 and sdss j2126 + 0035 were observed along the major axis , and sdss j1222@xmath300007 is within @xmath31 of the major axis . the others are observed at @xmath32 from the major axis . none were observed solely along the minor axis . cosmic - ray removal was performed using the spectroscopic version of lacosmic @xcite , and bias subtraction , flat - field correction , wavelength calibration , pattern - noise removal ( see paper i ) , and rectification were performed using the carnegie observatories reduction package cosmos . for the two - dimensional analysis discussed in this paper ( e.g. , the [ ] size determinations ) we additionally use the sky subtraction provided by cosmos . the flux calibration correction is determined from the extracted standard star using idl routines following methods described in @xcite and then applied in two dimensions . in the first paper we demonstrate that the absolute normalization of the flux calibration is reliable at the @xmath33 level . `` nuclear '' measurements refer to the 225 spatial extraction . the physical extent of the nlr provides one basic probe of the impact of the agn on the surrounding galaxy . we work with the rectified two - dimensional spectra . in order to boost the signal in the spatial direction , we collapse each spectrum in the velocity direction . we use a band with a velocity width that is twice the full width at half - maximum ( fwhm ) of the nuclear [ ] and centered on the nuclear [ ] line ( fig . [ fig : spat ] ) . the line width is measured from a continuum - subtracted spectrum , but we do not perform continuum subtraction on the two - dimensional spectra . this high signal - to - noise ( s / n ) spatial cut allows us to measure the nlr sizes much more sensitively than from typical narrow - band imaging . specifically , we measure the total spatial extent of the line emission down to a @xmath34 limit , where @xmath35 is determined from spatially - offset regions of the collapsed surface brightness profile . we are reaching typical depths of @xmath36 erg s@xmath3 cm@xmath17 arcsec@xmath17 . in three cases the nebular spectra are not spatially resolved ( i.e. , the spatial distribution matches that of a standard star ) . there are six objects for which we have multiple slit positions . the range in nebular size derived from cases with multiple slit positions is @xmath38 . in a few cases ( sdss j1356@xmath301026 , sdss j2126@xmath39 & sdss j2212@xmath40 ) , the line ratios change as a function of radius and [ ] /h@xmath41 falls below three . this changing ratio may reflect changes in the ionization parameter or gas - phase metallicity , or a transition from ionization dominated by the agn to regions ( e.g. , * ? ? ? * ) . by ionization parameter , we mean the ratio of the density of ionizing photons to the density of electrons . given the luminosities of quasars in our sample and the rates of star formation in their hosts @xcite , we expect that the number of ionizing photons from the quasars exceeds that from stars by about an order of magnitude . nevertheless , since quasar illumination is not necessarily isotropic and since photons from star formation are distributed more uniformly within the galaxy than those arising from the central engine , it is plausible that we may see gas excited by stars in the outer regions of the galaxy . shock excitation is unlikely since the linewidths are uniformly narrow in these outer regions . to be safe , we exclude the regions with [ ] /h@xmath41@xmath42 when calculating the nlr sizes . we have not applied any correction for reddening , which could be substantial ( e.g. , * ? ? ? @xcite show that deriving robust extinction corrections for the sdss obscured quasars is not straightforward , and we neglect such corrections here . one of the objects in our sample , sdss j1356 + 1026 , has a much more dramatic extended emission - line nebula than the rest ( figure [ fig : bubbleim ] ) . we will discuss the detailed kinematics and energetics of this object in more detail in a parallel paper ( greene & zakamska in preparation ) . for the present work , we explore the implications of detecting one single extended emission - line region in the sample ( 6 ) . we should note that deriving nebular sizes is an ill - defined task . first of all , ionized nebulae need not have regular shapes , and so the definition of size is not necessarily well - defined . this difficulty is only exacerbated when long - slit spectra are used to define the size , since our slit may well miss spatially extended regions . furthermore , the concept of size depends sensitively on the depth of the observation . deep observations probing depths of a few times @xmath43 erg s@xmath3 cm@xmath17 arcsec@xmath17 indeed reveal faint , extended gas with a range of morphologies ( e.g. , * ? ? ? thus , the primary size uncertainties are in these systematics , which dwarf the measurement errors . we quantify the uncertainties in the following manner . first of all , since we have not included surface brightness dimming , there is a dispersion of @xmath44 dex in the sizes due to distance . secondly , and more important , narrow - line regions are not strictly round . thus , depending on the position of the long - slit , we may derive a different answer . we have found , for the six objects with multiple slit positions , that the sizes agree within @xmath45 . finally , and most difficult to quantify , the shape will likely grow more irregular as we push to lower flux limits . we have attempted to quantify this dispersion in emission line profile using the ratio between the luminosity - weighted mean width of each spatial [ ] profile and the adopted radius measured down to a fixed surface brightness . were the nlrs all of the same shape , then the mean width would be a fixed fraction of the total size . instead , the ratio ranges from 0.2 to 4 , with a typical value of three . we thus adopt a factor of three as the overall uncertainty in the sizes . additionally , we flag as particularly uncertain those systems with a nearby massive companion galaxy , since there we are further contaminated by tidal gas . nlr sizes have been measured from narrow - band imaging @xcite and from long - slit spectroscopy ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? narrow - band imaging is preferable for studying the nlr morphology , but reaches shallower limits than the spectroscopy . -0 mm integral - field observations allow one to study two - dimensional kinematics @xcite , but for local objects only cover the inner nlr ( e.g. , * ? ? ? we compile a comparison sample of lower - luminosity obscured agns with measured nlr sizes from the literature @xcite . we include the bennert et al . ( 2002 ) and schmitt et al . ( 2003 ) measurements in fig . 2 for completeness , but note that the sizes can not be compared directly with those we measure here , because of the difference in depth . the limiting surface brightness values that we achieve in this work are at least a factor of 10 deeper than these narrow - band imaging studies from space , which range from @xmath46 erg s@xmath3 cm@xmath17 arcsec@xmath17 . for this reason , we do not include the space - based measurements in any analysis presented here ( e.g. , fitting of relationships ) . fraquelli et al . do not quote sizes but rather provide power - law fits to the surface brightness as a function of distance to the nucleus . taking their functional form , we calculate sizes that match our limiting surface brightness of @xmath47 erg s@xmath3 cm@xmath17 arcsec@xmath17 . for uniformity , we calculate sizes for bennert et al . ( 2006 ) in the same way , and we adopt their smaller radii in cases where star formation dominates in the outer parts . in cases of overlap between works , we prefer the @xcite observations , since they are both sensitive and take into account photoionization by starlight . the measurements for our sample are summarized in figure [ fig : sizes ] and table 2 , while the comparison samples are shown in figure [ fig : sizes ] . the observed distribution of nlr gas depends on the geometry and luminosity of the ionizing source , the geometry and kinematics of the host ism ( e.g. , disk , spherical , outflow , or infall ) , and the density distribution of the gas . while most of these are likely related to the morphology and dynamical state of the galaxy , the geometry of the ionizing source is tied to the orientation of the agn . in the simplest model , the galaxy ism is spherically distributed , while the ionizing radiation from the agn emerges anisotropically along lines of sight unaffected by the circumnuclear ` torus ' , as postulated by unified models of agn activity . in this case we expect to see ionization cones when the beam is not pointed directly at us , reflecting the geometry of circumnuclear obscuration . such cones are observed in images of nearby seyfert galaxies @xcite and more recently in the luminous obscured quasars studied here @xcite . -0 mm in this simplest geometry , we would expect to find smaller sizes in unobscured sources , when looking closer to the axis of the ionization cones . the difference in distributions depends on the expected opening angle of the torus , with larger difference for smaller opening angles . recent observations of obscured quasars suggest that the space densities of obscured and unobscured sources are @xmath48 equal @xcite , leading to opening angles of @xmath49 , but even if significantly smaller opening angles are assumed , the expected differences in the median projected size between the two populations is small ( @xmath50 dex ) . at low redshift and ( thus ) lower luminosity , ionization cones are also observed in unobscured sources ( e.g. , * ? ? ? * ; * ? ? ? * ) , while round nlrs are observed in both types @xcite . presumably , the ism is not always spherically distributed or relaxed @xcite . in figure [ fig : sizes]_a _ it appears that the nlr sizes of our obscured quasars are larger than the unobscured ones ( median difference 0.4 dex ) . however , this difference can be explained by differences in the depths of the observations , thus we can not address orientation differences in detail from these samples . it is interesting to note that at lower luminosities , @xcite do not see a significant size difference between the two populations . these observations , while shallow , are uniform between the obscured and unobscured populations . there has been some debate in the literature about the slope of a purported correlation between the nlr size and the agn luminosity ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? some correlation is expected , given that the agn is photoionizing the nlr gas , but the form it takes may tell us something about covering factor or density as a function of luminosity . it is clear from figure [ fig : sizes](_a _ ) that generally larger nlrs are found in more luminous objects ( kendall s @xmath51 with probability @xmath52 that no correlation is present ) . it is also clear that there is substantial scatter ; we find an rms scatter of 0.3 dex in radius at fixed @xmath53}}$ ] . we performed monte carlo simulations of ionization cones observed at random directions ( restricted to be outside the cones ) . they suggest that the orientation of the nlr axis relative to the line of sight is not a significant source of the observed scatter . at a fixed nlr size , orientation effects introduce a scatter of @xmath54 dex within each ( obscured or unobscured ) subpopulation , even when a wide range of opening angles is allowed for . therefore , the observed scatter is likely due to the combination of the true variance in nlr sizes at a given luminosity and to the differences in the definition of nlr `` size '' . for instance , @xcite derive sizes that are factors of @xmath55 larger than those based on _ hst_narrow - band imaging because of their increased sensitivity . given that the nlr is not always spherically symmetric or smooth , defining a meaningful size that is insensitive to depth is a difficult problem . for completeness , we fit a power - law relation between @xmath53}}$ ] and nlr size , using all narrow - line comparison samples as well as the objects considered here . because there are upper limits on the sizes , we calculate a linear regression using the binned schmitt method , from the astronomy survival analysis ( asurv ) software as implemented in iraf @xcite . the fit is shown in figure 2 . we find : @xmath56 } / 10^{42 } { \rm erg~s^{-1 } } ) \\ + ( 3.76 \pm 0.07 ) . \nonumber\end{aligned}\ ] ] the shallow slope we observe is consistent with a picture in which the nebulae are matter - bounded . at the distances from the quasar that we are probing with our observations , the density of material is low enough that the emissivity is no longer limited by the flux of photons by the quasar , but rather by the low density of the gas , and a large fraction of photons can escape into the intergalactic medium . note that the correlation between agn continuum luminosity and @xmath53}}$ ] in broad - line agns ( e.g. , * ? ? ? * ) suggests that the nebulae are limited by the number of photons in the bright central regions of the galaxy , but that the situation changes in the diffuse outer parts . if so , we would expect size to scale as the square - root of luminosity at low luminosities and then flatten out to at high luminosities , modulo differences in host galaxies . in addition to measuring the nebular sizes , we also parameterize the luminosity drop in the outer parts as a power - law and measure the power law slope ( @xmath53}}$]@xmath57 ) . the slopes range from @xmath58 ( table 2 ) . these slopes correspond to density profiles with slopes ranging from 1.3 to 2.4 , in good agreement with the _ hst _ observations of @xcite . one concern , as pointed out by @xcite , is that eventually the ionizing photons will run out of interstellar medium to ionize , particularly in the most luminous quasars . the nlr size can not in general grow indefinitely beyond the confines of the host galaxies . in figure [ fig : sizes](_b _ ) , we compare the continuum and nebular sizes . rather than using effective radii of host galaxies from photometry , we use the same method to measure the continuum extent as we used for the [ ] lines , collapsing the two - dimensional spectrum in the spectral direction over line - free regions to boost the signal . galaxy sizes are weakly correlated with nlr size ( kendall s @xmath59 with @xmath60 ) . we see that the nlr sizes are comparable to the galaxy continua . the exception is sdss j1356 + 1026 , which contains the spectacular bubble shown below . at these luminosities , the agn is effectively capable of photoionizing the entire galaxy ism , as well as companion galaxies out to several tens of kpc , as we saw in some of our previous long - slit observations @xcite . outflowing components of the nlr are routinely seen in radio galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ) , as well as in seyfert galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ) . on small scales , detailed modeling of the inner ( @xmath61 pc ) nlrs of a few local agns with _ hst _ indicates a surprising uniformity in behavior , with @xmath62 along an evacuated bicone @xcite . interestingly , we see similar qualitative behavior in sdss j1356 + 1026 ( greene & zakamska in prep ) . however , in general , nlr kinematics on larger scales are not as uniform , with mechanisms ranging from jet acceleration to radiation pressure driving ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? . for a complete review , see @xcite . we do find some correlation between fwhm and luminosity ( figure [ fig : sizes]_c _ ; kendall s @xmath63 ) . we will argue below based on the observed large velocity dispersions at large radius that the agn energy is stirring up the gas on large scales , thus explaining this correlation . -0 mm in this section we present the results of our two - dimensional analysis on the long - slit spectra . first we present velocity and dispersion profiles , as well as emission line ratios , as a function of position . to obtain these measurements , we extract spectra at uniform intervals as a function of spatial position along the slit . we start with rectified two - dimensional spectra from cosmos . each spectrum is extracted with a width of 095 ( 5 pixels ) to match the typical seeing of the observations . the central spatial position is determined by the spatial peak in the [ ] emission . the systemic velocity is determined from the absorption lines . galaxy continuum subtraction is performed for each spectrum using a scaled version of our best - fit model from the nuclear spectrum , with only the overall amplitude allowed to vary . while this is not strictly speaking a correct model , we have insufficient s / n in the off - nuclear spectra to constrain velocity or velocity dispersion , let alone changes in stellar populations . once the continuum - subtracted spectra are in hand , we fit the h@xmath41+[]@xmath64 lines for each spectrum as in paper i ( see also * ? ? ? * ; * ? ? ? . each line is modeled as a sum of gaussians ( a maximum of two for h@xmath41 and three for [ ] ) . the relative wavelengths of each transition and the ratio of the [ ] lines are fixed to their laboratory values , but the central velocity and line widths are allowed to vary from spectrum to spectrum . from these fits we are able to derive velocity , velocity dispersion , and line - ratio profiles as a function of spatial position . we report three measures of velocity at a given position , the peak in the [ ] line , the peak in the h@xmath41 line , and the flux - weighted mean velocity in the [ ] line . the velocity dispersion is measured as the fwhm of the [ ] model divided by 2.35 . at each spatial position we also measure the `` maximum '' and `` minimum '' velocities as the velocities at @xmath65 of the [ ] peak intensity ( e.g. , * ) relative to the systemic velocity of the stars ( shown as blue bullseyes and red crosses , respectively , in figure [ fig : rot1222 ] . ) errors are derived from monte carlo simulations . for each spectrum we generate 100 mock spectra using the best - fit parameters at that radius and the s / n of the original spectra . we fit each mock spectrum and the quoted parameter errors encompass @xmath66 of the mock fit values . in figure [ fig : rot1222 ] we present a representative radial velocity curve for sdss j1222@xmath67 . the remainder are shown in the appendix . first , we note that overall the radial velocity curves are flat . in paper i we presented detailed two - dimensional photometric fitting of these galaxies ( with the exception of sdss j1124 + 0456 and sdss j1142 + 1027 ) . using these fits , we divide the sample by the bulge - to - total ratio ( b / t ) , and call galaxies with b / t@xmath68 disks ( sdss j1106 + 0357 , sdss j1222@xmath300007 , sdss j1253@xmath300341 , and probably sdss j1124 + 0456 ) , while the rest are bulge dominated . additionally , those with clear tidal signatures are `` disturbed '' ( sdss j0841 + 0101 , sdss j1222@xmath300007 , sdss j1356 + 1026 , and sdss j2212@xmath300944 ) . we would expect to see the signature of rotation most clearly in disk - dominated galaxies . we note once again that sdss j1106 + 0357 and sdss j2126 + 0035 were observed along the major axis , sdss j1222@xmath300007 was within @xmath69 of the major axis , and the remaining two galaxies were observed at a @xmath70 angle to the major axis . we would expect to see the signature of rotation in most of these galaxies . instead , we only see rotation in the case of sdss j1106 + 0357 , sdss j1124 + 0456 , sdss j1142 + 1027 , and sdss j2212@xmath300007 . although with such a wide range of position angles , and such a small sample , it is hard to say for sure , we find it suggestive that neither sdss j1253@xmath300341 nor sdss j2126 + 0035 shows rotation . the sample galaxies showing rotation in their radial velocities also tend to show declines in @xmath71 by factors of two or more in the outer parts ( e.g. , sdss j1124 + 0456 ) . in contrast , those galaxies with flat radial velocity curves ( the majority in this sample ) also have notably flat @xmath71 distributions at kpc scales . again , this is strongly in contrast to the kinematics in the stars , even in bulge - dominated systems ( e.g. , * ? ? ? * ) . more to the point , it is in contrast to the kinematics of warm gas in inactive late - type ( e.g. , * ? ? ? * ) and early - type ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) spiral galaxies . in paper i we showed that @xmath71 in the nucleus is uncorrelated with @xmath72 . again , this behavior is in striking contrast not only to inactive galaxies but also to local , lower - luminosity active galaxies , for which it has been long been known that on average @xmath71/@xmath72@xmath73 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? here we are making a stronger statement . not only is the luminosity - weighted gas dispersion uncorrelated with the dispersion in the stars , but the dispersion in the gas stays high out to kpc scales in these galaxies . these observations provide new reason to doubt that gas velocity dispersions can be substituted for stellar velocity dispersions in luminous agns ( e.g. , * ? ? ? * ; * ? ? ? -0 mm this behavior is different from that seen in regular inactive galaxies . it is also different from that in local , well - observed seyfert galaxies . previous work looking at the kinematics of lower - luminosity local seyfert galaxies has found evidence for a two - tiered nlr structure ( e.g. , * ? ? ? * ) . in such objects , the inner or classical nlr extends to a few hundred pc and has linewidths of @xmath74 km s@xmath3 . at higher spatial resolution , there is clear evidence for outflow in the inner hundreds of pc in well - studied objects ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? in contrast , at larger radius , the linewidths drop and the kinematics of the nlr gas simply reflect that of the bulge or disk in which the gas sits ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . clearly , the observed kinematics of gas in hosts of obscured quasars are quite dissimilar from this picture . many of the host galaxies of our obscured quasars have nearby companions and/or show signs of recent interactions . it is therefore possible that the gas is being stirred by gravitational interactions with nearby galaxies . to explore that possibility further , we examine the analogous inactive ultra - luminous infrared galaxies . the integral - field spectra of @xcite show that even in these ongoing mergers the gas kinematics traces that of the stars . the @xmath71 profile is typically seen to decline in the outer parts as in non - merging systems , again in contrast to our findings for hosts of obscured quasars . of course , there are exceptions in the colina et al . sample , where the gas velocity dispersions are very complex . on the other hand , the mergers are more advanced in general than in our sample . thus , while we can not rule out gravitational effects in all cases , it seems most likely that the nuclear activity is directly responsible for stirring up the gas . we now address whether there is evidence for bulk motions ( e.g. , large - scale outflows ) in the gas based on the kinematics . -0 mm we have derived `` maximum '' red- and blueshifted velocities at @xmath65 of the line profile , relative to the systemic velocity of the stars . we examine the distribution of maximum velocities as a function of radius for the ensemble of spectra in figure [ fig : maxvel ] . while the emission extends to kpc scales for the majority of the targets , the gas velocities are not typically very high . the median maximum blue velocity at 8 kpc is @xmath75 km s@xmath3 while towards the red it is @xmath76 km s@xmath3 , where we quote errors in the mean . a few objects ( sdss j1253@xmath77 , sdss j1222@xmath67 ) have gas at velocities exceeding 500 km s@xmath3 . we note that the effective radii of these galaxies , for which we have well - resolved imaging , range from @xmath78 to @xmath79 kpc , with a median of @xmath80 kpc . these velocities exceed the velocity dispersions of the galaxies , but they do not compare to the @xmath48 thousands of km s@xmath3 outflow velocities seen by @xcite and postulated to be driven by recent agn activity . furthermore , they are not close to the escape velocity needed to actually unbind the gas . as we show in figure [ fig : o3maxvel ] , there is no evidence for a correlation between the nuclear @xmath53}}$ ] luminosity and the maximum observed velocity ( kendall s @xmath81 with a probability @xmath82 of no correlation ) . -0 mm we now quantitatively address whether any of the gas is approaching the escape velocity . following @xcite , we calculate an approximate escape velocity for each galaxy by assuming that the circular velocity scales with the velocity dispersion as @xmath83 . assuming the potential of an isothermal sphere , the escape velocity as a function of radius scales as : @xmath84^{0.5}.\ ] ] although @xmath85 is unknown , the escape velocity depends only weakly on its value . thus we assume @xmath86 kpc in all cases . the escape velocities thus estimated range from 500 to 1000 km s@xmath3 over the entire sample , but only vary by @xmath87 for an individual object over the range of radii that we probe . with escape velocities in hand , we can now address what fraction of the line emission comes from gas that is moving at or above the escape velocity . we first ask whether there is gas exceeding the escape velocity at each radius . with the same definition of systemic velocity as above , we integrate the line emission that exceeds the escape velocity to either the red or blue side of the systemic velocity . we then normalize by the total flux at that radius . these fractions are plotted as a function of radius in figure [ fig : fracesc ] for the 5 objects in which at least @xmath88 of the gas is nominally escaping for at least one radial position . for illustrative purposes , we focus here on the blue - shifted gas . in addition to calculating the escaping fraction at a given radius , we can also calculate an overall escaping fraction . they range from @xmath89 with a median value of 2% . nominally only a small fraction of the nlr gas is moving out of the galaxy at or around the escape velocity . however , the projection effects may be severe , and especially so because in obscured objects the gas motions are expected to occur largely in the plane of the sky . therefore , our estimates are a lower limit on the actual escape fractions ( see 6 for details ) . furthermore , as discussed further below , we have good reason to think that the medium is clumpy . depending on whether the outflowing component has the same clumping factor as the bound gas , it is difficult to translate these observed fractions into mass fractions . in addition to the escaping fraction , we would like to know how much mass is involved in the outflow . the standard method of estimating the density of the emission line gas uses density diagnostics such as the ratio of [ ] @xmath90 or [ ] @xmath91 . neither of these is available in the magellan spectra , and with several hundred km s@xmath3 velocities , the [ ] doublet is blended enough to be difficult to measure . the continuum - subtracted sdss spectra that integrate all emission within the 3 fiber yield a measurement of the [ ] @xmath90 ratio for all but the highest redshifts . using the iraf task _ temden _ , these can be translated into densities ranging from 250 - 500 @xmath92 , with a mean of 335 @xmath92 . these values are consistent with those commonly seen in spatially resolved observations of extended nlrs and used in mass estimations ( e.g. , @xcite and many others ) . however , such measurements can be highly biased toward high densities in clumpy gas . specifically , the recombination line luminosity depends on density as @xmath93 , whereas mass goes like @xmath94 , so the mass of the gas , its density and degree of clumpiness and its line luminosity are related through @xmath95 here we used a recombination coefficient @xmath96 @xmath97 s@xmath3 appropriate for a 20,000k gas , and @xmath98 is the degree of clumpiness , which by definition is @xmath99 and can be substantially greater . we have adopted a higher temperature than typical based on the [ ] @xmath100/[]@xmath101 line ratio . in general , the ratio ranges from @xmath102 in the central regions to @xmath103 further out . these ratios correspond to @xmath104k , and thus we adopt a temperature representative of the outer regions . the standard method of calculating the mass involved amounts to using this equation with @xmath105 and @xmath106 of a few @xmath107 @xmath92 , and produces an absolute minimum on the gas mass visible in the emission lines of a few @xmath108 . however , such high densities are in direct conflict with our observations . for one thing , we see high [ ] /h@xmath41 ratios , and thus high ionization parameters , and presumably low densities , at large radius . also , the observed extended scattering regions in obscured quasars place an independent constraint on gas densities @xcite . scattered light flux is @xmath109 , where @xmath110 is the density of scattering particles , electrons or dust . assuming purely electron scattering , _ hst_observations can be fit by density profiles that decline as @xmath111 and with density @xmath112 @xmath92 at a distance of about 3 kpc from the center ( figure [ fig : scatter ] ) . the scattering angle is not well known , but it introduces only about a factor of two uncertainty in this measurement . dust particles are even more efficient scatterers than electrons , so in the more realistic case of dust scattering , which is suggested by several lines of observational evidence @xcite , the implied mean density is constrained to be even smaller , @xmath113 @xmath92 . the uncertainties are larger in the case of dust scattering , because the density measurement is sensitive to the assumed gas - to - dust ratio and the scattering angle ( for this particular value , 90 and small magellanic cloud dust , @xcite ) , but nevertheless it is clear that the scattered light observations require much lower densities than those implied by [ ] ratios . the two measurements can be reconciled if the gas is highly clumped , so that most of the luminosity is coming from high - density clumps , whereas the mass and the scattering cross - section are dominated by low - density gas . while a detailed modeling of all observables is beyond the scope of this paper , we use a toy model in which the mass of the emitting gas at each density is a power - law function of the density , with power - law index @xmath114 between @xmath115 and @xmath116 , to estimate the clumping factor . since the [ ] line ratios are usually observed to be between the low - density and the high - density asymptotes , values of @xmath117 a few times the critical density are required ; we use @xmath118 @xmath92 . at the same time , for @xmath119 the minimal density is constrained to be @xmath120 by the scattering observations . with these constraints , the clumping factor is @xmath121 for example , for @xmath122 and @xmath123 @xmath92 , for each @xmath124 erg s@xmath3 of h@xmath41 emission , the mass of the emitting gas is @xmath125 . this estimate can only be considered very approximate , since the derived mass is quite sensitive to the specific assumptions about clumping ( for example , it varies by 2 dex as @xmath126 varies between 1 and 2 ) . nevertheless , we point out that the standard method of mass determination likely produces an underestimate of the true mass and that the scattering observations provide a valuable constraint on the physical conditions in the nlr . in short , we see compelling evidence that the nlr is clumpy . as a result , it is difficult to estimate robust gas masses , and thus difficult to determine what fraction of the gas may be expelled by potential agn outflows . many recent surveys have identified potential dual active galaxies ( i.e. , two active galaxies with @xmath48 kpc separations ) as narrow - line objects with multiple velocity peaks in the [ ] line in sdss spectra ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , as well as from the deep2 redshift survey @xcite . other candidates have been identified based on spatially offset nuclei @xcite . there are two intriguing objects in this sample that may contain dual agns . -0.5truein 0.1 in -0 mm the first is sdss j1356 + 1026 ( fig . [ fig : bubbleim ] ) , which has two clear continuum sources , each with associated high - ionization [ ] emission . their separation is @xmath127 kpc ( 11 ) . this object was highlighted as a potential dual agn by both @xcite based on multiple velocity peaks in the sdss spectrum and by @xcite from keck ao imaging . we have recently shown that @xmath7 of the double - peaked narrow - line candidates also have spatially resolved dual continuum sources @xcite . it seems natural that two galaxies would contain two bhs . on the other hand , there well may be a single radiating bh that is illuminating all of the surrounding gas . unfortunately , our long - slit spectra do not include [ ] or [ ] , which would give us a handle on the electron densities , and thereby whether a single ionizing source is plausible . given the projected separation of 2.5 kpc , if we assume that there is a single ionizing source associated with one of the two continua , we would expect to see the ionization parameter decrease by a factor of @xmath128 between the two targets . in fact , the [ ] /h@xmath41 ratios are within @xmath88 of each other , as are the [ ] fluxes . on the other hand , the very high ionization parameter seems to extend over the entire nebulosity ( @xmath6 kpc ) . of course , the accreting bh may sit between the two continuum sources . definitive proof requires the detection of x - ray or radio cores associated with each continuum source . sdss j0841 + 0101 shows much less ambiguous evidence for a pair of accreting bhs , with a projected separation of 38 ( 7.6 kpc ; fig . [ fig : binim ] ) . it would not be included in double - peaked samples assembled from the sdss because the separation on the sky between the two components is larger than the sdss 3 fibers . nevertheless , the component separations are comparable to those in the liu et al . sample . @xcite show that the double - peaked samples are probably dominated by single agns . these observations highlight that we are likewise missing dual agns with slightly larger separations . as is apparent from figure [ fig : binspec ] , the two agns are strikingly similar in spectroscopic properties . the [ ] luminosities ( @xmath129 erg s@xmath3 ) agree within @xmath130 dex , and the [ ] /h@xmath41 ratios ( @xmath6 ) agree within @xmath131 . the only clear difference is in the linewidths . the primary galaxy ( a ) has a @xmath132}$]@xmath133 km s@xmath3 , while the companion agn ( b ) is narrower , with @xmath132}$]@xmath134 km s@xmath3 . this difference most likely reflects the fact that a , with a stellar velocity dispersion of @xmath135 km s@xmath3 is more massive than b , with @xmath136 km s@xmath3 . taken at face value , this difference in dispersions corresponds to a difference of a factor of nearly 10 in bh mass between the two galaxies . accordingly , if the [ ] luminosity tracks the bolometric luminosity , then apparently b is accreting 10 times closer to its eddington limit than a. alternatively , there may be only a single radiating black hole . if there is only one quasar in galaxy a then we consider two scenarios . the first is that the quasar in a is unobscured as seen from b , so that the galaxy b is photoionized by the central engine in a. if we assume that most of the nlr emission in a is produced at a distance @xmath137 kpc from the nucleus , then in order to preserve the ionization parameter ( as evidenced by the similar spectra of a and b ) , the difference in electron density between the two galaxies would have to be a factor of @xmath138 . while the dust particles in galaxy b may scatter quasar spectrum , this emission can not dominate the observed spectrum ( otherwise we would see a broad - line agn in source b ) . the resulting estimates of the emerging equivalent width of the emission lines suggest that this scenario is possible , but has to be quite tuned in order to fit observations . the second scenario is that galaxy b is located along the obscured direction , just like the observer , but scatters some of the a s [ ] emission . however , in this scenario the ratio of [ oiii ] fluxes of b and a corresponds to the fraction of photons that b intercepts , @xmath139 , contrary to the observed similarity of fluxes . in conclusion , the picture of a single active black hole producing two objects with similar fluxes and ionization parameters appears unlikely . we are looking for direct signs of feedback in the two - dimensional spatial extents and kinematics of the nlrs of a sample of luminous obscured active galaxies . our conclusions are mixed . on the one hand , we see clear evidence that the agn is stirring up the galaxy ism . on the other hand , we do not see signs of galaxy - scale winds at high velocities . however , as we argue below , perhaps this is unsurprising . we see two distinct signatures of a luminous accreting bh on the ionized gas in these galaxies . the nlrs are much more extended at these high luminosities than in lower - luminosity seyfert galaxies . in fact , the agns are effectively photoionizing gas throughout the entire galaxy . this alone means that the agn is heating the ism on galaxy - wide scales . the impact of the agn is more directly seen in the kinematics . we see very few ordered radial velocity curves ; instead the velocity distributions are typically quite flat even at large radius . perhaps even more striking is that the gas velocity dispersions are high out to large radius . as we have argued , not only do inactive galaxies uniformly show a drop in gas ( and stellar ) velocity dispersion at large radius , but even in ultra - luminous infrared galaxies the gas velocity dispersions are observed to drop at large radius . we can not therefore attribute the gas stirring to gravitational effects such as mergers . it is most natural to implicate the accreting bh . on the other hand , overall the velocities we observe in the nlr gas are not very high ( a few hundred km s@xmath3 ) . taken at face value , our crude estimates suggest that very little of the ism is moving fast enough to escape the galaxy , although a clumpy nlr complicates our ability to estimate this fraction robustly . in only one case do we see the spectacular outflowing nebulosity one might imagine in thinking of agn feedback ( sdss j1356 + 1026 ) . before we can rule out that any gas is unbound from these galaxies , however , we should consider the impact of projection effects , potential observational biases , and some theoretical expectations . our observations suggest that ionized gas is ubiquitous within the galaxy but rare at larger ( e.g. , 10 kpc ) scales . as explained above , the observed ratio of obscured to unobscured objects leads us to assume an ionization cone opening angle of @xmath140 . with such a large opening angle , we would expect our slit to intercept the nlr nearly all the time , as we observe . on the other hand , we see extended gas on 10 kpc scales in only one case . furthermore , the _ hst _ continuum images show extended emission on these large scales , but with a much smaller opening angle of 20 - 60 . similarly , we have visually inspected the most luminous obscured agns from the reyes et al . sample with @xmath141 and found evidence for small opening angles from the broad - band images ( which have significant [ ] light in the @xmath142 band ) . probably we are seeing the effects of surface brightness dimming at the outer reaches of the bicone . although the true opening angle is large ( 120 ) , only a much narrower inner cone can be observed at 10 kpc . taking the smaller opening angles , we expect to see extra - galactic extended gas only 20 - 40% of the time . that fraction is not inconsistent with the number of objects that we observe with emission line regions extending beyond their host galaxies . projection effects also preferentially bias us against detecting the true outflowing velocities . these are obscured objects , and on large scales we see evidence for ionization cones in the _ hst _ continuum imaging . we thus expect the largest accelerations to occur in the plane of the sky . we perform a monte carlo simulation in which the nlr is modeled as a biconical outflow with constant velocity as a function of radius , assuming different opening angles for the bicone ( fig . [ fig : velproject ] ) . we sample random lines of sight outside of the bicone , and find that while the intrinsic velocity is uniformly high , we only expect to observe high ( e.g. , approaching escape ) velocities a small fraction of the time . these simulations take into account only the bias introduced by projection effects and assume constant velocity and uniform emissivity within the bi - cone . we also considered more realistic models , in which velocity varies as a function of distance ( @xmath143 ) from the center , mass conservation is satisfied and the emissivity correspondingly declines as @xmath144 . due to the decline of emissivity in these models , at a projected distance @xmath145 from the center the observed brightness is dominated by the location physically closest to the center ( that is , @xmath146 ) , and this gas is moving exactly in the plane of the sky exacerbating the projection bias . in the case @xmath147 the emissivity may be uniform , but the integral along the line of sight is dominated by gas that moves slower than the gas at @xmath145 because of the declining velocity profile . therefore , in these more realistic situations we find a radial velocity distribution that is more peaked at zero than shown in fig . [ fig : velproject ] . thus , while we observe small escaping fractions , once projection effects are accounted for , the observations may be consistent with high velocities in a large fraction of the gas . finally , there is the possibility that the outflows operate predominantly on small scales . in local low - luminosity sources outflows are observed only within the inner hundreds of pc ( e.g. , * ? ? ? in addition , recent simulations by @xcite suggest that bhs do self - regulate their own growth but do not generate galaxy - wide outflows . of course , other factors may be at play as well . there is the possibility that some fraction of the ionizing photons have escaped the galaxy ( e.g. , * ? ? ? * ) , or even that we are seeing galaxies in some pre - outburst phase , as may be expected if obscured accretion tends to accompany the late stages of merging and star formation activity ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? it is interesting to compare with simulations of galaxy - scale outflows . we start with the work of @xcite ( see also * ? ? ? * ; * ? ? ? these simulations focus on smaller scales than those we probe , extending no further than 10 pc . however , it is at least a starting point for comparison . the simulations include radiative heating by both an accretion disk and an x - ray corona , and look at the impact of varying the density and temperature structure , as well as rotation , of the gas . we highlight a few generic conclusions from their studies that are very relevant to our work . first of all , the final flow includes both an equatorial inflow and a bipolar outflow . consistent with our work , the opening angle of the outflowing cone can be quite wide ( up to 160 ) . also interesting to note is that the outflows can be dynamic , clumpy as we observe , and with multiple temperatures ( ranging from the 10@xmath148 k gas observed here all the way to x - ray emitting temperatures ) . it remains to be seen whether the outflows on pc scales will propogate to larger ( galaxy - wide scales ) . a recent study by hopkins et al . ( in preparation ) of outflows driven by agns in numerical simulations demonstrates several surprising similarities to the kinematics of the ionized gas we see in our study . the observations suggest that outflows are clumpy because the measurements of rms density and the mean density are highly discrepant . the simulations suggest that outflows are clumpy because they are subject to rayleigh - taylor instabilities . furthermore , the rate of the decline of mean density with distance from the center seen in scattering observations is similar to that seen in numerical simulations where the motion of the gas becomes ballistic at large distances . the masses and the velocities of the outflows that we find are quite similar to those seen in numerical simulations , and although the kinetic energies of the outflows ( @xmath149 erg s@xmath3 ) are just a small fraction of the total energy output of the agn , the simulations suggest that the wind is in fact driven by a much stronger coupling of the agn output to the gas . the small kinetic energies that we see at this late ( @xmath150 years ) stage are simply left - overs after much of the energy was efficiently radiated by the outflow . while these qualitative similarities are very encouraging , the specific mechanism responsible for coupling of the black hole output to the gas on much smaller spatial scales ( which then develops into the relic outflow we see now ) remains unidentified . in short , it is clear that the presence of the agn at the galaxy center impacts the entire galaxy . whether significant mass outflows are driven , particularly in the radio - quiet regime considered here , remains an open question . the next step for this type of analysis is already underway . integral - field unit observations ( e.g. , * ? ? ? * ) , particularly with a wider wavelength coverage , will remove some of the ambiguities we struggle with . we thank g. novak for numerous interesting discussions , and p. hopkins for sending us a manuscript in advance of publication . we thank the referee , sylvain veilleux , for a very prompt , careful and helpful report that significantly improved this manuscript . research by a.j.b . is supported by nsf grant ast-0548198 . this appendix includes all of the two - dimensional information for all galaxies that are spatially resolved in our observations ( figs . 12 - 20 ) . note in particular the high velocities and dispersions at large radius .

we use spatially resolved long - slit spectroscopy from magellan to investigate the extent , kinematics , and ionization structure in the narrow - line regions of 15 luminous , obscured quasars with @xmath0 . increasing the dynamic range in luminosity by an order of magnitude , as well as improving the depth of existing observations by a similar factor , we revisit relations between narrow - line region size and the luminosity and linewidth of the narrow emission lines . we find a slope of @xmath1 for the power - law relationship between size and luminosity , suggesting that the nebulae are limited by availability of gas to ionize at these luminosities . in fact , we find that the active galactic nucleus is effectively ionizing the interstellar medium over the full extent of the host galaxy . broad ( @xmath2 km s@xmath3 ) linewidths across the galaxies reveal that the gas is kinematically disturbed . furthermore , the rotation curves and velocity dispersions of the ionized gas remain constant out to large distances , in striking contrast to normal and starburst galaxies . we argue that the gas in the entire host galaxy is significantly disturbed by the central active galactic nucleus . while only @xmath4 @xmath5 worth of gas are directly observed to be leaving the host galaxies at or above their escape velocities , these estimates are likely lower limits because of the biases in both mass and outflow velocity measurements and may in fact be in accord with expectations of recent feedback models . additionally , we report the discovery of two dual obscured quasars , one of which is blowing a large - scale ( @xmath6 kpc ) bubble of ionized gas into the intergalactic medium .

one of the main open problems in asymptotic analysis of linear wave equations is to understand the accuracy of semiclassical approximations for large times . let @xmath4 be a riemannian manifold and let @xmath5 be the laplace beltrami operator on @xmath6 . in this paper we want to study the long time behaviour of solutions of the schrdinger equation @xmath7 with oscillatory initial conditions of the form @xmath8 where @xmath9 is smooth , @xmath10 is a smooth real valued function and @xmath11 is a small parameter . for small @xmath12 the solutions of can be approximated by a geometric optics like constructions which involves the geodesic flow @xmath13 . by classical results such approximations work well if one restricts the time to a fixed interval @xmath14 $ ] . but the joint limit @xmath0 , @xmath15 is much less well understood , in particular in the case we will be interested in , namely if the geodesic flow is hyperbolic , the accuracy of the approximations are currently only under control if @xmath16 where @xmath17 is a lyapunov exponent associated with the geodesic flow and @xmath18 is a constant . this time scale is called the ehrenfest time . rigorous results on propagation of coherent states up to this time have been recently derived in a series of papers , @xcite , and analogous results on egorov s theorem in @xcite . such results are interesting and useful because the dynamical properties of the geodesic flow become apparent only for large times , and one can use these results to relate the qualitative behaviour of wave propagation for large times and high frequencies to ergodic properties of the geodesic flow . for instance if the geodesic flow is anosov , it is rapidly mixing and this implies that propagated waves of the form become equidistributed for large times , under certain conditions on @xmath10 , see @xcite . this is in line with the conjecture that for classically chaotic systems propagated waves should for large times behave universally in the semiclassical limit like random superpositions of elementary plane waves , @xcite . estimates on time evolution on the scale of the ehrenfest time have as well recently been used to obtain strong estimates on the distribution of eigenfunctions , namely on the entropy of limit measures obtained from sequences of eigenfunctions , @xcite . but it would be very desirable to understand the behaviour beyond the ehrenfest time . propagation of waves is a very abundant physical phenomenon and can be observed and measured easily in many different situations . the ehrenfest time is rather short , and one would like to be able to use semiclassical approximations for much larger times . in addition to practical applications a better understanding of long time propagation could as well help to approach many open problems about the semiclassical behaviour of eigenfunctions and eigenvalues , e.g. , questions like the rate of quantum ergodicity or quantum unique ergodicity . the accuracy of semiclassical approximations in time evolution has been carefully studied numerically in @xcite for the stadium billiard , and they found no breakdown at the ehrenfest time . in addition they argued that semiclassical approximations should stay accurate up to a time scale of order @xmath19 , if the classical system has no singularities . the main problem one faces in the study of semiclassical approximation for chaotic systems is exponential proliferation . the approximations turn out to be a sum of oscillating terms whose number grows exponentially with time and so they are not absolutely convergent , and the error terms one obtains are of the same nature . the aim in this work is to show that indeed semiclassical approximations can be valid for times far beyond the ehrenfest time . we do this by developing techniques to control the size of the error term despite the exponential proliferation . this is only the first step to understand semiclassical approximations for large time in more detail , because the main term in the approximation is as well a sum of an exponentially growing number of terms , whose behaviour is not easy to understand . the system we will study is the schrdinger equation on a surface @xmath20 of constant negative curvature . let @xmath21 be the unit disk , equipped with the usual metric defined by the line element @xmath22 and @xmath23 the laplace beltrami operator on @xmath24 . @xmath20 can be represented as the quotient of @xmath24 by a group of isometries @xmath25 , @xmath26 and we will assume that @xmath25 is a fuchsian group , i.e. , acts discontinously on @xmath24 , this is equivalent to requiring that @xmath25 is a discrete subgroup of @xmath27 . the most interesting case is the one where @xmath25 is a fuchsian group of first kind , i.e. , @xmath20 is of finite volume , or even compact . functions on @xmath20 can be identified with functions on @xmath24 which are invariant under the action of @xmath25 . we can use summation over @xmath25 to build functions on @xmath20 from functions on @xmath24 : given a function @xmath28 we set @xmath29 which is a function on @xmath20 , provided the sum converges . since @xmath5 commutes with the action of isometries on @xmath24 , the time evolution operator @xmath30 , which is the solution to the schrdinger equation with initial condition @xmath31 , commutes with the action of @xmath25 and we have @xmath32 our strategy will be to construct first semiclassical approximations on @xmath24 and then use this relation to transfer them to @xmath20 . with examples of horocycles and geodesics associated with @xmath33 . the dashed circles tangent to @xmath34 at @xmath35 are horocycles and the geodesics emanating from @xmath35 are solid lines . the horocycles are the wavefronts associated with the phase - function @xmath36 , , and semiclassical wave propagation is described by transport along the geodesic spray emanating from @xmath35 ( which are the projections of trajectories on the unstable manifold associated with @xmath35 ) . , width=302 ] let us recall the definition of plane waves associated with horocycles . let @xmath37 be a point on the boundary of @xmath24 , a horocycle associated with @xmath35 is ( euclidean ) circle in @xmath24 tangent to @xmath34 at @xmath35 , given a point @xmath38 we denote by @xmath39 the unique horocycle associated with @xmath35 which passes through @xmath40 . furthermore let @xmath41 be the geodesic emanating from @xmath35 and passing through @xmath40 , see figure [ fig : unit - disk ] for illustration . let us write @xmath42 if @xmath43 lies inside @xmath39 , then given a @xmath37 we can define @xmath44 where @xmath45 denotes the hyperbolic distance between the two horocycles @xmath39 and @xmath43 . these function are used by helgason to define a set of plane waves on @xmath24 and develop harmonic analysis , @xcite . the initial states on @xmath24 we will consider are of the form @xmath46 with a smooth amplitude @xmath9 . such functions are known as lagrangian states , where the lagrangian submanifold of @xmath47 associated with them is the graph of @xmath48 , @xmath49 this manifold is an unstable manifold of the geodesic flow . let us denote by @xmath50 the geodesic flow over @xmath24 and by @xmath51 the restriction of the canonical projection @xmath52 to @xmath53 , then we can define an induced flow on @xmath24 by @xmath54 which we can then use to define a one parameter family of operators @xmath55 we will show in lemma [ lem : sl - unitary ] in the the next section that these operators actually form a unitary group . they are defined purely in terms of the geodesic flow , i.e. , the classical dynamical system associated with the schrdinger equation , and they will give the leading semiclassical approximation to the quantum propagation of an initial state of the form @xmath56 . let us see how @xmath57 is related to the classical picture of the geometric optics approximation to wave propagation at short wavelength , see , e.g. , @xcite for background . to an initial function @xmath58 one associates the wavefronts which are the level sets of the phase function @xmath10 , the propagated state at time @xmath59 is then of the same form @xmath60 ( provided there are no caustics ) , where the wavefronts of the new phase function @xmath61 are obtained by transporting the initial wavefronts along the geodesics perpendicular to them a time @xmath59 . this translates via the method of characteristics into a first order equation for @xmath61 , the hamilton jacobi or eikonal equation , and in addition the new amplitude @xmath62 is obtained by transporting the initial one along the same set of geodesics and multiplying it with a factor related to the expansion rate of the geodesics . now in our case the wavefronts of @xmath36 are the horocycles associated with @xmath35 and these are mapped onto themselves by transport along perpendicular geodesics , so @xmath36 stays invariant ( up to a simple time dependent constant ) , and only @xmath9 is transported , which is exactly described by the action of @xmath57 . so our first order semiclassical approximation to @xmath63 will be @xmath64 and to show that this is a good approximation even when we project it onto @xmath20 by summing over @xmath25 we have to place some conditions on the amplitude @xmath9 . [ def : h ] set @xmath65 and let @xmath66 and @xmath67 be functions of @xmath12 . then we define the norm @xmath68 and set @xmath69\times{\mathds{d}}\to{\mathds{c}}\ , : \,{\lverta(\hbar)\rvert}_{\alpha,\beta } < \infty\ } $ ] . we will usually omit the @xmath12-dependence from the notation . if @xmath70 then the functions in @xmath71 are analytic , and the factor @xmath72 makes them exponentially decaying for @xmath73 , i.e. , by simple sobolev imbedding we have for @xmath74 @xmath75 see lemma [ lem : pointw - est ] . this exponential decay ensures that the sum over @xmath25 converges , more precisely : for @xmath76 there is a constant @xmath77 such that @xmath78 a proof of this lemma with the explicit dependence of @xmath77 on @xmath79 will be given in section [ sec : quotient ] , see proposition [ prop : l2-h ] . for our applications we are mainly interested in the exponential decay for @xmath73 of the functions in @xmath71 , the analyticity will be necessary to obtain dispersive estimates in section [ sec : dispersive ] which show that these exponential decay properties are preserved under the action of certain operators . we can now state a special version of our main result . [ thm : main1 ] let @xmath80 , where @xmath25 is a fuchsian group , then for all constant @xmath81 there exist constants @xmath82 , @xmath83 such that for all @xmath74 and @xmath37 , @xmath84_{\gamma}\big\rvert}_{l^2(m)}\leq c{\lverta\rvert}_{\alpha,\beta}\hbar\ ] ] for @xmath85 where @xmath86 so the semiclassical approximation is accurate at least up to times of order @xmath3 . we will develop below as well higher order approximations which improve the error term in , but are valid on the same time range . we will as well make the dependence on @xmath87 explicit which will allow for @xmath12 dependent @xmath87 and @xmath9 . but before we do so let us outline the main ideas behind the proof of theorem [ thm : main1 ] . let @xmath88 be defined by @xmath89 this operator is self - adjoint so we can define the unitary operator @xmath90 as the solution of @xmath91 with initial condition @xmath92 . then we will show in section [ sec : cover ] that @xmath93 this relation is the main tool of our analysis , the propagation of a state on @xmath24 is expressed by the action of the two operators @xmath94 and @xmath90 on the amplitude @xmath9 . the first one , @xmath94 , induces propagation of the state along geodesics associated with @xmath35 . the second operator @xmath90 describes dispersion , which takes place on a scale of order @xmath95 , and this is responsible for the error term in . using the unitarity of @xmath90 and we can rewrite the leading semiclassical approximation as @xmath96 and so @xmath97{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi_b}\big)$ ] . since @xmath98 commutes with the action of @xmath25 and is unitary we then find @xmath99_{\gamma}\rvert}_{l^2(m ) } = { \lvert\big([{v_{b}}^*(t)a - a]{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi_b}\big)_{\gamma}\rvert}_{l^2(m)}\ ] ] and now the right hand side contains only the dispersive part @xmath90 . then from integrating we get @xmath100 and so @xmath101 which is of order @xmath95 . so using the unitarity of @xmath98 we got rid of the propagating part @xmath94 which would have lead to exponential proliferation in the sum over @xmath25 and are left with the dispersive part which is easier to control . what we need now to conclude the proof is to show that @xmath102 decays sufficiently fast for @xmath73 so that its sum over @xmath25 is bounded . to this end we have introduced the spaces @xmath103 , we will show in section [ sec : dispersive ] that for times up to we can control the action of @xmath90 on these spaces well enough to ensure the necessary convergence of the sum . but as we will discuss further in section [ sec : dispersive ] it is likely that our estimates are not optimal and could be extended to time scales up to @xmath104 . this would then imply correspondingly larger times in theorem [ thm : main1 ] above and theorem [ thm : main ] below . we will describe now higher order semiclassical approximations and more refined estimates . to this end let @xmath105 for @xmath106 and @xmath107 , and for @xmath74 set @xmath108 and @xmath109 then our main result is [ thm : main ] let @xmath80 , where @xmath25 is a fuchsian group , then for all @xmath79 , with @xmath110 and @xmath111 there exist constants @xmath82 , @xmath83 such that for all @xmath74 and @xmath37 , and any @xmath112 , @xmath113_{\gamma}-u^{(k)}(t)_{\gamma}\big]\big\rvert}_{l^2(m)}\leq c^{n+k+1 } \frac{n^{2n}k!}{(\alpha\hbar)^{2n+4}}\bigg(\frac{\hbar t}{\alpha^2}\bigg)^k\ , { \lverta\rvert}_{\alpha,\beta}\,\ , .\ ] ] for @xmath114 where @xmath115 is given by . by choosing @xmath116 optimally one can obtain an exponentially small remainder term . [ cor : maini ] assume the same conditions as in theorem [ thm : main ] are satisfied , then for any @xmath117 we have for @xmath118 $ ] @xmath119_{\gamma}-u^{(k)}(t)_{\gamma}\big]\big\rvert}_{l^2(m ) } \leq \frac{1}{{\varepsilon}^{1/2}}c^{n+1 } \frac{n^{2n}}{(\alpha\hbar)^{2n+4}}{\mathrm{e}}^{-\frac{1-{\varepsilon}}{c}\frac{\alpha^2}{\hbar t}}\ , { \lverta\rvert}_{\alpha,\beta}\,\ , , \ ] ] where @xmath18 is the same constant as in and @xmath59 has to satisfy @xmath120 the reason for the introduction of @xmath121 is that it allows us to use sobolev imbedding to pass to point - wise estimates . [ cor : mainii ] assume the same conditions as in theorem [ thm : main ] are satisfied , then @xmath122_{\gamma}-u^{(k)}(t)_{\gamma}\big]\big\rvert}\leq c^{n+k+1 } \frac{1}{(\alpha\hbar)^{2n+8 } } ( \hbar t)^k(2n)!k!\ , { \lverta\rvert}_{\alpha,\beta}\,\ , , \ ] ] for @xmath123 furthermore if @xmath118 $ ] for @xmath117 then @xmath124_{\gamma}-\psi^{(k)}(t)_{\gamma}\big]\big\rvert } \leq \frac{1}{{\varepsilon}^{1/2}}c^{n+1 } \frac{n^{2n}}{(\alpha\hbar)^{2n+8}}{\mathrm{e}}^{-\frac{1-{\varepsilon}}{c}\frac{\alpha^2}{\hbar t}}\ , { \lverta\rvert}_{\alpha,\beta } , \ ] ] for @xmath123 so the semiclassical approximations are even point - wise close to the true evolved states . let us make a couple of remarks about these results . the @xmath87 dependence : if @xmath87 and @xmath125 are constant , then we have a time range up to @xmath126 . but we can let @xmath87 depend as well on @xmath12 , this allows to use amplitude functions @xmath9 which depend on @xmath12 and become , e.g. , localised for @xmath0 . an example would be @xmath127 for @xmath128 . for @xmath129 this function is in @xmath103 with @xmath130 and @xmath131 and so the semiclassical approximations work at least up to @xmath132 . this means that we have to have @xmath133 to be able to reach large times . one can improve this by refining the semiclassical approximations and write @xmath134 when applied to a function localised at @xmath135 as a product of a metaplectic operator times another unitary operator . this allows to treat coherent states for which @xmath136 , and we hope to discuss this in more detail in the future . one can as well allow larger spaces than @xmath103 , in particular gevrey type spaces defined by the norm @xmath137 could be useful , because they contain functions of compact support . for these spaces with constant @xmath87 and @xmath125 we would expect that with the mollification introduced in section [ sec : dispersive ] to be able to control semiclassical approximations up to @xmath138 . our semiclassical approximations are of the form @xmath139 for some @xmath140 and the action of @xmath141 increases the effective support of @xmath9 at an exponential rate in @xmath59 , so if @xmath25 is a fuchsian group of the first kind then one can show that , even if the sum is absolutely convergent for @xmath142 , we still have @xmath143 so the phase factors @xmath144 are absolutly crucial to ensure uniformly bounded @xmath145-norms for large @xmath59 . as we mentioned already in the discussion after theorem [ thm : main1 ] our methods can possibly be improved to extend the time range from @xmath3 to @xmath19 . in order to do so we would need some stronger estimates on the action of the operator @xmath134 . in order to keep the presentation as simple as possible we have restricted ourselves here to two - dimensional manifolds of constant negative curvature , but it should be possible to generalise the results . the generalisation to higher dimensional manifolds of constant curvature should be straightforward and we expect the same results to hold , in particular the time scales our methods give do not depend on the dimension . a natural general time scale in semiclassical problems is the heisenberg time @xmath146 which is related to the mean spacing of the eigenvalues , it is the time scale on which the system starts to resolve individual eigenvalues . we see that the optimal time range we can hope to reach with our methods coincides in two - dimensions with the heisenberg time but is shorter in higher dimensions . it is not clear if this is an artefact of the method , or some change of behaviour can happen at that time . since our constructions are mainly of a geometric nature paired with some general analytic estimates on the action of pseudodifferential operators , one should be able to generalise them to riemannian manifolds of non - constant negative curvature . the phase functions @xmath36 are busemann functions and the operators @xmath141 and @xmath134 together with the decomposition @xmath147 can be constructed in exactly the same way . but some of the ensuing estimates become more complicated since the operator @xmath148 can have coefficients which become highly oscillatory , although with a very small amplitude . similar results should hold for other hyperbolic problems , e.g. , the standard wave equation and the dirac equation , with oscillatory initial conditions . the methods developed here can probably be generalised to such cases . the plan of the paper is as follows . in section [ sec : cover ] we discuss time evolution on the universal cover and prove the decomposition . in section [ sec : quotient ] we study the action of differential and pseudodifferential operators on the spaces @xmath103 and show how they can be used together with sobolev imbeddings to get precise estimates on functions @xmath149 on the quotient in terms of @xmath9 . we then proceed in section [ sec : dispersive ] to discuss the crucial properties of the action of @xmath150 on @xmath103 , and in section [ sec : proofs ] we finally use the material collected in the previous sections to prove our main theorems and some related results . some auxiliary material on pseudodifferential operators on @xmath24 has been collected in the appendix . * note on notation * : we will denote by @xmath18 a generic constant which can change from line to line . we write as well sometimes @xmath151 if there is a constant @xmath82 such that @xmath152 . it will be useful to choose special coordinates adapted to the phase - function @xmath36 . since any rotation around the origin is an isometry on @xmath24 , there is an isometry @xmath153 such that @xmath154 , where @xmath155 is the point on @xmath34 at @xmath156 . composing @xmath153 with the standard mapping @xmath157 from the unit disk model to the upper half plane @xmath158 , the geodesics emanating from @xmath35 are mapped to straight lines parallel to the @xmath159-axis and the corresponding horocycles are horizontal lines . the phase function @xmath36 takes in these coordinates the simple form @xmath160 and in order to keep the notation light we will from now on fix the point @xmath37 and drop the reference to it from the notation . we recall as well the expressions for the metric @xmath161 , the laplacian @xmath162 and the volume element @xmath163 in these coordinates . the geodesics emanating from @xmath35 are given in the adapted coordinates by @xmath164 and the flow on @xmath165 induced by shifting with constant speed along these geodesics can be easily seen to be @xmath166 therefore the action of the operator @xmath167 defined in is given in these coordinates by @xmath168 [ lem : sl - unitary ] the operator @xmath169 is unitary and @xmath170 furthermore @xmath171s(t)a\,\ , .\ ] ] i.e. , the generator of @xmath167 is @xmath172 $ ] . the unitarity follows using a simple change of coordinates @xmath173 and is a straightforward computation . using we find that the generator @xmath174 of the unitary operator @xmath175 , see , has as well a simple explicit expression in the adapted coordinates @xmath176 [ prop : reduced - schr ] let @xmath177 and @xmath178 then we have the identity @xmath179 as operators on @xmath180 . since @xmath98 is a solution of @xmath181 , @xmath182 satisfies @xmath183 with the initial condition @xmath184 and where @xmath185 . now a short calculation gives @xmath186 where @xmath187 is the generator of @xmath167 from lemma [ lem : sl - unitary ] . on the other hand we have @xmath188&= \bigg[\frac{1}{2}+\hbar y\bigg]{\mathrm{e}}^{-\frac{{\mathrm{i}}}{\hbar}\frac{t}{2}}s(t)v(t ) -\frac{\hbar^2}{2}{\mathrm{e}}^{-\frac{{\mathrm{i}}}{\hbar}\frac{t}{2}}s(t)\delta(t)v(t)\\ = & \bigg[\frac{1}{2}+\hbar y-\frac{\hbar^2}{2}s(t)\delta(t)s^*(t)\bigg]{\mathrm{e}}^{-\frac{{\mathrm{i}}}{\hbar}\frac{t}{2}}s(t)v(t ) \end{split}\ ] ] and since @xmath189 we find that @xmath190 and @xmath191 satisfy the same first order differential equation with the same initial condition , so they coincide . thus we have separated the action of @xmath98 on oscillatory states @xmath58 into two parts . the part described by @xmath94 is the propagation which is induced by the classical dynamics , note that @xmath94 does not depend on @xmath12 . the second part , coming from @xmath90 , is responsible for dispersion which takes place on a scale of order @xmath95 as we will see in section [ sec : dispersive ] . in this section we will discuss how to use the spaces @xmath103 to obtain precise estimates when passing from @xmath24 to a quotient @xmath80 . recall that @xmath20 is the quotient of @xmath24 by the fundamental group @xmath25 , @xmath80 . given a function @xmath192 on @xmath24 we defined a function @xmath193 on @xmath20 , provided that the sum converges . we will now discuss some conditions on @xmath192 which ensure convergence of @xmath194 . these are based on sobolev imbeddings combined with the following simple estimate for the @xmath195 norm : [ lem : l1-l1 ] let @xmath196 , then @xmath197 and @xmath198 let @xmath199 be a fundamental domain for @xmath20 , then @xmath200 since @xmath201 . let us recall two of the standard sobolev imbedding results . for every @xmath202 there is a constant @xmath203 such that @xmath204 and for every @xmath205 there is another constant @xmath206 such that @xmath207 combining lemma [ lem : l1-l1 ] with the sobolev imbedding gives [ prop : sob ] assume that @xmath196 and @xmath208 , then @xmath209 and there is a constant @xmath82 such that @xmath210 to obtain an estimate on @xmath211 we use the following simple lemma . note that we continue to use the notation @xmath167 instead of @xmath94 , since there is no @xmath35 dependence in the estimates . [ lem : s - l1 ] for @xmath196 we have @xmath212 this follows from @xmath213 where we have used @xmath214 . [ cor : sob ] assume that @xmath196 and @xmath215 for @xmath216 , then there is a constant @xmath82 such that for @xmath216 @xmath217 by proposition [ prop : sob ] we have to estimate @xmath218 and @xmath219 . but by lemma [ lem : s - l1 ] @xmath220 and @xmath221 the drawback of working with @xmath222 on the universal cover is that the action of @xmath175 on @xmath222 is difficult to control , this is the reason that we introduced the spaces @xmath71 . we now analyse the action of pseudodifferential operators on the spaces @xmath103 . the classes of pseudodifferential operators we use are a semiclassical version of the ones developed by zelditch in @xcite based on helgason s harmonic analysis on @xmath24 . the small semiclassical parameter will be denoted by @xmath117 and for reference we have collected the definitions and basic properties in appendix [ app : pseudos - on - d ] . [ prop : pso - action ] assume @xmath223 , @xmath224 , has an analytic symbol , then there are @xmath225 and a constant @xmath226 such that for all @xmath227 with @xmath228 and @xmath229 , and for all @xmath230 $ ] , @xmath231 for @xmath232 . we have @xmath233 and we write @xmath234 with @xmath235 so that @xmath236 . therefore we have to estimate the @xmath145 norm of @xmath237 . we first observe that by theorem [ lem : eqc - egorov ] @xmath238 and since @xmath239 we can write @xmath240 because @xmath241 , since @xmath242 is elliptic . now the operator @xmath243 has symbol @xmath244 which can be bounded using the following auxiliary lemma whose proof we leave to the reader : [ lem : aux ] let @xmath245 and set @xmath246 , then for every @xmath247 there is a constant @xmath82 such that @xmath248 for @xmath249 and @xmath250 . so using this lemma and the calderon vallaincourt theorem we see that @xmath251 for @xmath252 small enough , and this gives @xmath253 in case @xmath254 or that we have products of @xmath255 different operators we would like to determine how the norms depend on @xmath255 . [ cor : prod - est ] let @xmath256 , @xmath257 , and assume that @xmath258 with the same @xmath18 for all @xmath259 , then for @xmath260 we have @xmath261 set @xmath262 for @xmath263 , then @xmath264 and since @xmath265 , @xmath266 we find @xmath267 we can use proposition [ prop : pso - action ] as well to estimate the @xmath195 norm of @xmath268 in terms of @xmath269 . in order to do so we need an auxiliary lemma . [ lem : weight - est ] for @xmath110 we have @xmath270 for some @xmath82 . in geodesic polar coordinates @xmath271 centred at the origin of @xmath24 the riemannian volume element is @xmath272 , and so we find @xmath273 for some @xmath82 . [ cor : l1-h - est ] let @xmath274 and assume @xmath275 and @xmath110 , then there is a @xmath82 such that @xmath276 furthermore if @xmath260 with @xmath256 ( uniformly , i.e. , with the same constants in ) then there is a constant @xmath18 independent of @xmath255 such that @xmath277 we write @xmath278 and apply the cauchy schwarz inequality @xmath279 by lemma [ lem : weight - est ] the first factor on the right hand side is finite since @xmath110 , and we notice that the second is @xmath280 and so the results follow from proposition [ prop : pso - action ] and corollary [ cor : prod - est ] with @xmath281 . if we combine this with the estimates in proposition [ prop : sob ] and corollary [ cor : sob ] this implies the [ prop : l2-h ] assume that @xmath282 , then there is a constant @xmath82 such that for any @xmath74 @xmath283 and @xmath284 furthermore if @xmath285 for some @xmath286 , then there is a @xmath82 such that for all @xmath287 @xmath288_{\gamma}\big\rvert}_{l^2(m)}\leq c^{n+1}\frac{n^{2n}}{(\alpha\hbar)^{2n+4}}{\lverta\rvert}_{\alpha,\beta}\ ] ] and @xmath289_{\gamma}\big\rvert}_{l^2(m)}\leq c^{n+1}\frac{n^{2n}}{(\alpha\hbar)^{2n+4 } } { \lverta\rvert}_{\alpha,\beta}{\mathrm{e}}^{t/2}\,\ , .\ ] ] the first two estimates , and , follow directly by combining proposition [ prop : sob ] and corollary [ cor : sob ] with corollary [ cor : l1-h - est ] . to prove we first use that @xmath5 commutes with the action of @xmath25 and proposition [ prop : sob ] @xmath290_{\gamma}\big\rvert}_{l^2(m)}&= { \big\lvert\big[\delta^n(s(t)a){\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big]_{\gamma}\big\rvert}_{l^2(m)}\\ & \leq c\big({\lvert\delta^{n+2}(s(t)a){\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\rvert}_{l^1({\mathds{d}})}+{\lvert\delta^{n}(s(t)a){\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\rvert}_{l^1({\mathds{d}})}\big ) \end{split}\ ] ] now @xmath291 with @xmath292 and since by @xmath293 where @xmath187 is the generator of @xmath167 , we have @xmath294 uniformly for @xmath216 . then with lemma [ lem : s - l1 ] we find @xmath295 and applying corollary [ cor : l1-h - est ] gives then @xmath296 this , together with the same estimate for @xmath297 proves then , and follows from by setting @xmath142 . we will need as well some point - wise estimates on @xmath9 for @xmath140 , these follow again from sobolev imbedding . [ lem : pointw - est ] there is a @xmath82 such that for all @xmath275 we have @xmath298 for @xmath140 by sobolev imbedding , , we have @xmath299 and applying this to @xmath300 gives @xmath301 but @xmath302 and @xmath303 since @xmath304 by lemma [ lem : weight - est ] and in the last step we used the equivalence of different expressions for the norm @xmath305 in proposition [ prop : norm - equ ] . this lemma is the main tool in the proof of [ prop : oscill - est ] there is a @xmath82 such that for all @xmath306 and @xmath140 with @xmath275 and @xmath307 we have @xmath308 we have @xmath309 and by lemma [ lem : pointw - est ] @xmath310 but by proposition [ prop : sob ] @xmath311 and now using polar coordinates as in the proof of lemma [ lem : weight - est ] gives @xmath312 and @xmath313 since the derivatives of @xmath314 are bounded . this proposition is quite similar to in proposition [ prop : l2-h ] for @xmath315 , but we have no powers of @xmath19 on the right hand side , instead we had to increase the lower bound on the value of @xmath125 from @xmath316 to @xmath317 . we will study in this section how to control that action of @xmath175 on @xmath71 . we have to discuss now some a priori estimates on the action of unitary groups generated by second order operators on functions from the spaces @xmath103 . these belong to the family of energy estimates which are a standard tool . but we will think of the particular estimates we need rather as estimates on the rate of dispersion , and to explain this let us first describe what we need these estimates for . in proposition [ prop : reduced - schr ] we have shown how to write the action of the time evolution operator @xmath98 on oscillatory functions @xmath58 in terms of the action of two operators @xmath167 and @xmath175 on the amplitude @xmath9 . here the operator @xmath167 described transport along geodesics , whereas @xmath175 is the dispersive part . using this partition we are able , as sketched after theorem [ thm : main1 ] , to get rid of @xmath167 in the remainder estimates and reduce them to expressions involving only @xmath175 . the problem is now that we have to estimate the sum over @xmath25 of @xmath318 . using sobolev imbedding we could reduce this to @xmath195-estimates of @xmath175 , but these seem to be very difficult , so we decided to pose the problem in the following form ; assume that @xmath9 satisfies @xmath319 for @xmath110 ( which ensures by proposition [ prop : l2-h ] that @xmath149 is convergent ) , under which conditions ( on @xmath59 and @xmath9 ) do we have then @xmath320 ( so that @xmath321 is still convergent ) . to answer this question it is natural to look at the action of @xmath175 on weighted @xmath145-sobolev spaces , with a weight @xmath322 , which in turn leads to the study of the operator @xmath323 on @xmath324 which satisfies the equation @xmath325 where @xmath326 now @xmath327 is no longer unitary and @xmath328 not selfadjoint . in order to understand the consequences of this let us look at a simple model problem . let @xmath5 be the laplacian on @xmath329 and let us conjugate it with @xmath330 , where @xmath331 is fixed with @xmath332 , then @xmath333 is not selfadjoint due to the term @xmath334 . the equation @xmath335 can easily be solved using fourier transformation which gives @xmath336 , for the fourier - transformed @xmath9 , where @xmath337 denotes the initial condition . so we have an exponentially growing factor @xmath338 and in order to balance this exponential growth we require that for our initial function @xmath337 we have @xmath339 , then @xmath340 is well behaved for @xmath341 but this requirement on the fourier transformation of @xmath337 is equivalent to requiring analyticity and leads directly to the definition of the norms @xmath342 . these heuristic arguments lead us to the following [ conj : dispersive ] for @xmath343 and @xmath344 there exist @xmath345 such that @xmath346 for @xmath140 and @xmath347 since a proof of this conjecture remained elusive , we have to work around it by mollifying the generator of @xmath175 , this will be described in the rest of this section . for the mollified operator we obtain a result similar to conjecture [ conj : dispersive ] but the time scale we eventually reach is of order @xmath3 . conjecture [ conj : dispersive ] would allow us to extend the time scales in theorem [ thm : main ] from @xmath3 to @xmath19 . as support for the conjecture let us show that it is rather easy to prove for @xmath98 . there exist @xmath348 such that for @xmath343 and @xmath344 we have @xmath349 for @xmath140 and @xmath350 by using that @xmath98 is unitary and commutes with @xmath5 we have @xmath351 so we have to estimate the operator @xmath352 . let us set @xmath353 , we have @xmath354 and therefore we have to consider @xmath355 . using the schrdinger equation for @xmath98 we find @xmath356\ ] ] and integrating this equation gives @xmath357\,\ , { \mathrm{d}}t'\\ & = \frac{\hbar}{2}\int_0^t{{\mathcal u}}^*(t ' ) { \mathrm{i}}[\delta,\psi]{{\mathcal u}}(t')\,\ , { \mathrm{d}}t'\,\ , . \end{split}\ ] ] now @xmath358 $ ] is a symmetric first order operator , and since @xmath5 is elliptic there exists a constant @xmath82 such that @xmath358\leq 2c(1+\sqrt{-\delta})$ ] and so @xmath359 this yields @xmath360 and by lemma [ prop : norm - equ ] @xmath361 for some @xmath362 . so we get the condition @xmath363 or @xmath364 let us choose a @xmath365 with @xmath366 $ ] and @xmath367 for @xmath368 $ ] , furthermore let @xmath369 and set @xmath370 . we define @xmath371 then @xmath372 is analytic and exponentially small in @xmath373 for @xmath374 outside any neighbourhood of @xmath375 $ ] . our mollifying operator will then be @xmath376 @xmath377 is a smoothed analytic version of @xmath378 and so for @xmath379 we have @xmath380 . in the next lemma we quantify how fast this limit is reached on @xmath103 . notice that the symbol of @xmath381 is @xmath382 , see appendix a. [ lem : mollifier - est ] we have for any @xmath140 and @xmath344 @xmath383 we have @xmath384 with @xmath385 and so we have to estimate the @xmath145 norm of @xmath386 . to begin with we note that @xmath387 is an analytic @xmath388-pseudodifferential operator with symbol @xmath389 which is analytic and satisfies @xmath390 for some constants @xmath391 . on the other hand in local normal coordinates the standard full symbol of @xmath237 is a function @xmath392 which satisfies similar estimates ( with different @xmath393 , see @xcite for a calculus on non - compact manifolds based on local normal coordinates ) , i.e. , the integral kernel of @xmath237 can be locally written as @xmath394 and so @xmath395 has integral kernel @xmath396 } b(z,\xi)\,\ , { \mathrm{d}}\xi\,\ , .\ ] ] now we use the kuranishi trick and expand @xmath397 using taylors theorem , and so the phase function becomes @xmath398 and then the coordinate change @xmath399 gives @xmath400 with the amplitude @xmath401 . but @xmath402 is bounded and @xmath35 is analytic , so the amplitude @xmath403 satisfies for @xmath404 small enough the same estimate and so by the calderon vallaincourt theorem the @xmath145-norm of @xmath386 is bounded by @xmath405 and therefore @xmath406 let @xmath407 be the mollifier introduced in , and set @xmath408 and let @xmath409 be the unitary operator generated by @xmath410 , i.e. , the solution to @xmath411 since @xmath412 uniformly for @xmath216 and @xmath413 we have [ lem : moll - bounded ] we have @xmath414 where @xmath415 is uniformly bounded for all @xmath216 . in this subsection we want to prove the following dispersive estimate [ thm : dipsersive ] there exist constants @xmath348 such that for any @xmath416 with @xmath417 and @xmath344 we have @xmath418 for @xmath140 and @xmath216 satisfying @xmath419 if the condition @xmath417 is not fulfilled , then the theorem remains true if one replaces by @xmath420 as follows from the proof . since we use this theorem mostly for the case that @xmath421 , @xmath422 and @xmath423 , we do nt need this case . the proof gives as well a larger time range if we have @xmath424 , for @xmath281 we can actually get @xmath425 but this transition of the time scales takes place on a @xmath426 scale , so we need very small @xmath427 to see it . to prepare the proof we need several lemmas [ lem : delta - conj ] there exist a @xmath428 such that @xmath429 let us introduce the operator @xmath430 , then @xmath431+\frac{\hbar/{\varepsilon}}{1+\hbar t/{\varepsilon}}\sqrt{-\delta } \bigg]v_{{\varepsilon}}(t)\,\ , , \ ] ] and we rewrite the term in brackets as @xmath432+\frac{\hbar/{\varepsilon}}{1+\hbar t/{\varepsilon}}\sqrt{-\delta}\\ & = ( -\delta)^{1/4}\bigg[{\mathrm{i}}\hbar(-\delta)^{-1/4}[\delta_{{\varepsilon}}(t),\sqrt{-\delta}](-\delta)^{-1/4}+\frac{\hbar/{\varepsilon}}{1+\hbar t/{\varepsilon } } \bigg](-\delta)^{1/4}\,\ , . \end{split}\ ] ] with lemma [ lem : moll - bounded ] we have @xmath433 and since @xmath434 the pseudodifferential calculus gives @xmath435\in \psi_{{\varepsilon}}^{0,2}$ ] and @xmath436(-\delta)^{-1/4}\in\psi_{{\varepsilon}}^{-1,1}$ ] . therefore @xmath437(-\delta)^{-1/4}\in \psi^{-1,0}_{{\varepsilon}}\ ] ] and hence is bounded , so there is a constant @xmath438 such that @xmath439+\frac{\hbar/{\varepsilon}}{1+\hbar t/{\varepsilon}}\sqrt{-\delta } & \leq \frac{\hbar}{{\varepsilon } } \bigg[c-1+\frac{1}{1 + \hbar t/{\varepsilon}}\bigg]\sqrt{-\delta}\\ & \leq c\frac{\hbar}{{\varepsilon}}\ , \sqrt{-\delta } \end{split}\ ] ] and @xmath440+\frac{\hbar/{\varepsilon}}{1+\hbar t/ { \varepsilon}}\sqrt{-\delta } & \geq \frac{\hbar}{{\varepsilon } } \bigg[-c+\frac{1}{1+\hbar t/{\varepsilon}}\bigg]\sqrt{-\delta}\\ & \geq -c\frac{\hbar}{{\varepsilon}}\ , \sqrt{-\delta}\,\ , . \end{split}\ ] ] so by using the estimate in we get @xmath441 which implies @xmath442 , i.e. , @xmath443 on the other hand side , if we use in we have @xmath444 which implies @xmath445 , i.e. , @xmath446 since @xmath447 . [ lem : psi - conj ] let @xmath448 be a smooth function with @xmath449 for some @xmath82 , then @xmath450 we have @xmath451v_{{\varepsilon}}(t)$ ] and integrating this equation gives @xmath452v_{{\varepsilon}}(t')\,\ , { \mathrm{d}}t'\,\ , .\ ] ] but there is a constant @xmath82 such that @xmath453\leq c\sqrt{-\delta}\ ] ] and so by lemma [ lem : delta - conj ] we find @xmath454v_{{\varepsilon}}(t')\,\ , { \mathrm{d}}t'\leq \hbar t c\sqrt{-\delta}\ ] ] for @xmath455 . we can now prove theorem [ thm : dipsersive ] by proposition [ prop : norm - equ ] the norm @xmath305 is equivalent to @xmath456 , and we will work with that norm . so we have to estimate @xmath457 and using unitarity of @xmath409 we have @xmath458 now @xmath459 and lemma [ lem : delta - conj ] and lemma [ lem : psi - conj ] give @xmath460 and so we have @xmath461 if @xmath462 we will now analyse this inequality , in order to simplify the notation let us introduce @xmath463 and @xmath464 , then can be rewritten as @xmath465 , and this is certainly satisfied if we have @xmath466 the first of these inequalities easily reduces to @xmath467 by convexity of the log , the second inequality is satisfied if we have @xmath468 but @xmath469 for @xmath470 and so we obtain the condition @xmath471 , which is @xmath472 so we have shown that if and are satisfied , that then also holds . but is and for @xmath417 implies . this proves the upper bound in . on the other hand we obtain from lemma [ lem : delta - conj ] and lemma [ lem : psi - conj ] as well that @xmath473 and so we find @xmath474 provided @xmath475 we analyse this inequality along the same lines as , with the same abbreviations it can be rewritten as @xmath476 which follows from the two separate inequalities @xmath477 the first one reduces again to , the second one is equivalent to @xmath478 and by convexity this holds if @xmath479 . but like above we have @xmath480 and so the second inequality follows as well from . so under these conditions we have the lower bound @xmath481 . from this we obtain @xmath482 which is . in this section we combine the semiclassical approximations on the upper half plane developed in section [ sec : cover ] with the dispersive estimates from section [ sec : dispersive ] and the estimates on @xmath103 from section [ sec : quotient ] to provide the proof of the main theorems . the first step is to show that we can replace the operator @xmath175 with its mollified version @xmath409 . to this end we show first that the generators are close on @xmath103 . [ lem : comp - deltas ] we have @xmath483 we write @xmath484 and then applying lemma [ prop : pso - action ] and lemma [ lem : mollifier - est ] gives @xmath485 { \lverta\rvert}_{\alpha,\beta } \end{split}\ ] ] and with the choice @xmath486 the claim follows . now we can proceed to show that @xmath175 and @xmath409 are close . [ lem : vmol - v ] there exist constants @xmath348 such that for @xmath110 , @xmath487 , @xmath140 and @xmath488 @xmath489a{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big)_{\gamma}\rvert}_{l^2(m ) } \leq { \lverta\rvert}_{\alpha,\beta } \frac{c^{n+1 } n^{2n}}{\alpha^3(\hbar\alpha)^{2n+3}}{\mathrm{e}}^{-\frac{1}{4}(\alpha/{\varepsilon}-2 t ) } \,\ , , \ ] ] for @xmath490 . notice that the right hand side of is small if @xmath491 , whereas we have as well the condition @xmath492 , so we see that the largest time range for which @xmath409 is close to @xmath175 on @xmath103 is obtained if we choose @xmath493 then @xmath409 is close to @xmath175 on @xmath103 if @xmath494 we have @xmath495v_{{\varepsilon}}(t)\ ] ] and integrating this equation gives @xmath496v_{{\varepsilon}}(t')\ , \ , { \mathrm{d}}t'\,\ , .\ ] ] if we set @xmath497v_{{\varepsilon}}(t')a\,\ , , \ ] ] and use @xmath498{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}={\mathrm{e}}^{-\frac{{\mathrm{i}}}{\hbar}\frac{t'}{2 } } { { \mathcal u}}^*(t')[(s(t')b){\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}]\,\ , , \ ] ] which follows from proposition [ prop : reduced - schr ] , we find @xmath499{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi } = { \mathrm{i}}\frac{\hbar}{2 } \int_0^t{\mathrm{e}}^{-\frac{{\mathrm{i}}}{\hbar}\frac{t'}{2 } } { { \mathcal u}}^*(t')[(s(t')b){\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}]\,\ , { \mathrm{d}}t'\ ] ] which gives @xmath500{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big)_{\gamma}\rvert}_{l^2(m ) } \leq \frac{\hbar}{2 } \int_0^t{\lvert\delta^n[(s(t')b(t')){\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}]_{\gamma}\rvert}_{l^2(m)}\,\ , { \mathrm{d}}t'\ ] ] since @xmath98 commutes with action of @xmath25 and is unitary . but by proposition [ prop : l2-h ] @xmath501_{\gamma}\rvert}_{l^2(m ) } \leq c^{n+1}\frac{n^{2n}}{(\hbar\alpha_0)^{2n+4 } } { \lvertb(t')\rvert}_{\alpha_0,\beta}{\mathrm{e}}^{t'/2}\,\ , , \ ] ] an furthermore by lemma [ lem : comp - deltas ] and the dispersive estimate in theorem [ thm : dipsersive ] @xmath502 for @xmath503 and @xmath504 ( and of course @xmath18 changes from line to line ) . combining these estimates gives @xmath501_{\gamma}\rvert}_{l^2(m ) } \leq c^{n+1}\frac{n^{2n}}{(\hbar\alpha_0)^{2n+4}}\frac{{\mathrm{e}}^{-(\alpha_1-\alpha_0)/{\varepsilon}}}{(\alpha_1-\alpha_0)^2 } { \lverta\rvert}_{\alpha_2,\beta}{\mathrm{e}}^{t'/2}\ ] ] and if we choose now @xmath505 , @xmath506 and @xmath507 this is @xmath501_{\gamma}\rvert}_{l^2(m ) } \leq c^{n+1}\frac{n^{2n}}{\alpha^2(\hbar\alpha)^{2n+4 } } { \lverta\rvert}_{\alpha,\beta}{\mathrm{e}}^{-(\alpha/{\varepsilon}-2t')/4}{\lverta\rvert}_{\alpha,\beta}\ ] ] for @xmath508 and so finally @xmath500{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big)_{\gamma}\rvert}_{l^2(m ) } \leq \frac{c^{n+1}\hbar n^{2n}}{\alpha^2(\hbar\alpha)^{2n+4 } } { \lverta\rvert}_{\alpha,\beta}{\mathrm{e}}^{-(\alpha/{\varepsilon}-2t)/4}{\lverta\rvert}_{\alpha,\beta}\,\ , .\ ] ] the operator @xmath409 can be approximated recursively by a volterra series as follows , [ lem : dyson - series ] let @xmath509 , then for any @xmath510 @xmath511 where @xmath512 @xmath513 for @xmath106 and @xmath514 this is a standart argument . we integrate equation @xmath515 and iterating this equation gives the lemma . we now estimate the terms in this expansion and the remainder . [ lem : pk - est ] there exists a constant @xmath82 such that for @xmath516 with @xmath517 and @xmath518 we have for all @xmath140 @xmath519 and for every @xmath83 there is a constant @xmath520 such that @xmath521 if @xmath522 furthermore @xmath523 if @xmath524 . we can view @xmath410 as an operator in @xmath525 for all @xmath224 , this allows to balance the powers of @xmath404 and @xmath526 appearing in the estimate , we will choose @xmath527 , which gives the estimate @xmath528 then by corollary [ cor : prod - est ] @xmath529 and together with @xmath530 and @xmath531 this gives . from we directly obtain @xmath532 and if @xmath533 the sum is uniformly bounded . finally the same argument leading to gives @xmath534 and by theorem [ thm : dipsersive ] @xmath535 if @xmath536 and @xmath487 , and so choosing @xmath486 proves . we have now collected most of the material we need to prove theorem [ thm : main ] . we will do this in two steps . we first prove a theorem similar to theorem [ thm : main ] but with the semiclassical approximation done in terms of the volterra series defined by the mollified operator @xmath409 . and then we will show that the volterra series defined by the mollified operator and the original operator @xmath175 are close . for @xmath140 let us set @xmath537 and @xmath538 [ thm : main - moll ] there are constants @xmath348 such that for @xmath140 , with @xmath110 , @xmath487 , and @xmath539 we have @xmath540\big\rvert}_{l^2(m)}\\ & \qquad \qquad\leq c^{n+1}\frac{n^{2n}}{(\hbar\alpha)^{2n+4}}\bigg[c^k\bigg(\frac{\hbar { \lvertt\rvert}}{\alpha^2}\bigg)^{k+1 } + \frac{\hbar}{\alpha^2}{\mathrm{e}}^{-\frac{1}{8}(\alpha/{\varepsilon}-4t)}\bigg ] { \lverta\rvert}_{\alpha,\beta } \end{split}\ ] ] for @xmath508 . we start by using proposition [ prop : reduced - schr ] to write @xmath541{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big ) \end{split}\ ] ] and so @xmath542_{\gamma}\rvert}_{l^2(m ) } & = { \lvert\delta^n[{{\mathcal u}}(t)((v^*(t)a^{(k)}_{{\varepsilon}}){\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi})]_{\gamma}\rvert}_{l^2(m)}\\ & = { \lvert\delta^n[(v^*(t)a^{(k)}_{{\varepsilon}}-a){\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}]_{\gamma}\rvert}_{l^2(m ) } \end{split}\ ] ] since @xmath98 commutes with @xmath121 and the action of @xmath25 and is unitary . in the next step we want to replace @xmath543 with the mollified version @xmath544 , to this end we write @xmath545 and so with lemma [ lem : vmol - v ] we find @xmath546_{\gamma}\rvert}_{l^2(m ) } \\ & \qquad \qquad \qquad\leq c^{n+1}\frac{n^{2n}}{\alpha^3(\hbar\alpha)^{2n+3}}{\mathrm{e}}^{-\frac{1}{8}(\alpha/{\varepsilon}-4 t ) } { \lvertv^*_{{\varepsilon}}(t)a^{(k)}_{{\varepsilon}}\rvert}_{\alpha/2,\beta } \end{split}\ ] ] for @xmath490 . and then theorem [ thm : dipsersive ] and lemma [ lem : pk - est ] finally give @xmath547 for @xmath548 . so we have @xmath549_{\gamma}\rvert}_{l^2(m ) } & = { \lvert\delta^n[(v^*_{{\varepsilon}}(t)a^{(k)}_{{\varepsilon}}-a){\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}]_{\gamma}\rvert}_{l^2(m)}\\ & \quad + o\bigg({\lverta\rvert}_{\alpha,\beta}\frac{c^{n+1 } n^{2n}}{\alpha^3(\hbar\alpha)^{2n+3}}{\mathrm{e}}^{-\frac{1}{8}(\alpha/{\varepsilon}-4t)}\bigg)\,\ , . \end{split}\ ] ] now we can use the volterra series for @xmath409 from lemma [ lem : dyson - series ] @xmath550 and so with from proposition [ prop : l2-h ] , the dispersive estimates from theorem [ thm : dipsersive ] and the estimates for @xmath551 from lemma [ lem : pk - est ] we obtain @xmath552_{\gamma}\rvert}_{l^2(m ) } & = \bigg(\frac{\hbar}{2}\bigg)^k{\lvert\delta^n[(v^*_{{\varepsilon}}(t)r_k(t)a){\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}]_{\gamma}\rvert}_{l^2(m)}\\ & \leq c^{n+1}\frac{\hbar^k n^{2n}}{(\hbar\alpha)^{2n+4}}{\lvertv^*_{{\varepsilon}}(t)r_k(t)a\rvert}_{\alpha/3,\beta}\\ & \leq c^{n+1}\frac{\hbar^k n^{2n}}{(\hbar\alpha)^{2n+4}}{\lvertr_k(t)a\rvert}_{2\alpha/3,\beta}\\ & \leq c^{k+n+1}\frac{n^{2n}}{(\hbar\alpha)^{2n+4}}\bigg(\frac{\hbar { \lvertt\rvert}}{\alpha^2}\bigg)^k { \lverta\rvert}_{\alpha,\beta}\,\ , . \end{split}\ ] ] for @xmath553 and @xmath554 . let us discuss for which choice of @xmath404 we obtain the maximal time range for which the right hand side of is small . in order that the exponential term @xmath555 we must have @xmath491 . this must hold together with @xmath492 , and these two upper bounds are equal if @xmath556 with this choice of @xmath404 we have @xmath557 and then @xmath558 , using these bounds together with @xmath559 the estimate becomes @xmath560_{\gamma}\rvert}_{l^2(m ) } \leq { \lverta\rvert}_{\alpha,\beta}c^{n+k+1}\frac{n^{2n}k!}{(\hbar\alpha)^{2n+4}}\frac{1}{\alpha^{k+1 } } \bigg(\frac{\hbar } { \alpha}\bigg)^{\frac{k+1}{2}}\ ] ] if @xmath561 with a sufficiently small constant @xmath286 this gives us already a good approximation for @xmath562 , but it is defined in terms of the mollified operator @xmath563 . in the final step we replace @xmath563 by @xmath5 in the approximations . but before doing so we want to show how to prove theorem [ thm : main1 ] using . we set @xmath315 in which gives @xmath564_{\gamma}\rvert}_{l^2(m ) } \leq { \lverta\rvert}_{\alpha,\beta}c^{k+1}\frac{k!}{(\hbar\alpha)^{4}}\frac{1}{\alpha^{k+1 } } \bigg(\frac{\hbar } { \alpha}\bigg)^{\frac{k+1}{2}}\ ] ] for all @xmath140 and @xmath565 . we would like to use this with @xmath566 , but the factor @xmath567 on the right hand side prevents us from doing so . instead we will write @xmath568 as a sum of terms to which we can apply with large @xmath116 . to this end we use for @xmath569 let us set @xmath570 and furthermore @xmath571 if @xmath572 and @xmath573 for @xmath566 . using these operators we then set for @xmath140 , @xmath106 @xmath574 and @xmath575 . then we have for all @xmath576 @xmath577 we have for all @xmath116 @xmath578 and this can be rewritten as @xmath579 by iterating this relation we arrive at . but in order to prove that is actually correct it is easier to use with @xmath116 replaced by @xmath580 and @xmath9 by @xmath581 which gives @xmath582 by . summing this over @xmath583 then yields @xmath584 using this lemma we now set @xmath585 and we notice that this is close to @xmath586 by theorem [ thm : main - moll ] , and therefore we rewrite @xmath587 as @xmath588+\sum_{k=0}^k\bigg(\frac{-{\mathrm{i}}\hbar}{2}\bigg)^k{{\mathcal u}}(t ) \big(a_k^{(k)}{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big ) \end{split}\ ] ] which finally gives @xmath589\\ & + \sum_{k=1}^k\bigg(\frac{-{\mathrm{i}}\hbar}{2}\bigg)^k{{\mathcal u}}(t ) \big(a_k^{(k)}{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big)\,\ , . \end{split}\ ] ] now we can take the @xmath145-norms of the projections to @xmath20 and with the unitarity of @xmath590 , the estimate and proposition [ prop : oscill - est ] we obtain @xmath591_{\gamma}\rvert}_{l^2(m ) } & \leq \sum_{k=0}^k\bigg(\frac{\hbar}{2}\bigg)^k { \lvert\big[u_{{\varepsilon},k}^{(k - k)}-{{\mathcal u}}(t ) \big(a_k^{(k)}{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big)\big]_{\gamma}\rvert}_{l^2(m)}\\ & \qquad \qquad + \sum_{k=1}^k\bigg(\frac{\hbar}{2}\bigg)^k{\lvert[a_k^{(k)}{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}]_{\gamma}\rvert}_{l^2(m ) } \\ & \leq \sum_{k=0}^k\bigg(\frac{\hbar}{2}\bigg)^k{\lverta_k^{(k)}\rvert}_{\alpha,\beta } \frac{(k - k)!}{(\hbar\alpha)^{4 } } \frac{c^{k - k+1}}{\alpha^{k - k+1 } } \bigg(\frac{\hbar } { \alpha}\bigg)^{\frac{k - k+1}{2 } } \\ & \qquad \qquad + \frac{c\beta^4}{\beta-1}\bigg(\frac{1}{\alpha^4}+1\bigg ) \sum_{k=1}^k\bigg(\frac{\hbar}{2}\bigg)^k{\lverta_k^{(k)}\rvert}_{\alpha,\beta}\,\ , . \end{split}\ ] ] we have by assumption @xmath592 ( independent of @xmath12 ) and @xmath307 fixed , so then the second sum is for finite @xmath116 of order @xmath593 . in the first sum the power of @xmath12 in the @xmath583th term is @xmath594 and so if we choose @xmath595 this sum is as well of order @xmath593 . therefore we have @xmath596_{\gamma}\rvert}_{l^2(m ) } \ll { \lverta\rvert}_{\alpha,\beta}\hbar\ ] ] for @xmath597 . what is left now in order to complete the proof of our main result , theorem [ thm : main ] , is to estimate the difference between the semiclassical approximations in terms of the mollified operator @xmath563 and the original @xmath5 . let us set @xmath598 and @xmath599 then we have [ lem : a - aeps ] there is a constant @xmath82 such that for @xmath600 , @xmath110 , @xmath111 , @xmath140 , @xmath572 and @xmath601 @xmath602_{\gamma}\rvert}_{l^2(m ) } \leq c^{n+k+1}\frac{n^{2n}k!}{(\alpha\hbar)^{2n+4 } } \bigg(\frac{{\lvertt\rvert}\hbar}{\alpha^2}\bigg)^{k}{\lverta\rvert}_{\alpha,\beta}\ ] ] if @xmath603 the proof of this proposition relies on two lemmas . in the first we estimate the difference between the expansions of @xmath175 and @xmath409 on @xmath103 . [ lem : a - aeps ] there is a constant @xmath82 such that for all @xmath517 we have @xmath604 we start by estimating the norm of @xmath605 to this end we introduce for @xmath606 @xmath607 and then write @xmath608 now by combining lemma [ lem : comp - deltas ] and proposition [ prop : pso - action ] we see that @xmath609 and so therefore @xmath610 taking the @xmath59-integral into account as in this leads to @xmath611 now what remains to do is to estimate the sum in . [ lem : aux - sum - est ] for every @xmath612 there is are constants @xmath348 such that for @xmath613 we have @xmath614 we write @xmath615 and setting @xmath616 and @xmath617 the sum becomes @xmath618 now by lemma [ lem : aux ] we have @xmath619 and using @xmath620 we have @xmath621 so using lemma [ lem : aux ] once more to see that @xmath622 we finally have @xmath623 for @xmath624 and some @xmath82 . the first part of the proof is similar to the proof of theorem [ thm : main - moll ] . let us set @xmath625 and write @xmath626{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\\ & = [ s(t)v(t)v^*(t)b]{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\\ & = { { \mathcal u}}(t)\big([v^*(t)b]{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big ) \end{split}\ ] ] and since @xmath98 is unitary and commutes with @xmath5 and the action of @xmath25 we find @xmath627_{\gamma}\rvert}_{l^2(m ) } = { \lvert\delta^n\big([v^*(t)b]{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big)_{\gamma}\rvert}_{l^2(m)}\,\ , .\ ] ] now we want to replace @xmath628 by @xmath629 as in the proof of theorem [ thm : main - moll ] . to this end we write @xmath630 and then by proposition [ prop : l2-h ] and lemma [ lem : vmol - v ] we obtain @xmath631{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big)_{\gamma}\rvert}_{l^2(m)}&\leq { \lvert\delta^n\big([v_{{\varepsilon}}^*(t)b]{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big)_{\gamma}\rvert}_{l^2(m)}\\ & \qquad+{\lvert\delta^n\big([(v^*(t)v_{{\varepsilon}}(t)-1)v_{{\varepsilon}}^*(t)b]{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi}\big)_{\gamma}\rvert}_{l^2(m)}\\ & \leq c^{n+1}\frac{n^{2n}}{(\alpha\hbar)^{2n+4 } } { \lvertv_{{\varepsilon}}^*(t)b\rvert}_{\alpha/3,\beta}\\ & \qquad+c^{n+1}\frac{n^{2n}}{\alpha^3(\alpha\hbar)^{2n+3 } } { \mathrm{e}}^{-\frac{1}{4}(\frac{\alpha}{{\varepsilon}}-2t)}{\lvertv_{{\varepsilon}}^*(t)b\rvert}_{\alpha/3,\beta}\\ & = c^{n+1}\frac{n^{2n}}{(\alpha\hbar)^{2n+4 } } \bigg(1+\frac{\hbar}{\alpha^2 } { \mathrm{e}}^{-\frac{1}{4}(\frac{\alpha}{{\varepsilon}}-2t)}\bigg){\lvertv_{{\varepsilon}}^*(t)b\rvert}_{\alpha/3,\beta}\,\ , . \end{split}\ ] ] but by the dispersive estimate in theorem [ thm : dipsersive ] we have @xmath632 for @xmath492 and @xmath633 . now we can apply lemma [ lem : a - aeps ] to @xmath634 , which gives @xmath635 and then lemma [ lem : aux - sum - est ] allows to estimate the sum which yields @xmath636 if @xmath637 . if we require in addition that @xmath491 then the exponential term @xmath638 is bounded and the optimal choice for @xmath404 is then @xmath639 with this choice for @xmath404 the condition @xmath487 becomes @xmath111 . combining the successive estimates gives then finally @xmath627_{\gamma}\rvert}_{l^2(m ) } \leq c^{n+k+1}\frac{n^{2n}k!}{(\alpha\hbar)^{2n+4 } } \bigg(\frac{{\lvertt\rvert}\hbar}{\alpha^2}\bigg)^{k}{\lverta\rvert}_{\alpha,\beta}\ ] ] for @xmath600 , @xmath111 and @xmath603 notice that for @xmath116 fixed one actually can obtain an error estimate of order @xmath640 . now we can prove our main theorem . if we combine the estimates from theorem [ thm : main - moll ] and proposition [ lem : a - aeps ] and set @xmath600 we obtain @xmath641_{\gamma}\rvert}_{l^2(m ) } & \leq { \lvert\delta^n[u^{(k)}-u^{(k)}_{{\varepsilon}}]_{\gamma}\rvert}_{l^2(m)}\\ & \qquad\quad + { \lvert\delta^n[u^{(k)}_{{\varepsilon}}-{{\mathcal u}}(t)(a{\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\varphi})]_{\gamma}\rvert}_{l^2(m)}\\ & \leq c^{n+k+1}\frac{n^{2n}k!}{(\alpha\hbar)^{2n+4 } } \bigg(\frac{{\lvertt\rvert}\hbar}{\alpha^2}\bigg)^{k}{\lverta\rvert}_{\alpha,\beta } \end{split}\ ] ] for @xmath603 finally the condition @xmath417 from theorem [ thm : dipsersive ] together with the choice @xmath600 gives @xmath111 . the proof of corollary [ cor : maini ] is now a standard estimate . we have @xmath642\big\rvert}_{l^2(m)}\leq c^{n+k+1 } \frac{n^{2n}k!}{(\alpha\hbar)^{2n+4}}\bigg(\frac{\hbar t}{\alpha^2}\bigg)^k\ , { \lverta\rvert}_{\alpha,\beta}\,\ , , \ ] ] let us set @xmath643 , then using sterlings formula we find @xmath644 if @xmath645 . but if @xmath646 then @xmath647 and so @xmath648 finally corollary [ cor : mainii ] follows from sobolev imbedding . we use the standard relation @xmath649 applying this to gives , and to gives . notice that if @xmath307 we could use as well use proposition [ prop : oscill - est ] which would reduce the power of @xmath650 in corollary [ cor : maini ] and [ cor : mainii ] . we collect here some elements of a semiclassical calculus of pseudodifferential operators on @xmath24 , which is a simple extension of the calculus developed in @xcite . we denote by @xmath651 the space of uniformly bounded smooth functions on @xmath24 , i.e. , @xmath652 if for every @xmath653 there is a constant @xmath654 such that @xmath655 let @xmath656 $ ] be small parameter , we say a family of operators @xmath657 has symbol @xmath658 if @xmath659 for all @xmath660 . , this is due to the fact that we use the metric @xmath661 instead of @xmath662 , in order to have curvature @xmath663 . ] for @xmath664 we have the non - euclidean fourier - transform @xmath665 and the inversion formula @xmath666 applying the definition of the symbol to the inversion formula gives an integral formula for the action of the operator @xmath667 , @xmath668 pseudodifferential operators are defined by requiring conditions on the symbol of an operator . we will view @xmath669 as coordinates on the co - tangent bundle @xmath47 via the mapping @xmath670 let @xmath671 be the sasaki metric on @xmath47 , @xmath672 the restriction to the unit cotangent bundle @xmath673 and @xmath674 the corresponding laplace beltrami operator on @xmath673 . we say that @xmath675 if for all @xmath676 there are constants @xmath77 such that @xmath677 the corresponding class of operators are defined by will be denoted by @xmath678 . these classes of pseudodifferential operators satisfy the usual properties * product - formula : for @xmath679 and @xmath680 we have @xmath681 and @xmath682\in \psi^{m+m'-1,k+k'-1}_{{\varepsilon}}({\mathds{d}})$ ] * boundedness : the calderon vallaincourt theorem holds : the @xmath145 norm of operators can be estimated by a finite number of derivatives of the symbol , in particular the operators in @xmath683 are bounded on @xmath324 . in particular we have @xmath684 since its symbol is @xmath685 in this appendix we sketch a proof of we have @xmath695{\mathrm{e}}^{tp_{\alpha,\beta}}\ ] ] and @xmath696\in \psi^{0,0}_{{\varepsilon}}$ ] is analytic , so by theorem [ lem : eqc - egorov ] @xmath697{\mathrm{e}}^{tp_{\alpha,\beta}}\in \psi^{0,0}_{{\varepsilon}}\ ] ] for @xmath698 and therefore by integrating in @xmath59 we find @xmath699 let us define @xmath700 by @xmath701 i.e. , @xmath702 , and taking the derivative with respect to @xmath59 gives @xmath703{\mathrm{e}}^{-tp_{\alpha,0}}{\mathrm{e}}^{-tp_{0,\beta}}\\ & = { \mathrm{e}}^{tp_{\alpha,\beta}}[p_{\alpha,\beta}-p_{\alpha,0}-{\mathrm{e}}^{-tp_{\alpha,0}}p_{0,\beta}{\mathrm{e}}^{tp_{\alpha,0}}]{\mathrm{e}}^{-tp_{\alpha,\beta}}b(t)\,\ , . \end{split}\ ] ] now we have @xmath704 by lemma [ lem : com - small ] and so by theorem [ lem : eqc - egorov ] @xmath705 and from @xmath706 a comparison argument gives that @xmath707 is bounded from above and below , which proves the equivalence of the norms .

we study solutions of the time dependent schrdinger equation on riemannian manifolds with oscillatory initial conditions given by lagrangian states . semiclassical approximations describe these solutions for @xmath0 , but their accuracy for @xmath1 is in general only understood up to the ehrenfest time @xmath2 , and the most difficult case is the one where the underlying classical system is chaotic . we show that on surfaces of constant negative curvature semiclassical approximations remain accurate for times at least up to @xmath3 in the case that the lagrangian state is associated with an unstable manifold of the geodesic flow .

the question of the time that it takes for stochastic process to reach a specific point or state by the first time is central in many applications of stochastic modeling in physics ( kramers problem @xcite ) , chemistry ( reaction kinetics @xcite ) , biology ( neural activity models @xcite ) and economics ( estimation of the ruin time @xcite ) . for a random walk sequence , a nontrivial theorem due to sparre - andersen @xcite states that the asymptotic form of the probability @xmath0 of not crossing the boundary within the first @xmath1 steps after starting the motion at @xmath2 ( otherwise named the survival probability on the positive semi - axis ) does not depend on the form of a jump length distribution if only it is symmetric , continuous and markovian . for a large number of steps , one invariably has regardless of the exact jump length distribution type ) @xmath3 , where the prefactor @xmath4 depends on the initial position . the result can be easily generalized for the continuous - time version of the process . for unbiased , continuous gaussian random walk its first passage time density ( fptd ) from @xmath5 to @xmath6 can be easily calculated explicitly @xcite : let us denote @xmath7 the probability of motion from a position @xmath5 to @xmath8 in time @xmath9 with @xmath6 denoting the position on the way from @xmath5 to @xmath8 , i.e. @xmath10 . by taking @xmath11 as the probability of arriving for the first time at @xmath6 at time @xmath12 , the equation for @xmath13 reads @xmath14 with its laplace transform given by @xmath15 for unbiased gaussian random walk we have @xmath16 . with the laplace transform @xmath17 one obtains @xmath18 which , by inverse laplace transform , yields the lvy - smirnov distribution @xmath19 where @xmath5 represents the initial condition . this `` inverse gaussian distribution '' decays for long times as @xmath20 and does not have a first moment , i.e. the mean first passage time from @xmath5 to @xmath6 diverges . on the other hand , since @xmath21 , the particle performing the one dimensional gaussian random walk will certainly hit any point @xmath6 during its motion . assuming the absorbing boundary located at the origin , i.e. at @xmath22 , formula ( [ eq : l - s ] ) with @xmath22 gives the probability density of the first passage time from the positive semi - axis for a gaussian random walk . it should be stressed , however that for generally non - gaussian noises , the knowledge of the boundary location may be insufficient to specify in full the corresponding conditions for absorption or reflection @xcite . in particular trajectories of lvy walks may exhibit discontinuous jumps and in a consequence , the location of the boundary itself is not hit by the majority of sample trajectories . in order to properly take care of possible excursion of the trajectories beyond the location of the boundary ( at , say @xmath22 ) with subsequent re - crossings into the interval ( @xmath23 ) , the whole semi - line ( @xmath24 ) has to be assumed `` absorbing '' . this nonlocal definition of the boundary conditions secures proper evaluation of the first passage time distribution and survival probability @xcite , see below . the very same scenario , see eq . ( [ eq : l - s ] ) , dictated by the sparre - andersen theorem holds also true for `` paradoxical '' diffusion - like processes studied in terms of ctrw ( continuous time random walks ) where kinetics of the walker is determined by the distribution of jump lengths and distribution of waiting times before a next jump to occur @xcite . if the process is regular in time but with nontrivial jump distribution following the ( symmetric ) lvy law of stability ( so called symmetric lvy flight ) , the first passage time density ( fptd ) follows the sparre - andersen universality @xcite . notably , however , if subordinating the number of steps @xmath1 to the physical clock time @xmath9 such that the number of steps @xmath1 per unit of physical time follows some distribution with a power - law tail @xmath25 with @xmath26 , the deviations from the universality can be observed @xcite . to further elucidate the nature of deviation from the `` standard '' sparre - andersen scaling in subordinated scenarios , we consider the process @xmath27 , for which the parental process @xmath28 is described by a langevin equation @xcite @xmath29 driven by a symmetric @xmath30-stable lvy motion @xmath31 with the fourier transform @xmath32 . here @xmath12 stands for the operational time scale which is changed to the physical time scale @xmath9 by subordination @xmath33 . the subordinator @xmath34 is defined as @xmath35 with @xmath36 denoting a strictly increasing @xmath37-stable lvy motion ( @xmath38 ) and is assumed to be independent from the noise term @xmath31 . the above setup has been recently proved @xcite to give a proper stochastic realization of the random process described otherwise by a fractional diffusion equation @xcite @xmath39 p(x , t ) . \label{eq : ffpe}\ ] ] here @xmath40 denotes the riemann - liouville fractional derivative defined as @xmath41 with @xmath42 and @xmath43 with @xmath44 stands for the riesz fractional derivative with the fourier transform @xmath45=-|k|^{\alpha}\hat{f}(k)$ ] @xcite . occurrence of the operator @xmath40 is due to the heavy - tailed waiting times between successive jumps and presence of the riesz fractional derivative @xmath43 is a consequence of the lvy - flight character of the jumps . in this paper , instead of seeking an analytical solution to eq . ( [ eq : ffpe ] ) , we switch to a monte carlo method @xcite which allows generating trajectories of the subordinated process @xmath46 with the parent process @xmath28 . furthermore , we study the potential free case , see eq . ( [ eq : ffpe ] ) , i.e. we assume @xmath47=0 . the assumed algorithm provides means to investigate the competition between subdiffusion ( controlled by @xmath37-parameter ) and lvy flights characterized by a stability index @xmath30 . for markov processes , the sparre - andersen scaling @xcite presents a universal law which is independent of detailed properties of the jump length distribution ( if it is only continuous and symmetric ) . in particular , for continuous times , the scaling predicts the @xmath48 decay of the survival probability , independently of whether the moments of the underlying jump process exist or not . for example , for @xmath49 , the moments of the process @xmath46 ( cf . ( [ eq : def ] ) ) exist only for @xmath50 with obvious divergence of moments of order @xmath51 , i.e. @xmath52 this divergence can be easily demonstrated in the case of ( pure ) lvy flights described by eq . ( [ eq : def ] ) , for which the operational time @xmath12 and physical time @xmath9 is equivalent , i.e. @xmath53 and consequently @xmath54 , see below . in such a case , the @xmath55 is a lvy stable density ( whose width is growing with time ) and @xmath56 stays infinite for @xmath57 . clearly , for finite time @xmath9 and finite number of representative trajectories @xmath58 ( otherwise called realizations of the process @xmath46 ) , variance @xmath59 of ( symmetric ) lvy flights stays finite , see ( * ? ? ? * eq . ( 1.19 ) ) and @xcite . in fact , finite number of realizations ( to be distinguished from the number of steps @xmath1 used in simulation of a single trajectory of time duration @xmath60 ) and finite time introduce an effective cutoff to the jump length distribution . in contrast , for any fixed time the variance diverges with increasing number of simulated trajectories @xmath58 . analogously , for any fixed @xmath58 , the variance diverges with increasing time ( scaling like @xmath61 , see ( * ? ? ? * eq . ( 1.19 ) ) and @xcite ) . the problem of mathematical divergence can be resolved either by introducing spatiotemporal coupling ( typical for so called lvy walks @xcite ) or by proper truncation of the jump length distribution @xcite . several truncation methods have been proposed @xcite to retain the finite second moment . in particular , paralleling the simulation studies of mantegna and stanley @xcite , a smooth exponential cutoff has been introduced by koponen @xcite . instead of truncating tails of a distribution , this approach is based upon the exponential tempering of the lvy density and preserves the infinite - divisibility @xcite of the distribution . the classical tempered stable distribution has been further generalized by rosiski ( for more detailed discussion see ( * ? ? ? * chapter 5.7 ) and @xcite ) . for systems driven by symmetric process the generalized sparre - andersen scaling @xcite can be used to discriminate between markovian and non - markovian situations . more precisely , according to the sparre - andersen theorem for a stochastic processes driven by any symmetric white noises , the first passage time densities , @xmath62 , from the real half line asymptotically behave like @xmath63 consequently the survival probability @xmath64 , i.e. the probability of finding a particle starting its motion at @xmath65 in the real ( positive ) half line , scales like @xmath66 therefore , any deviation of the survival probability from @xmath48 dependence indicates violation of assumptions assuring the proof of the theorem . it can mean either that a system is driven by non - symmetric or not `` memory - less '' driving . in consequence , for symmetric drivings , analysis of data based on ( assumed a priori ) sparre - andersen scaling may reveal deviations from markovianity . we study statistical properties of a symmetric free lvy motion eq . ( [ eq : def ] ) constrained to the initial position @xmath67 . to achieve the goals , we use the scheme of stochastic subordination @xcite , i.e. we obtain the process of primary interest @xmath46 as a function @xmath27 by randomizing the time clock of the process @xmath68 using a different clock @xmath34 . the parent process @xmath28 is composed of increments of symmetric @xmath30-stable motion described in an operational time @xmath12 and in every jump moment the relation @xmath69 is fulfilled . the ( inverse - time ) subordinator @xmath34 is ( in general ) non - markovian hence , as it will be shown , the diffusion process @xmath33 possesses also some degree of memory . the survival probability , see eq . ( [ eq : sparre ] ) , was estimated from ensemble of trajectories of the process @xmath46 starting at @xmath5 ( @xmath65 ) . for @xmath49 , in order to correctly account for non - local boundary conditions @xcite we have excluded multiple recrossing events , i.e. every time the particle reached any point @xmath6 beyond the boundary it was removed from the system . ) for @xmath70 ( top panel ) and @xmath71 ( bottom panel ) with various @xmath30 . the process was numerically approximated by subordination techniques with @xmath72 and averaged over @xmath73 realizations , @xmath74 . ] in figs . [ fig : sa_fixed_sub][fig : sa_fixed_alpha ] the survival probability @xmath75 is depicted for various stability indices @xmath30 and various subdiffusion parameters @xmath37 . it is clearly visible that the survival probability @xmath64 behaves like a power - law for all considered values of the subdiffusion parameter @xmath37 and stability index @xmath30 . however , the exponent characterizing the power - law dependence is equal to @xmath76 , as predicted by the ( standard ) sparre - andersen theorem , only for the markovian case ( @xmath70 ) . in more general case the power - law is characterized by the exponent @xmath77 @xmath78 which differs from @xmath79 for @xmath80 with any @xmath37 ( @xmath42 ) , the first passage time distribution is one sided lvy distribution characterized by the stability index @xmath81 @xcite , i.e. @xmath82 furthermore , in the general case , the value of the exponent @xmath77 does not depend on the stability index @xmath30 of the jump length distribution @xcite . figs . [ fig : sa_fixed_sub][fig : sa_fixed_alpha ] confirm that the value of the exponent @xmath77 depends on the subdiffusion parameter , @xmath37 , only . [ fig : sa_fixed_alpha ] shows results for @xmath83 . results for others values of stability index @xmath30 are the same as those one for @xmath83 . finally , fig . [ fig : exponents ] presents value of the exponent @xmath77 , see eq . ( [ eq : exponent ] ) , as the function of the subdiffusion parameter @xmath37 and stability index @xmath30 . fig . [ fig : exponents ] confirms that exponent @xmath77 depends on the subdiffusion parameter @xmath37 and the influence of the stability index @xmath30 is negligible . furthermore , @xmath77 depends linearly on @xmath37 : @xmath84 , what agrees with earlier findings @xcite , see fig . [ fig : exponents ] . value of the exponent @xmath77 is the decreasing function of the subdiffusion parameter @xmath37 leading to the slowest decay of the survival probability for small values of the @xmath37 parameter , i.e. when the exponent @xmath37 deviates the most from its markovian `` memory - less '' value 1 . the deviation of the exponent @xmath77 from @xmath76 clearly indicates a typical slowing down of the subdiffusive process in comparison to its ( markov ) regular diffusion analogue . the @xmath30-independence of the survival probability @xmath64 in this case shows that the properties of the decay kinetics are determined by the subdiffusive part of the process only . this observation is different from the results obtained by sokolov and metzler for a class of lvy random processes subordinated ( via the relation connecting distribution of number of jumps @xmath1 in physical time @xmath9 ) to lvy flights or to brownian random walks . in particular , in their derivation of subordination , the authors are using the markovian lvy flight process @xmath46 transformed to the process @xmath85 by use of the operational time @xmath86 which , by itself , is called the directing process @xmath87 . the density for the process @xmath85 assumes the form @xmath88 with @xmath55 , @xmath89 representing densities of a lvy flight process and the density of the directing process , respectively . if @xmath46 is a stable process with a stability parameter @xmath30 and @xmath87 is a one - sided stable process with exponent @xmath37 , the subordinated process @xmath85 becomes a stable process with the stability index @xmath90 . in contrast , in more general terms of the ctrw scenario , after waiving the assumption about independent increments of the @xmath87 process , the asymptotic form of the distribution @xmath91 can be derived by use of tauberian theorems @xcite and is known to be @xmath92 self - similar , i.e. @xmath93 @xcite . ) for @xmath83 with various @xmath37 . simulation parameters as in fig . [ fig : sa_fixed_sub ] . in figs . [ fig : sa_fixed_sub ] [ fig : sa_fixed_alpha ] initial position was set to @xmath67 . however , due to the sparre - andersen theorem , results with other values of @xmath94 are perfectly coherent with results for @xmath67 ( not shown ) and lead to the same values of exponent @xmath77 , see eq . ( [ eq : exponent ] ) . ] , see eq . ( [ eq : exponent ] ) , characterizing power - law behavior of the survival probability @xmath64 as a function of the stability index @xmath30 ( top panel ) and subdiffusion parameter @xmath37 ( bottom panel ) . simulation parameters as in fig . [ fig : sa_fixed_sub ] . ] we have discussed effects of the subordination scheme leading to the fractional diffusion equation eq . ( [ eq : ffpe ] ) . by use of the monte carlo method we have created trajectories of the process @xmath27 with @xmath34 being the inverse time @xmath30-stable subordinator . since the @xmath34 process appears as an asymptotic one in the ctrw scheme with heavy - tailed waiting time distribution between successive jumps and the parental process @xmath68 is assumed symmetric @xmath30-stable , the proposed subordination @xcite leads to @xmath95 self - similar process whose survival probabilities are governed by the stability exponent @xmath37 . information gained from the analysis of generated trajectories brings around further confirmation of non - markov property of the motion @xcite . moreover , due to the interplay between the subdiffusion in time and superdiffusion in step lengths , the resulting process violates the ergodicity ( in the weak sense ) so that the long time average is different from the average taken over the ensemble of trajectories @xcite . this issue is of special interest in the context of single - particle measurements @xcite which require analysis of time series representative for the motion . in this work we demonstrate that subdiffusive and non - markovian character of the motion can be grasped by analyzing survival probabilities which deviate from the ( standard ) sparre - andersen scaling also in those cases when the ensemble averages suggest a brownian diffusion with @xmath96 @xcite . the research has been supported by the marie curie tok cocos grant ( 6th eu framework program under contract no . mtkd - ct-2004 - 517186 ) . additionally , bd acknowledges the support from the foundation for polish science .

we are discussing long - time , scaling limit for the anomalous diffusion composed of the subordinated lvy - wiener process . the limiting anomalous diffusion is in general non - markov , even in the regime , where ensemble averages of a mean - square displacement or quantiles representing the group spread of the distribution follow the scaling characteristic for an ordinary stochastic diffusion . to discriminate between truly memory - less process and the non - markov one , we are analyzing deviation of the survival probability from the ( standard ) sparre - andersen scaling .

the manipulating bose - einstein condensates ( becs ) in double wells provides us a versatile tool to explore the underlying physics in various nonlinear phenomena since almost each parameter can be tuned experimentally @xcite . there have been many studies on the fascinating features of nonlinear effect , such as rabi oscillation @xcite , josephson oscillation @xcite , self trapping @xcite , and measure synchronization @xcite , in terms of becs in double wells . because the becs in double wells can be regarded as a two - level system , it is also expected either to be employed as a possible qubit or to simulate certain issues @xcite in quantum computation and information . recently , the effect of decoherence of becs in double wells was investigated experimentally by means of interference between becs @xcite and studied theoretically in terms of single - particle density matrix @xcite . in order to exhibit the phenomena of decoherence , they @xcite need to introduce the condensate - environment coupling because one - species bec in double wells were merely considered there . in comparison to one - species bec system , the two - species system can exhibit distinct decoherence phenomena due to the existence of the interspecies interaction . for example , the degree of coherence in a two - species system can evolve with time without the application of condensate - environment coupling , which is useful for one to get a system with desired degree of coherence . it is therefore worthwhile to study the coherence dynamics of two - species becs in double wells . in this paper , we study the effects of the inter - well tunneling strength on the coherence dynamics for a system of two - species becs in double wells . with the help of the reduced single - particle density matrix , we show that such a system can exhibit decoherence phenomena without condensate - environment coupling . we also propose an experimental strategy to prepare a bec system with any desired degree of coherence through a time - dependent tunneling strength . in the next section , we model the two - species bec system and introduce the reduced single - particle density matrix . in sec . [ sec : decoherence ] , we study the time evolution of the degree of coherence for a time - independent inter - well tunneling strength . in sec . [ sec : contro ] , we investigate the time evolution of the degree of coherence for a rosen - zener form of tunneling strength and discuss the possibility of preparing a bec system with any degree of coherence . then we briefly give our conclusion in sec . [ sec : sum ] . we consider a two - species bose - einstein condensate system confined in double wells . the hamiltonian is given by @xmath0 where @xmath1 ( @xmath2 ) and @xmath3 ( @xmath4 ) creates and annihilates a bosonic atom of species @xmath5 ( @xmath6 ) in the @xmath7th well , respectively ; @xmath8 ( @xmath9 ) denotes the particle number operator of species @xmath5 ( @xmath6 ) . here the parameters @xmath10 and @xmath11 denote the tunneling strengths of species @xmath5 and @xmath6 between the two wells , @xmath12 and @xmath13 are the intraspecies interaction strengths , and @xmath14 is the interspecies interaction strength . the hamiltonian ( [ eq : hamil ] ) can describe a bec mixture confined in a double well potential consisting of different atoms , or different isotopes , or different hyperfine states of the same kind of atom . the coherence dynamics of the above model has not been investigated although its dynamical properties , like josephson oscillation , stability and measure synchronization etc . , have been studied in earlier works @xcite . whereas , we know that the role of coherence of the system is very important since the first obstacle attempted to be avoided is the decoherence when a condensate in double wells is expected to be employed as a qubit . so the coherence dynamics of the model ( [ eq : hamil ] ) is worthy of study . as we know , under the semiclassical limit @xcite , the dynamics of this system is conventionally studied in mean - field approach by replacing the expectation values of annihilators with complex numbers , _ i.e. _ , @xmath15 and @xmath16 . with the help of heisenberg equation of motion for operators , one can easily get the dynamical equations for @xmath17 and @xmath18 . these equations guarantee the conservation law @xmath19 and @xmath20 with @xmath21 being the total particle number of species @xmath5 and @xmath6 . to simplify the calculation , one usually assumes @xmath22 and @xmath23 . then the aforementioned dynamical equations for @xmath17 and @xmath18 can be rewritten as , @xmath24 with @xmath25 where @xmath26 , @xmath27 , and @xmath28 . here the wave function @xmath29 refers to @xmath30 where the state @xmath31 or @xmath32 specifies the two different species while @xmath33 or @xmath34 specifies the two wells . note that the dynamical properties of the system can be determined by eq . ( [ eq : dynapure ] ) if the system is in a completely coherent state ( _ i.e. _ , a pure state ) . whereas , if the system is in a mixed state , the equation ( [ eq : dynapure ] ) becomes insufficient . in order to study the coherence dynamics of the system , we introduce the single - particle density matrix @xmath35 whose elements are @xmath36 . from this definition , we can see that , as a @xmath37 matrix , @xmath38 describes a pure state . their diagonal elements @xmath39 ( @xmath40 ) and @xmath41 ( @xmath42 ) represent the population of species @xmath5 ( @xmath6 ) in the first and second well , respectively . in this paper , we only focus on the distribution of the total particle numbers in the two wells but not distinguish the particle species , so the system can be described by the reduced density matrix @xmath43 whose elements are @xmath44 , @xmath45 , @xmath46 , and @xmath47 . clearly , the matrix @xmath48 can describe a mixed state . its diagonal elements @xmath49 and @xmath50 represent the total population probability in the first and second well , respectively . to investigate the coherence dynamics of the system , we can introduce the definition of degree of coherence according to ref . @xcite , @xmath51 from eq . ( [ eq : reducematrix ] ) and eq . ( [ eq : coherence ] ) , we can see that the time evolution of the degree of coherence depends on that of the elements @xmath52 of the single - particle density matrix @xmath38 . since @xmath38 describes a pure state , the time evolution of @xmath52 can be determined by eq . ( [ eq : dynapure ] ) . thus one can solve eq . ( [ eq : dynapure ] ) to get the evolution of @xmath53 and @xmath54 firstly , and then gives the time evolution of @xmath52 according to the definition of @xmath35 . then in the following sections , we will study the coherence dynamics of the system with the help of eq . ( [ eq : dynapure ] ) . note that in the following calculations , we take @xmath55 for simplicity without losing the generality . we know that the degree of coherence does not change for an isolated bec system @xcite . to study the effect of decoherence of bec system , several authors considered the condensate - environment coupling @xcite . whereas , we will show that the degree of coherence of the two - species bec system in double wells can still change with time even without the condensate - environment coupling . here we consider the case of time - independent inter - well tunneling strength , _ i.e. _ , @xmath56 is a constant in the calculation . due to the fact that eq . ( [ eq : dynapure ] ) can not be analytically solved , we solve eq . ( [ eq : dynapure ] ) numerically to get the time evolution of @xmath53 and @xmath54 . then we give the time evolution of the degree of coherence with the help of the definition of @xmath38 and @xmath57 and eq . ( [ eq : coherence ] ) . the corresponding results are summarized in fig . [ fig : coherence ] . ( color online ) time evolution of the degree of coherence for different initial states . the parameters are @xmath58 ( a ) , @xmath59 and @xmath60 ( b ) , @xmath61 and @xmath62 ( c ) , and @xmath61 and @xmath60 ( d ) . , width=340 ] in fig . [ fig : coherence ] , we plot the time evolution of the degree of coherence for different initial states and parameters . the initial states are @xmath63 and @xmath64 for the blue ( top ) line , @xmath65 and @xmath64 for the red ( middle ) line , and @xmath66 and @xmath67 for the black ( bottom ) line . from the fig . [ fig : coherence ] ( a ) , we can find that the degree of coherence of the two - species bec system in double wells does not change for any initial states when @xmath68 . that can be easily understood because this two - species system is equivalent to the one - species system once @xmath69 . and according to ref . ( @xcite ) , the degree of coherence of the one - species bec system in double wells does not change without the condensate - environment coupling . so the result shown in fig . [ fig : coherence ] ( a ) is reasonable . the figure [ fig : coherence ] ( b ) shows that when @xmath70 , the degree of coherence does not change for the case of @xmath71 at the initial time , _ i.e. _ , the initial population distributions of species @xmath5 and @xmath6 are the same , but that changes for the other initial states . note that for the case of @xmath70 , the time evolutions of species @xmath5 and @xmath6 are the same if @xmath71 at the initial time . so the degree of coherence does not change due to the two species having the same symmetry . additionally , from the fig . [ fig : coherence ] ( c ) and ( d ) , we can find that the degree of coherence changes for any initial states for the case of @xmath72 . comparing fig . [ fig : coherence ] ( c ) and ( d ) , we can see that for the same initial state , the variation tendency of the degree of coherence depends on the parameters of the system . in summary , whether the degree of coherence can change with time depends on both the parameters and the initial states . so does the variation tendency of the degree of coherence . this fact indicates that one can control the degree of coherence without introducing the effect of environment , which will be discussed in the next section . in previous section , we show that the time evolution of the degree of coherence depends on the initial states and the parameters of the system , so one can change it by varying either the initial states or the parameters of the system . in the following , we will show how to control the degree of coherence through a rosen - zener form of @xmath56 , @xmath73 _ i.e. _ , @xmath56 increases from zero to its maximum value @xmath74 and then decreases to zero again in the end of the calculation . in the numerical calculation in this section , we take the initial state @xmath75 , and @xmath64 . ( color online ) time evolution of the degree of coherence for the rosen - zener form of inter - well tunneling strength . the parameters are @xmath76 . , width=188 ] the dependence of the maximum value of the degree of coherence ( left panel ) and the corresponding time ( right panel ) on the period of the inter - well tunneling strength . the parameters are @xmath77 . , width=340 ] the dependence of the maximum value of the degree of coherence ( left panel ) and the corresponding time ( right panel ) on the maximum value of the inter - well tunneling strength . the parameters are @xmath77 . , width=340 ] the time evolution of the degree of coherence for the rosen - zener form of inter - well tunneling strength is plotted in fig . [ fig : evorosen ] . we can see that the degree of coherence changes with time and can reach the maximum value @xmath78 at time @xmath79 . note that the values of @xmath78 and @xmath79 depend on both the period @xmath80 and the maximum value @xmath74 of the inter - well tunneling strength . in fig . [ fig : omega ] , we plot the dependence of @xmath78 and @xmath81 on @xmath80 for a fixed @xmath74 in the left and right panel , respectively . meanwhile , we plot the dependence of @xmath78 and @xmath81 on @xmath74 for a fixed @xmath80 in fig . [ fig : tuj ] . from fig . [ fig : omega ] and fig . [ fig : tuj ] , we can see that the values of @xmath80 and @xmath74 affect the maximum value of the degree of coherence sensitively . in order to get a system with large value of degree of coherence , one must choose the form of @xmath56 with suitable period @xmath80 and maximum value @xmath74 . note that although one can get a system with large value of degree of coherence through tuning the value of @xmath80 and @xmath74 , it is difficult to control the time evolution of the degree of coherence due to the fact that the degree of coherence does not evolve periodically , which can be confirmed by fig . [ fig : evorosen ] . in order to overcome the aforementioned problem , we consider the following form of @xmath56 @xmath82 where @xmath81 can be obtained from fig . [ fig : omega ] and fig . [ fig : tuj ] . ( color online ) time evolution of the degree of coherence for the inter - well tunneling strength given in eq . ( [ eq : jform ] ) . the parameters are @xmath83 . , width=188 ] for the form of @xmath56 given in eq . ( [ eq : jform ] ) , the time evolution of the degree of coherence is plotted in fig . [ fig : evo ] . from this figure , we can find that the degree of coherence reaches its maximum value which is determined by the values of @xmath80 and @xmath74 , and then oscillates periodically at the following time . since eq . ( [ eq : dynapure ] ) can be analytically solved for the case of @xmath84 , we can give the oscillation period in analytical method . solving eq . ( [ eq : dynapure ] ) , we obtain @xmath85 , and @xmath86 where @xmath87 , and @xmath88 . substituting the expressions of @xmath53 and @xmath54 into the eq . ( [ eq : coherence ] ) , we can get the oscillation period of the degree of coherence @xmath89 . for the parameters taken in fig . [ fig : evo ] , we find @xmath90 which is consistent with the numerical result . from the above discussion , we know that the degree of coherence oscillates periodically after time @xmath81 , and both the maximum value of the degree of coherence and the oscillation period can be changed by tuning the values of @xmath80 and @xmath74 . once the values of @xmath80 and @xmath74 are fixed , the time evolution of the degree of coherence is well defined , so that we can know the degree of coherence of the system at any time . then one can easily get a system with any desired degree of coherence through controlling the evolution time . in the above , we investigated the coherence dynamics for a system of two - species bose - einstein condensate in double wells . in mean field approximation , we studied the influence of the inter - well tunneling strength on the coherence features with the help of the reduced single - particle density matrix . since we need not distinguish particle species in the system , we only focused on the distribution of the total particle numbers in the two wells , which can be described by a 2@xmath912 reduced density matrix . after studying the time evolution of the degree of coherence for a time - independent inter - well tunneling strength , we found that the degree of coherence of the two - species bec system changes for some parameters and initial states even without the condensate - environment coupling , which differs from the case of refs . motivated by the fact that the variation tendency of the degree of coherence depends on both the parameters and the initial states of the system , we considered a system with a rosen - zener form of inter - well tunneling strength to control the degree of coherence . although its tendency is not periodical , the degree of coherence can reach a maximum value @xmath92 at time @xmath93 which are dependent of the period @xmath80 and the maximum value @xmath74 of the inter - well tunneling strength . the dependence of @xmath78 and @xmath81 on @xmath80 and @xmath74 we obtained is helpful for one to get a system with large value of degree of coherence utilizing a rosen - zener form of inter - well tunneling strength . we also gave a useful form of the inter - well tunneling strength for one to easily get a system with any degree of coherence by controlling the evolution time . 99 y. shin , g. b. jo , m. saba , t. a. pasquini , w. ketterle , and d. e. pritchard , phys 95 , 170402 ( 2005 ) ; g. b. jo , y. shin , s. will , t. a. pasquini , m. saba , w. ketterle , d. e. pritchard , m. vengalattore , and m. prentiss , ibid . 98 , 030407 ( 2007 ) ; r. gati , b. hemmerling , j. flling , m. albiez , and m. k. oberthaler , ibid . 96 , 130404 ( 2006 ) . g. j. milburn , j. corney , e. m. wright , and d. f. walls , phys . rev . a * 55 * , 4318 ( 1997 ) . j. m. choi , g. n. kim , and d. cho , phys . rev . a * 77 * , 010501 ( 2008 ) . l. h. lu and y. q. li , phys . a * 80 * , 033619 ( 2009 ) . b. sun and m. s. pindzola , phys . a * 80 * , 033616 ( 2009 ) . s. raghavan , a. smerzi , s. fantoni , and s. r. shenoy , phys . a * 59 * , 620 ( 1999 ) . m. albiez , r. gati , j. flling , s. hunsmann , m. cristiani , and m. k. oberthaler , phys . 95 , 010402 ( 2005 ) . d. sokolovski and s. a. gurvitz , phys . a * 79 * , 032106 ( 2009 ) . t. schumm , s. hofferberth , l. m. andersson , s. wildermuth , s. groth , i , bar - joseph , j. schmiedmayer , and p. krger , nat . phys . * 1*,57 ( 2005 ) . s. hofferberth , i. lesanovsky , b. fischer , t. schumm , and j. schmiedmayer , nature ( london ) * 449 * , 324 ( 2007 ) .

coherence dynamics of two - species bose - einstein condensates in double wells is investigated in mean field approximation . we show that the system can exhibit decoherence phenomena even without the condensate - environment coupling and the variation tendency of the degree of coherence depends on not only the parameters of the system but also the initial states . we also investigate the time evolution of the degree of coherence for a rosen - zener form of tunneling strength , and propose a method to get a condensate system with certain degree of coherence through a time - dependent tunneling strength .

the _ metric boundary _ of a metric space was defined by m. rieffel in @xcite as the boundary of a _ metric compactification_. the metric compactification of a metric space @xmath0 with the base point @xmath1 is the compactification given via gelfand s theorem as the maximal ideal space of the @xmath2-algebra generated by constant functions , continuous functions vanishing at infinity , and continuous functions which form @xmath3 for all @xmath4 . he observed that the metric compactification is naturally identified with the compactification given by m. gromov in @xcite , which recently called the _ horofunction compactification _ 4 in @xcite . see also 8.12 of chapter ii in @xcite ) . in @xcite , he also defined geodesic - like sequences in a metric space with the base point , which called _ almost geodesics _ ( cf . [ subsec : almost_geodesics ] ) . he observed that any almost geodesic admits the limit in the metric boundary . he defined _ busemann points _ in the metric boundary as the limits of almost geodesics , and posed a question which asks to determine whether every point in the metric boundary of a given metric space is a busemann point ( see the paragraph after definition 4.8 in @xcite ) . for this problem , c. webster and a. winchester @xcite gave geometric conditions which determine whether or not every point on the metric boundary of a graph with the standard path metric is a busemann point , and an example of a graph which admits non - busemann points in its metric boundary . let @xmath5 be a riemann surface of type @xmath6 with @xmath7 . the _ teichmller space _ @xmath8 of @xmath5 is a quasiconformal deformation space of marked riemann surfaces with same type as @xmath5 . teichmller space @xmath8 admits a canonical distance , called the _ teichmller distance _ @xmath9 ( cf . [ subsec : teichmuller - space ] ) . the aim of this paper is to show the following . [ thm : main ] when @xmath10 , the metric boundary of the teichmller space with respect to the teichmller distance contains non - busemann points . when @xmath11 , the teichmller space equipped with the teichmller distance is isometric to the poincar hyperbolic disk . hence , every point in the metric boundary is a busemann point . furthermore , in this case , the metric boundary of the teichmller space equipped with the teichmller space coincides with the thurston boundary ( cf . e.g. @xcite ) . recently , in @xcite , c. walsh defined the horofunction boundaries for asymmetric metric spaces , and observed that the horofunction boundary of the teichmller space with respect to the thurston s ( non - symmetrized ) lipschitz metric is canonically identified with the thurston boundary . he also showed that every point in the thurston boundary is a busemann point with respect to the thurston s lipschitz metric ( cf . theorem 4.1 of @xcite ) . the thurston s lipscthiz metric is the length spectrum asymmetric metric with respect to the hyperbolic lengths of simple closed curves , meanwhile the teichmller distance is recognized as the length spectrum metric with respect to the extremal lengths of simple closed curves via kerckhoff s formula ( cf . . see also @xcite ) . since hyperbolic lengths and extremal lengths are fundamental geometric quantities in the teichmller theory , it is natural to compare properties of these two distances . theorem [ thm : main ] and walsh s results above imply that the asymptotic geometry with respect to the teichmller distance is more complicated than that with respect to the thurston s lipschitz metric . it is known that the metric boundary of a complete @xmath12-space consists of busemann points ( cf . corollary ii.8.20 of @xcite ) . therefore , we conclude the following which is already well - known ( cf . @xcite ) . [ coro : not_cat0 ] when @xmath10 , the teichmller space equipped with the teichmller distance is not a @xmath12-space . let @xmath13 be the set of homotopy classes of non - trivial and non - peripheral simple closed curves on @xmath5 . we denote by @xmath14 the _ extremal length _ of @xmath15 for @xmath16 ( cf . [ subsubsec : teichmuller - distance ] ) . in a beautiful paper @xcite , f. gardiner and h. masur proved that the mapping @xmath17 \in { \rm p}\mathbb{r}_+^\mathcal{s}\ ] ] is an embedding and the image is relatively compact , where @xmath18 and @xmath19 . the closure of the image is called the _ gardiner - masur compactification _ and the _ gardiner - masur boundary _ @xmath20 is the complement of the image from the gardiner - masur compactification . they showed that the gardiner - masur boundary contains the space @xmath21 of projective measured foliations ( cf . theorem 7.1 in @xcite ) . in @xcite , l. liu and w. su have shown that the horofunction boundary with respect to the teichmller distance is canonically identified with the gardiner - masur boundary of teichmller space . hence , to conclude theorem [ thm : main ] , we will show the following . [ thm : main2 ] when @xmath10 , the projective class of a maximal rational measured foliation can not be the limit of any almost geodesic in the gardiner - masur compactification . in contrast , from theorem 7.1 in @xcite and theorem 3 in @xcite , when a measured foliation @xmath22 is either a weighted simple closed curve or a uniquely ergodic measured foliation , the projective class @xmath23 $ ] is the limit of the teichmller ray associated to @xmath23 $ ] , and hence it is a busemann point with respect to the teichmller distance . in @xcite , the author have already observed that any teichmller geodesic ray does not converge to the projective class @xmath23 $ ] when @xmath22 is a rational foliation whose support consists of at least two curves . however , the author does not know whether this induces theorem [ thm : main2 ] . this paper is organized as follows . in 2 , we recall the definitions and properties of ingredients in the teichmller theory , including the extremal length and the teichmller distance . in 3 , we discuss the metric boundaries of metric spaces , and check that any almost geodesic converges in the gardiner - masur compactification . though this convergence follows from properties of the metric boundary and liu and su s work in @xcite , we shall give a simple proof of the convergence from the teichmller theory for the completeness of readers . we treat measured foliations whose projective classes are the limits of almost geodesics in 4 and 5 . indeed , in 5 , we will observe that when a measured foliation whose projective class is the limit of an almost geodesic has a foliated annulus as its component , any simple closed curve is not so _ twisted _ in the characteristic annulus corresponding to the foliated annulus through the almost geodesic ( cf . lemma [ lem : twisting_number ] ) . this is a key for getting our result . in 6 , we give the proof of theorem [ thm : main2 ] by contradiction . indeed , under the assumption that the projective class of maximal measured foliation @xmath22 is the limit of an almost geodesic , we calculate the limit of a given almost geodesic , but we can check that the limit can not be equal to the boundary point induced from the intersection number function with respect to @xmath22 . for getting the limit , we will apply the kerckhoff s calculation in @xcite of the extremal length along the teichmller ray . one of the reason why the kerckhoff s calculation works is such _ non - twisted property _ of simple closed curves along the core curve of the characteristic annuli discussed in 5 ( see [ subsec : idea ] ) . let @xmath24 be a family of rectifiable curves on a riemann surface @xmath25 . the _ extremal length _ of @xmath24 ( on @xmath25 ) is defined by @xmath26 where supremum runs over all measurable conformal metric @xmath27 and @xmath28 the extremal length is a _ conformal invariant _ in the sense that @xmath29 for a @xmath30-quasiconformal mapping @xmath31 , a riemann surface @xmath25 , and a family @xmath24 of rectifiable curves on @xmath25 . [ prop : extremal_length ] let @xmath32 and @xmath33 be two families of rectifiable curves on a riemann surface @xmath25 . * if any curve in @xmath32 is contained in a subdomain @xmath34 of @xmath25 , the extremal length of @xmath32 on @xmath25 is equal to the extremal length of @xmath32 on @xmath34 . * if any curve in @xmath33 contains a curve in @xmath32 , @xmath35 . * if curves of @xmath32 and @xmath33 are mutually disjoint , @xmath36 . where @xmath37 . for an annulus @xmath38 , we denote by @xmath39 the extremal length of the family of simple closed curves which homotopic to the core curve of @xmath38 . the _ modulus _ of @xmath38 is the reciprocal of the extremal length of @xmath38 . if @xmath38 is conformally equivalent to the flat annulus @xmath40 , it holds that @xmath41 . [ prop : extremal_length_upper ] let @xmath38 be an annulus . let @xmath42 be mutually disjoint jordan arcs joining components of @xmath43 such that @xmath44 and @xmath45 divides @xmath46 from the other arcs ( set @xmath47 ) . let @xmath48 be the set of paths in @xmath49 connecting @xmath46 and @xmath45 . let @xmath50 be the extremal metric for @xmath39 on @xmath38 such that @xmath51 . suppose that the @xmath50-length of @xmath46 is bounded for all @xmath52 . then , @xmath53 where @xmath54 is the totality of @xmath50-lengths of @xmath46 s . for a riemann surface @xmath55 and a simple closed curve @xmath56 on @xmath55 , we define the _ extremal length _ @xmath57 of @xmath56 on @xmath55 is the extremal length of the family of rectifiable closed curves on @xmath55 homotopic to @xmath56 . the extremal length is characterized geometrically as @xmath58 where @xmath38 runs all annuli on @xmath55 whose core is homotopic to @xmath56 ( cf . e.g. @xcite and @xcite ) . the formal product @xmath59 is embedded into @xmath60 via the intersection number function : @xmath61\in \mathbb{r}_+^{\mathcal{s}}.\ ] ] the closure @xmath62 of the image in @xmath60 is called the _ space of measured foliations _ on @xmath5 . the _ space @xmath63 of projective measured foliations _ is the quotient space @xmath64 . it is known that @xmath65 and @xmath21 are homeomorphic to @xmath66 and @xmath67 , respectively ( cf . it is also known that when we put @xmath68 for @xmath69 , the intersection number function extends continuously on @xmath70 . to a measured foliation @xmath22 , we associate a singular foliation and a transverse measure to the underlying foliation ( cf . @xcite ) . in this paper , we denote by @xmath71 the integration of the corresponding transverse measure over a path @xmath56 . a measured foliation @xmath22 is called _ rational _ if @xmath22 satisfies @xmath72 for some @xmath73 and @xmath74 such that @xmath75 and @xmath76 for @xmath77 with @xmath78 . we write @xmath79 for such measured foliation . a rational measured foliation @xmath79 is _ maximal _ if any component of @xmath80 is a pair of pants . in this case , @xmath81 . in @xcite , s. kerckhoff showed that when we put @xmath82 for @xmath83 , the extremal length extends continuously on @xmath65 . we define @xmath84 which is homeomorphic to @xmath21 via the projection @xmath85 . in @xcite , y. minsky showed the following inequality , which recently called the _ minsky s inequality _ : @xmath86 for all @xmath87 ( cf . lemma 5.1 of @xcite ) . from theorem 5.1 in @xcite , minsky s inequality is sharp in the sense that for any @xmath88 , there is an @xmath89 which satisfies the equality in . the _ teichmller space _ @xmath8 of @xmath5 is the set of equivalence classes of marked riemann surfaces @xmath90 where @xmath55 is a riemann surface and @xmath91 a quasiconformal mapping . two marked riemann surfaces @xmath92 and @xmath93 are _ teichmller equivalent _ if there is a conformal mapping @xmath94 which homotopic to @xmath95 . throughout this paper , we consider the teichmller space as a pointed space with the base point @xmath96 . the _ teichmller distance _ between @xmath97 and @xmath98 is , by definition , the half of the logarithm of the extremal quasiconformal mapping between @xmath99 and @xmath100 preserving markings . in @xcite , s. kerckhoff gave the geometric interpretation of the teichmller distance by using the extremal lengths of measured foliations as follows . for @xmath89 and @xmath101 , we define the extremal length of @xmath102 on @xmath103 by @xmath104 then , the following equality holds : @xmath105 the teichmller space is topologized with the teichmller distance . under this topology , the extremal length of a measured foliation varies continuously on @xmath8 from the conformal invariance . for a holomorphic quadratic differential @xmath106 on a riemann surface @xmath55 , we define a singular flat metric @xmath107 . we call here this metric the _ @xmath108-metric_. in @xcite , hubbard and masur observed that for @xmath101 and @xmath88 , there is a unique holomorphic quadratic differential @xmath109 on @xmath55 whose vertical foliation is equal to @xmath110 . namely , @xmath111 holds for all @xmath112 . in this case , we can see that @xmath113 namely , the extremal length is the area of the @xmath109-metric . when @xmath114 , we call the differential @xmath115 the _ jenkins - strebel differential for @xmath56_. let @xmath116 and @xmath23\in \mathcal{pmf}$ ] . by ahlfors - bers theorem , we can define an isometric embedding @xmath117 with respect to the teichmller distance by assigning the solution of the beltrami equation defined by the teichmller beltrami differential @xmath118 for @xmath119 . we call @xmath120 the _ teichmller ( geodesic ) ray associated to @xmath23\in \mathcal{pmf}$]_. notice that the differential depends only on the projective class of @xmath22 . it is known that @xmath121,t ) \mapsto r_{g , x_0}(t)\in t(x)\ ] ] is a homeomorphism ( cf . one can see that @xmath122 for @xmath123 . for @xmath16 , we let @xmath124 . consider a continuous function on @xmath65 @xmath125 for @xmath16 . then , in @xcite , the author observed that for any @xmath126 , there is a function @xmath127 on @xmath65 such that the function @xmath128 represents @xmath129 and when a sequence @xmath130 converges to @xmath129 in the gardiner - masur compactification , there are @xmath131 and a subsequence @xmath132 such that @xmath133 converges to @xmath127 uniformly on any compact set of @xmath65 . let @xmath0 be a locally compact metric space . let @xmath134 be the space of continuous functions on @xmath135 , equipped with the topology of uniform convergence on compact subsets of @xmath135 . let @xmath136 be the quotient space of @xmath134 via constant functions . for @xmath4 we set @xmath137 . then , @xmath138 is a continuous embedding into @xmath134 . this embedding descends a continuous embedding into @xmath136 . the closure @xmath139 of the image of this embedding is called the _ horofunction compactification _ and the complement @xmath140 is said to be the _ horofunction boundary _ of @xmath135 ( cf . @xcite , @xcite , and @xcite ) . m. rieffel pointed out that the metric boundary of @xmath135 is canonically identified with the horofunction boundary of @xmath135 as discussed in the introduction ( cf . 4 in @xcite ) . in @xcite , l. liu and w. su showed that the horofunction compactification of the teichmller space with the teichmller distance is identified with the gardiner - masur compactification . let @xmath0 be a metric space . let @xmath141 be an unbounded set with @xmath142 . a mapping @xmath143 is said to be an _ almost geodesic _ if for any @xmath144 there is an @xmath145 such that for all @xmath146 with @xmath147 , @xmath148 ( cf . definition 4.3 of @xcite ) . by definition , any geodesic ray is an almost geodesic . when @xmath0 is a pointed metric space , we assume in addition that @xmath149 is the base point ( cf . the assumption of lemma 4.5 in @xcite ) . by definition , for an unbounded subset @xmath150 with @xmath151 , the restriction @xmath152 is also an almost geodesic . we call the restriction a _ subsequence _ of an almost geodesic @xmath143 . a point of the metric boundary or the horofunction boundary of @xmath135 is said to be a _ busemann point _ if it is the limit of an almost geodesic ( cf . definition 4.8 of @xcite ) . in this section , we shall check that any almost geodesic in @xmath8 converges in the gardiner - masur compactification . though this follows from a fundamental property of the metric boundary ( cf . @xcite ) and liu and su s work @xcite , we now try to give a simple proof from the teichmller theory and it seems to be intriguing in itself . notice that the author observed in @xcite that any teichmller ray @xmath153 admits the limit for all @xmath23\in \mathcal{pmf}$ ] by the different idea . let @xmath154 be an almost geodesic with the base point @xmath155 . by definition , @xmath156 satisfies that @xmath157 and for any @xmath144 , there is an @xmath158 such that latexmath:[\[\label{eq : almost_geodesic } all @xmath147 . from kerckhoff s formula , is equivalent to @xmath160 in particular , we have @xmath161 when we set @xmath162 in . therefore , we deduce @xmath163 and hence @xmath164 for all @xmath165 and @xmath147 . we set @xmath166 for @xmath89 . from , for all @xmath112 , the limit of any converging subsequence in @xmath167 coincides with @xmath168 , which implies that @xmath154 converges in the gardiner - masur compactification as @xmath169 . let @xmath170 . suppose that the projective class @xmath23 $ ] is a busemann point in the horofunction compactification of teichmller space with respect to the teichmller metric . by definition and liu and su s work @xcite , there is an almost - geodesic @xmath154 such that @xmath171 $ ] in the gardiner - masur closure . this means that there is a @xmath131 such that @xmath172 converges to @xmath173 uniformly on any compact sets of @xmath65 . we take @xmath174 with @xmath175 . [ lem : t_0_is_one ] under the notation above , it holds @xmath176 . let @xmath177 be an accumulation point of @xmath178 . by taking a subsequence if necessary , we may assume that @xmath179 converges to @xmath177 . let @xmath180 . from and , we have @xmath181 since @xmath144 is taken arbitrary , we get @xmath182 for all @xmath180 . thus , it follows from the marden - strebel s minimal norm property that @xmath183 and hence @xmath184 ( see theorem 3.2 of @xcite . see also @xcite ) . from , by dividing every term in by @xmath185 and letting @xmath169 , we get @xmath186 for @xmath187 . from minsky s inequality and kerckhoff s formula , we have @xmath188 and @xmath189 hence , we get @xmath190 on the other hand , from the distortion property , @xmath191 holds in general . therefore , we have @xmath192 and @xmath193 , which is what we wanted . from the proof of the lemma above , we also observe the following . [ lcoro : limit_is_g ] @xmath179 converges to @xmath22 as @xmath169 . let @xmath177 be an accumulation point of @xmath178 as above . recall from and lemma [ lem : t_0_is_one ] above that @xmath194 for all @xmath180 . since @xmath195 , by the calculation in and the conclusion from the equality of the minimal norm property , we get @xmath196 and @xmath197 . notice from and lemma [ lem : t_0_is_one ] that @xmath198 in this section , we devote to give asymptotic behaviors of moduli of characteristic annuli corresponding to foliated annuli and the twisting number of closed geodesics on the characteristic annuli . these observations will be used for proving theorem [ thm : main2 ] in the next section . as the previous section , we continue to suppose that the projective class @xmath23 $ ] of @xmath170 is the limit of an almost geodesic @xmath199 . throughout this section , we suppose in addition that @xmath22 has a component of a foliated annulus with core @xmath180 . namely , @xmath200 for some @xmath201 and @xmath89 . for the simplicity , we set @xmath202 . let @xmath203 for @xmath204 and @xmath205 be the characteristic annulus of @xmath206 for @xmath15 . we now fix a notation . for two functions @xmath207 and @xmath208 with variable @xmath209 , @xmath210 means that @xmath207 and @xmath208 are comparable in the sense that there are positive numbers @xmath211 and @xmath212 independent of the parameter @xmath209 such that @xmath213 . the asymptotic behavior of the modulus of @xmath214 is given as follows . [ lem : char_annulus_modulus ] @xmath215 as @xmath169 . by , @xmath216 for all @xmath204 . in addition , by , @xmath217 as @xmath169 . we here define the _ twisting numbers _ of proper paths in flat annuli . let @xmath218 be the euclidean circle of length @xmath219 . let @xmath220\times \mathbb{s}^1_l$ ] be a flat annulus . let @xmath221 be an ( unoriented ) path connecting components of @xmath43 . take a universal cover @xmath222\times \mathbb{r}\to a$ ] . let @xmath223 be a lift of @xmath56 . let @xmath224\times \mathbb{r}$ ] be the endpoints of @xmath223 . then , we define a _ twisting number _ @xmath225 of @xmath56 in @xmath38 by @xmath226 one can easily check that the twisting number is defined independently of the choice of lifts . let @xmath112 with @xmath227 . for @xmath204 , we set @xmath228 be the geodesic representative of @xmath56 in @xmath229 with respect to the @xmath230-metric . if @xmath206 admits a flat annulus whose core is homotopic to @xmath56 , we choose one of closed trajectories in the flat annulus to define @xmath228 . let @xmath231 be the set of straight segment in @xmath228 in the part of @xmath214 counting multiplicity , where @xmath232 . let @xmath233 be a collection of maximal straight segments in @xmath234 , counting multiplicity . in this section , for a measured foliation @xmath102 and a path @xmath235 transverse to the underlying foliation of @xmath102 , we define @xmath236 as the infimum of the integrals of the transversal measure of @xmath102 over all paths homotopic to @xmath235 rel endpoints . [ lem : twisting_number ] for @xmath237 , the twisting number of @xmath238 in @xmath239 satisfies @xmath240 as @xmath169 . when @xmath241 , the geodesic representative @xmath228 does not intersect the interior of @xmath214 . hence , the conclusion automatically holds . therefore , we may assume that @xmath242 . let @xmath243 . then , the vertical foliation @xmath244 of @xmath245 is equal to @xmath246 for all @xmath204 . especially , the @xmath245-height @xmath247 of the characteristic annulus @xmath214 is equal to @xmath248 . let @xmath249 be the horizontal foliation of @xmath245 . since each @xmath238 is a @xmath245-straight segment , @xmath250 for @xmath251 . hence , @xmath252 since @xmath253 , @xmath254 from . therefore , @xmath255 thus , we obtain @xmath256 from the assumption , lemma [ lem : t_0_is_one ] and , @xmath257 tends to @xmath258 as @xmath169 . since @xmath259 we deduce from that the summation @xmath260 tends to zero as @xmath261 . since every term in is non - negative , we get @xmath262 for @xmath237 . we now fix @xmath237 . let @xmath263\times \mathbb{r}\to [ 0,w_t]\times \mathbb{s}^1_{\ell_t}\cong a_{t}$ ] be the universal cover , where @xmath264 is the @xmath245-circumference of @xmath239 . let @xmath265 and @xmath266 be the endpoints of a lift of @xmath238 . from the definition , @xmath267 since @xmath268 from lemma [ lem : char_annulus_modulus ] , we obtain @xmath269 for @xmath270 . thus , it follows from that @xmath271 which implies what we wanted . in this section , we shall recall a canonical quasiconformal mapping of the twisting deformations along the core curve on a flat annulus ( cf . @xcite ) . let @xmath272 be a flat annulus of modulus @xmath273 . for @xmath274 , we consider a quasiconformal self - mapping @xmath275 of @xmath38 by @xmath276 then , the beltrami differential of @xmath275 is equal to @xmath277 we can check that @xmath278 especially , when a proper path @xmath235 in @xmath38 has the twist parameter @xmath279 , @xmath280 . in this section , we shall show theorem [ thm : main2 ] . throughout this section , we assume that @xmath281 is a maximal rational foliation and @xmath282 . as before , we also assume that the projective class @xmath23 $ ] is the limit of an almost geodesic @xmath154 . we continue to use symbols given in the previous sections . let @xmath283 be the characteristic annulus of @xmath243 for @xmath284 . let @xmath285 be the critical graph of @xmath245 and consider the @xmath286-neighborhood @xmath287 of @xmath285 in @xmath229 with respect to the @xmath245-metric . let @xmath288 ( cf . figure [ fig : foliation ] ) . let @xmath296 . since @xmath297 consists of closed leaves in @xmath292 and the heights of the remaining annuli in @xmath298 are at most @xmath286 , from , the moduli of remaining annuli in @xmath298 are uniformly bounded , and hence @xmath299 as @xmath169 . let @xmath301 be the characteristic annulus of the jenkins - strebel differential @xmath302 for @xmath56 . fix @xmath303 . the intersection @xmath304 contains at least @xmath305-components @xmath306 such that @xmath307 contains a path connecting @xmath308 and @xmath309 . let @xmath310 be the family of rectifiable curves in @xmath307 connecting @xmath308 and @xmath309 . let @xmath311 be the restriction of the @xmath245-metric to @xmath312 . from , any curve in @xmath313 has @xmath311-length at most @xmath314 . since the critical graph of the jenkins - strebel differential of @xmath56 on @xmath229 has measure zero , @xmath315 by the definition of the extremal length , we have @xmath316 since any non - trivial simple closed curve in @xmath301 traverses each @xmath307 between @xmath317 and @xmath318 , such simple closed curve contains a curve in @xmath313 . therefore , from ( 3 ) of proposition [ prop : extremal_length ] , we conclude @xmath319 as @xmath169 . before discussing the upper bound , we deform @xmath229 slightly as follows . for @xmath303 , we fix a component @xmath320 of @xmath321 . we put the beltrami differential on each flat annulus @xmath292 with @xmath322 . we extend the beltrami differential to @xmath229 by putting @xmath323 on the remaining part . then , we obtain a quasiconformal deformation of @xmath229 with respect to the beltrami differential to get @xmath324 . by lemmas [ lem : char_annulus_modulus ] and [ lem : twisting_number ] , @xmath325 as @xmath261 for all @xmath326 , and hence , @xmath327 when @xmath261 . this means that @xmath328 has the same limit as that of @xmath329 in the gardiner - masur compactification . thus , for simplifying of the notation , we may suppose that @xmath330 . notice from that after this deformation , the twist parameter of each @xmath320 is zero . hence , any segment in @xmath321 has the twisting number at most one in @xmath292 for all @xmath326 , because @xmath56 is a simple closed curve and any two segments in @xmath321 do not intersect transversely in @xmath292 . by taking a subsequence , we may assume that there is a ( non - connected ) graph @xmath331 on @xmath5 such that the making @xmath332 induces an isomorphism @xmath331 and @xmath285 . to give an upper estimate , from , it suffices to construct a suitable annulus @xmath301 on @xmath229 whose core is homotopic to @xmath333 . the procedure given here is originally due to s. kerckhoff in @xcite , when a given almost geodesic @xmath156 is actually a geodesic ( see also 9 of @xcite ) . we briefly recall the case when @xmath156 is a geodesic . we first cut each characteristic annuli @xmath292 of @xmath230 into @xmath305-congruent horizontal rectangles . the annulus @xmath301 is made by composing appropriately such ( slightly modified ) @xmath334-congruent horizontal rectangles and ties ( quadrilaterals ) in @xmath287 ( cf . ) . we can take such ties with uniform extremal length ( cf . claim [ claim : b_s ] ) . then , by applying proposition [ prop : extremal_length_upper ] , we obtain an upper bound of the extremal length of @xmath301 . one of the essential reason why we can get an appropriate upper bound in the case above is that , through the teichmller ray associated to the projective class of @xmath335 , there are no " twisting deformation along @xmath284 on the characteristic annulus , because the teichmller deformation is done by stretching in the horizontal and vertical directions . indeed , the major part of the upper bound comes from the extremal length of congruent rectangles ( cf . ) . the ` no - twisting ' property implies that the totality of the extremal lengths of such rectangles is equal to the major part of the lower estimate ( cf . ) . in the case when @xmath156 is an almost geodesic , we have already observed in that @xmath56 is not so twisted on the characteristic annuli too much . hence , we can apply the similar argument for getting an appropriate upper bound of @xmath336 . since @xmath22 is maximal , any component @xmath289 ( @xmath338 ) of @xmath287 is one of the three types : a pair of pants , an annulus with one distinguished point ( a singularity of angle @xmath339 or a flat point ) , or a half - pillow with two cone singularities of angle @xmath339 ( cf . figure [ fig : pants ] ) . in the the case when @xmath289 is either an annulus or a half - pillow , we can deal with the same manner , and hence we now assume that @xmath289 is a pair of pants . notice from that the length of any component of @xmath340 is of order @xmath286 with respect to the metric @xmath341 . for simplifying of the notation , we assume that components of @xmath340 are @xmath342 , @xmath343 , and @xmath344 . then , the critical graph @xmath345 forms one of the graph in figure [ fig : pants ] ( cf . @xcite ) . we make equally spaced @xmath346-cuts in @xmath347 where @xmath348 ( @xmath349 ) . let @xmath350 be a component of @xmath351 which contains @xmath347 in the boundary . let @xmath352 be a subannulus of @xmath350 with height @xmath353 and @xmath354 . we cut @xmath352 along the vertical slits with endpoints in the @xmath346-cuts in @xmath347 and get a family of euclidean rectangles . since the circumference and the height of @xmath352 are of order @xmath355 , the moduli of such euclidean rectangles are uniformly bounded above and below . * @xmath358 is a rectangle above for all @xmath359 and @xmath360 , * the arc system given by correcting cores of @xmath361 s is homotopic to @xmath362 , where the _ core _ of @xmath361 is a path in @xmath363 connecting between facing arcs in @xmath364 and * the extremal length of family of paths in @xmath361 homotopic to the core is uniformly bounded above . notice from and the uniformity the moduli of @xmath365 that the conformal structure of @xmath289 is precompact in the reduced teichmller space . since the intersection numbers @xmath366 are independent of @xmath209 , we can take @xmath361 such that the width of each @xmath361 with respect to the @xmath245-metric are comparable with @xmath286 . by definition , the @xmath367-area of each @xmath361 is @xmath368 . from the reciprocal relation between the module and the extremal length for quadrilateral or rengel s type inequality , the extremal length @xmath369 of the family of paths in @xmath361 homotopic to the core satisfies @xmath370 for all @xmath359 ( see 4 in chapter i of @xcite ) . we divide each @xmath292 into congruence @xmath305-rectangles @xmath371 via proper horizontal segments . we may assume that for any @xmath360 and @xmath372 , there is an @xmath359 such that @xmath373 is congruent to @xmath374 . we set @xmath375 ( cf . figure [ fig : annulusr ] ) . since twisting numbers of segments in @xmath321 on each @xmath292 are at most one for all @xmath326 , from and the dehn - thurston s parametrization of simple closed curves ( cf . @xcite ) , we can glue all @xmath292 and @xmath287 appropriately at the part @xmath377 to get a riemann surface @xmath378 and an annulus @xmath379 such that after deforming @xmath378 by a quasiconformal mapping with maximal dilatation @xmath380 , we obtain @xmath229 and the core of the image @xmath301 of the annulus @xmath381 is homotopic to @xmath333 . thus , we conclude @xmath382 as @xmath169 . therefore , to get the upper estimate of the extremal length of @xmath56 on @xmath383 , it suffices to give an upper estimate of the extremal length of @xmath381 . let @xmath384 be the extremal metric on @xmath381 for the extremal length @xmath385 with @xmath386 . let @xmath387 be a collection of all rectangles of form @xmath388 for all @xmath389 . by the same argument as claim 1 in 9.6 , we can see the following . let us continue the calculation . let @xmath394 be a collection of components of @xmath395 . by labeling correctly , @xmath396 contains @xmath391 and @xmath397 , where @xmath391 is labeled cyclically in @xmath390 . by definition , each @xmath398 is contained in either @xmath399 or @xmath361 for some @xmath400 . let @xmath401 be the family of paths connecting vertical segments @xmath391 and @xmath397 . let @xmath402 be the family of paths in @xmath399 connecting vertical boundary segments . since @xmath403 by ( 1 ) and ( 2 ) of proposition [ prop : extremal_length_upper ] and claim [ claim : upper_bound_1 ] , we have @xmath404 thus we get an upper bound of the extremal length of @xmath381 as we desired . from lemma [ lem : char_annulus_modulus ] , by taking a subsequence if necessary , we may assume that @xmath405 tends to a positive number @xmath406 for any @xmath303 . from , , and , we deduce that @xmath407 for all @xmath112 . from the density of @xmath408 in @xmath65 , the above also holds for all measured foliations . thus , for @xmath409 and @xmath410 with @xmath411 , by substituting @xmath412 to , we get @xmath413 where @xmath414 . therefore , the discriminant of the quadratic form above is zero . namely , we have @xmath415 for all such @xmath410 . hence , two vectors @xmath416 are parallel for all @xmath410 with @xmath411 . however , this is impossible as we already observed in 6 of @xcite . a. douady , a. fathi , d. fried , f. laudenbach , v. ponaru , and m. shub , _ travaux de thurston sur les surfaces _ , sminaire orsay ( seconde dition ) . astrisque no . 66 - 67 , socit mathmatique de france , paris ( 1991 ) .

in this paper , we shall show that the metric boundary of the teichmller space with respect to the teichmller distance contains non - busemann points when the complex dimension of the teichmller space is at least two .

suppose that @xmath1 , with @xmath2 , is a smooth function supported in the half space @xmath3 , and let @xmath4 be some real number . we study the problem of reconstructing the function @xmath5 from the integrals @xmath6 for @xmath7 and @xmath8 . ( here @xmath9 is the unit sphere in @xmath10 and @xmath11 the surface measure on @xmath9 ) . we call the function @xmath12 the conical radon transform of @xmath5 . as illustrated in figure [ fig : cone ] , @xmath13 is the integral of @xmath5 over the one sided conical surface @xmath14 having vertex @xmath15 on the plane @xmath16 , symmetry axis @xmath17 , and half opening angle @xmath18 . the product @xmath19 is the standard surface measure on @xmath14 , and @xmath20 is an additional radial weight that can be adapted to a particular application at hand . for @xmath21 , the function @xmath22 may be considered as a conical projection of @xmath5 onto @xmath23 . integrates a function with support in the upper half space over one sided conical surfaces @xmath14 centered at @xmath24 having symmetry axis @xmath25 and half - angle @xmath26 . any point on @xmath14 can be written in the form @xmath27 with @xmath28 and @xmath29 . the @xmath30 dimensional surface measure on @xmath14 is given by @xmath19 , with @xmath11 denoting the standard surface measure on @xmath31.,scaledwidth=80.0% ] inversion of the conical radon transform in three spatial dimension is important for computed tomography taking compton scattered photons into account @xcite . in @xcite fourier reconstruction formulas have been derived for the cases @xmath32 . for two spatial dimensions , @xmath33 has been studied with @xmath32 in @xcite , where reconstruction formulas of the back - projection type have been derived . in dimensions @xmath34 , the conical radon transform has , to the best of our knowledge , not been studied so far . in this paper we study @xmath33 for any @xmath2 and any @xmath35 . we derive explicit reconstruction formulas of the back - projection type ( see theorem [ thm : fbp ] ) as well as a fourier slice identity ( see theorem [ thm : fourierslice ] ) similar to the one of the classical radon transform . before we present our main results we introduce some notation . by @xmath36 we denote the space of all functions defined on @xmath0 , that are @xmath37 and have compact support in @xmath38 . likewise @xmath39 denotes the space of all infinitely smooth functions defined on @xmath40 . as can easily be seen , the conical radon transform defined by ( [ eq : c - radon ] ) is well defined as an operator @xmath41 . points in @xmath0 will be written in the form @xmath42 with @xmath43 and @xmath44 . the fourier transform of a function @xmath45 with respect to the first component is denoted by @xmath46 for @xmath47 . the hankel transform of order @xmath48 in the second argument is denoted by @xmath49 for @xmath50 , where @xmath51 is the bessel function of the first kind of order @xmath48 . note that for @xmath52 , we have @xmath53 and hence @xmath54 is closely related to the cosine transform . similarly , we denote by @xmath55 the fourier transform of a function @xmath56 with respect to the first argument . finally , we denote by @xmath57 the riesz potential of @xmath58 , defined by @xmath59 the riesz potential is well defined if @xmath60 for every @xmath61 , which will always be the case in our considerations . the central results of this paper are the following explicit reconstruction formulas for inverting the conical radon transform . for[thm : fbp ] every @xmath35 , every @xmath62 and every @xmath63 , we have @xmath64 here @xmath65 is the riesz potential defined by ( [ eq : riesz ] ) . see sections [ sec : fbp ] and [ sec : fbp2 ] . the reconstruction formulas ( [ eq : fbp ] ) , ( [ eq : fbp2 ] ) are of the filtered back - projection type : the riesz potential can be interpreted as a filtration step in the first argument and the integrations actually sum over all conical surfaces that pass through the reconstruction point @xmath63 . in analogy to the classical radon transform the integration process may therefore be called _ conical back - projection_. note that ( [ eq : fbp ] ) , ( [ eq : fbp2 ] ) only differ up to a different parametrization of the set of all conical surfaces passing through the reconstruction point . for practical applications , the two and three dimensional situations are the most relevant ones . in these cases the formulas of theorem [ thm : fbp ] read as follows . 1 . [ it : fbp-2d ] suppose @xmath52 . then , for every @xmath66 and every @xmath67 , @xmath68 here @xmath69 and @xmath70 denote the derivative and the hilbert transform in first argument . [ it : fbp-3d ] suppose @xmath71 . then , for every @xmath72 and every @xmath73 , @xmath74 here @xmath75 denotes the laplacian in the first component . for @xmath52 we have the fourier representation @xmath76 and @xmath77 of the hilbert transform and the one dimensional derivative , respectively . this shows @xmath78 . hence item [ it : fbp-2d ] follows from theorem [ thm : fbp ] . similarly , for @xmath71 , we have @xmath79 and hence item [ it : fbp-3d ] again follows from theorem [ thm : fbp ] . for @xmath52 , formulas equivalent to the ones of theorem [ thm : fbp-23d ] [ it : fbp-2d ] have been first derived in @xcite . the three dimensional reconstruction formulas of theorem [ thm : fbp-23d ] [ it : fbp-3d ] ( as well as the higher dimensional generalizations of theorem [ thm : fbp ] ) are new . one notes , that in three spatial dimensions the reconstruction formulas are particularly simple and further local : the reconstruction of @xmath5 at some reconstruction point @xmath42 only requires the integrals over cones passing through an arbitrarily small neighbourhood of @xmath42 . since for any odd @xmath80 , the riesz potential satisfies @xmath81 , the reconstruction formulas ( [ eq : fbp ] ) , ( [ eq : fbp2 ] ) are in fact local for every odd space dimensions . contrary , in even space dimension ( [ eq : fbp ] ) , ( [ eq : fbp2 ] ) are non - local : recovering a function at a single point requires knowledge of the integrals over all conical surfaces . this behaviour is similar to the one of the classical radon transform , where also the inversion is local in odd and non - local in even dimensions ( see , for example , @xcite ) . theorem [ thm : fbp ] will be established using the following theorem [ thm : fourierslice ] , which an analogon of the well known fourier slice identity ( * ? ? ? * chapter 1 , theorem 1.1 ) satisfied by the classical radon transform . for every @xmath35 , every @xmath82 and every @xmath83 , we have @xmath84 here @xmath85 is the function @xmath86 , @xmath87 the fourier transform in the first argument , and @xmath88 the hankel transform of order @xmath89 in the second argument . see section [ sec : fs ] . the fourier slice identity is of course of interest on its own . the argument @xmath90 , for @xmath91 and @xmath92 , appearing on the left hand side of ( [ eq : fourierslice ] ) , fills in the whole upper half - space , which is required to invert the fourier - hankel transform using well known explicit and stable inversion formulas . hence the function @xmath5 can be reconstructed based on ( [ eq : fourierslice ] ) by means of a @xmath30-dimensional fourier transform , followed by an interpolation , and finally performing an inverse @xmath80-dimensional fourier - hankel transform . the remainder of the paper is mainly devoted to the proofs of theorems [ thm : fbp ] and [ thm : fourierslice ] that we will establish in the following section [ sec : proofs ] . we will first derive the fourier slice identity of theorem [ thm : fourierslice ] , which will then be used to proof the reconstruction formulas of theorem [ thm : fbp ] . the paper ends with a discussion in section [ sec : discussion ] . in this section we derive theorems [ thm : fbp ] and [ thm : fourierslice ] . the following elementary lemma shows that it suffices to derive these results for the special case @xmath93 . for[lem : munu ] every @xmath35 , every @xmath82 and every @xmath94 , we have @xmath95 here @xmath96 stands for the operator that multiplies a function @xmath97 by @xmath96 and likewise @xmath98 stands for the operator that multiplies @xmath99 by @xmath100 . the definition of @xmath33 and the substitution @xmath101 yield @xmath102 comparing the last expression for @xmath103 with the corresponding expression for @xmath104 obviously shows ( [ eq : munu ] ) . we start by showing ( [ eq : fourierslice ] ) for the special case @xmath93 . the general case will then be a consequence of lemma [ lem : munu ] . the definition of the conical radon transform , the definition of the fourier transform and some basic manipulations yield @xmath105 } { \mathrm d}s\,.\end{aligned}\ ] ] now we use the identity ( see , for example , @xcite ) , @xmath106 application of ( [ eq : ps ] ) with @xmath107 followed by the substitution @xmath108 yields @xmath109 the last displayed equation is recognised as the hankel transform of order @xmath110 of @xmath111 in the second argument . we conclude , that @xmath112 this shows ( [ eq : fourierslice ] ) for the special case @xmath93 . for general @xmath35 we use the relation @xmath113 from lemma [ lem : munu ] . together with ( [ eq : fs-0 ] ) this yields @xmath114 this is ( [ eq : fourierslice ] ) for the case of general @xmath35 and concludes the proof of theorem [ thm : fourierslice ] . we start with the proof of ( [ eq : fbp ] ) for @xmath93 . application of the inversion formulas for the fourier and the hankel transform followed by the substitution @xmath115 shows @xmath116 application of the fourier slice identity ( theorem [ thm : fourierslice ] ) with @xmath93 and interchanging the order of integration then yields @xmath117 by ( [ eq : ps ] ) , we have @xmath118 therefore , @xmath119 } \\ & = { \left(2\pi\right)}^{\frac{{d}-1}{2 } } \int_{s^{{d}-2 } } { \left(\tan{\left(\theta\right ) } { y}\right)}^{\frac{{d}-3}{2 } } { \left({\mathcal i}^{(1-{d})}{\mathcal r}^{(0 ) } f\right)}{\left({\mathbf x}- \tan{\left(\theta\right ) } { y}{\mathbf n},\theta\right ) } { \mathrm d}{\mathbf n}\ , . \end{aligned}\end{gathered}\ ] ] together with ( [ eq : aux1 ] ) this further implies @xmath120 this shows formula ( [ eq : fbp ] ) for the special case @xmath93 . to show ( [ eq : fbp ] ) in the general case @xmath35 , we again use the relation @xmath121 from lemma [ lem : munu ] . hence application of the reconstruction formula for the special case @xmath93 to @xmath122 yields @xmath123 this shows ( [ eq : fbp ] ) in the general case @xmath124 . finally we derive ( [ eq : fbp2 ] ) as an easy consequence of ( [ eq : fbp ] ) . to that end we first substitute @xmath125 with @xmath126 . then @xmath127 and @xmath128 . consequently , ( [ eq : fbp ] ) implies @xmath129 now we substitute @xmath130 ( polar coordinates in the plane @xmath131 around the center @xmath132 ) . then @xmath133 and @xmath134 . consequently , @xmath135 this is the reconstruction formula ( [ eq : fbp2 ] ) . in this paper we derived explicit reconstruction formulas for the conical radon transform , which integrates a function in @xmath80 spatial variables over all cones with vertices on a hyperplane and symmetry axis orthogonal to this plane . the derived formulas are of the back - projection type and are theoretically exact . further , they are local for odd @xmath80 , and non - local for even @xmath80 . among others , inversion of the conical radon transform is relevant for emission tomography using compton cameras as proposed in @xcite . such a device measures the direction as well as the scattering angle of an incoming photon at the front of the camera . the location of the photon emission can therefore be traced back to the surface of a cone . recovering the density of the photon source therefore yields to the inversion of the conical radon transform in a natural manner . radon transforms are the theoretical foundation of many medical imaging and remote sensing application . certainly the most well known instance is the classical radon transform , which integrates a function over hyperplanes . among others , inversion of the classical radon transform is important for classical transmission computed tomography and has been studied in many textbooks ( see , for example , @xcite ) . closed form reconstruction formulas are known for a long time and have first been derived already in 1917 by j. radon @xcite . another radon transform that has been studied in detail more recently is the spherical radon transform . this transform integrates a function over spherical surfaces ( for some restricted centers of integration ) and is , among others , important for photo- and thermoacoustic tomography @xcite . closed form reconstruction formulas for planar and spherical center sets have been found in @xcite . the conical radon transform , on the other hand , is much less studied . in particular , closed form reconstruction formulas have only been known for the case @xmath136 , see @xcite . in this paper we derived such reconstruction formulas for arbitrary dimension @xmath2 . for computed tomography with compton cameras @xcite , the three dimensional case is of course the most relevant one . in this case , our reconstruction formulas have a particularly simple structure and consist of an application of the laplacian followed by a conical back - projection . the numerical implementation seems quite straight forward following the ones of the classical or the spherical radon transform ( see , for example , @xcite ) . numerical studies , however , will be subject of future research .

inversion of radon transforms is the mathematical foundation of many modern tomographic imaging modalities . in this paper we study a conical radon transform , which is important for computed tomography taking compton scattering into account . the conical radon transform we study integrates a function in @xmath0 over all conical surfaces having vertices on a hyperplane and symmetry axis orthogonal to this plane . as the main result we derive exact reconstruction formulas of the filtered back - projection type for inverting this transform . * keywords . * radon transform , conical projections , computed tomography , inversion formula , filtered back - projection . * ams subject classifications . * 44a12 , 45q05 , 92c55 .

monte carlo simulations of qcd with dynamical quarks are done in most cases at relatively large quark masses ( typically two quark flavours with @xmath9 ) . this makes the extrapolation to the physical point @xmath10 rather uncertain . the extrapolation is done by using ( pq)chpt typically to nlo ( 1-loop ) order . estimates @xcite show that one should perform simulations in the range @xmath11 in order to see the expected logarithmic dependence . matching the predicted functional dependence is a crucial test for lattice qcd . until recently the comparison between lattice data and chpt was not satisfactory . in a recent paper @xcite we suggested that the agreement can be found when going to light enough dynamical quarks . here we supplement some integration of the analysis of @xcite and provide some further comments . for our simulation we used the algorithm described in @xcite and further improved as in @xcite and references therein . in our range of parameters we found that the cost for producing one independent gauge field configuration roughly goes as : @xmath12 where @xmath13 is the quark mass in lattice unit , @xmath14 is the number of lattice points and the factor @xmath15flop for the plaquette , but one order of magnitude smaller for @xmath16 and @xmath17 . we produced three sets of @xmath18 thermalized configurations for @xmath19 unimproved wilson fermions at vol@xmath20 , @xmath6 and @xmath21 . from these points we extrapolated a value of @xmath22 at @xmath23 which is @xmath24 . this corresponds to an uv cutoff @xmath25fm @xmath26 , and to a physical volume @xmath27fm . in those three points we found a ratio @xmath28 equal to @xmath29 , @xmath30 and @xmath31 respectively . here @xmath32 represents the sea quark mass defined as @xmath33 , when the strange quark mass corresponds to @xmath34 @xcite . for the lightest valence quarks the ratios @xmath35 become respectively @xmath36 , @xmath37 and @xmath38 . even with these small quark masses finite volume effects are expected to be under control since we have always @xmath39 . of course this is payed with a very low uv cutoff , and one expects large lattice artifacts . these are taken into account in the analysis . our first goal is to confront the results of numerical simulations with the ( pq)chpt formulas @xcite . in order to cancel the @xmath40-factors of multiplicative renormalization , which in the case of a mass - independent renormalization scheme only depend on the gauge coupling and not on the quark mass , we considered ratios of quark masses ( @xmath41 ) , pion masses ( @xmath42 ) and pion decay constants ( @xmath43 ) . here @xmath44 stand for the flavor indices of valence ( @xmath45 ) or sea ( @xmath46 ) quarks . if we assume that there are no lattice artifacts , no nnlo corrections and we take for @xmath47 its tree level estimate , @xmath48 , then the ratios ( here @xmath49 ) @xmath50 , \\ rrn \hspace{-7pt } & \equiv & \hspace{-7pt } \frac{4\xi m_{vs}^4}{(\xi\hspace{-3pt}+\hspace{-3pt}1)^2m_{vv}^2 m_{ss}^2 } \hspace{-3pt}= \hspace{-3pt}1 + \hspace{-3pt}\frac{\chi_s^{est } [ \log(\xi)\hspace{-3pt}-\hspace{-3pt}\xi \hspace{-3pt}+\hspace{-3pt}1]}{16n_s\pi^2 } \end{aligned}\ ] ] are non trivial and parameterless predictions of chpt . this provides a strong check of how far we are from the nlo chpt regime . in figure [ fig : rrf ] and [ fig : rrn ] we superimpose the predicted functions to the data . the agreement is of course not complete ( there are indeed @xmath51 and nnlo effects ) , but the corrections are sub - dominant contributions . moreover the agreement improves when the sea quark masses decrease . encouraged by this results , we systematically compared our data with those ratios of pion masses and coupling constants which determine the gasser - leutwyler @xcite coefficients @xmath2 , @xmath3 , @xmath4 and @xmath5 . since we expected large lattice artifacts in the data , the comparison was done with w(pq)chpt @xcite , including @xmath51 lattice artifacts in the effective continuum theory . besides that , we also included the relevant contributions from nnlo chpt @xcite . although this involve many parameters , one can obtain enough constraints from partially quenched simulations . the details of the fitting procedure and the results are described in @xcite . here we simply add that , following the analysis of @xcite , we can now include also @xmath52 lattice artifacts in our fit of the pion mass ratios . it turns out that this does not add new parameters to the fit , but it amounts to a redefinition of the wilson - chpt coefficients @xmath53 . in order to have a consistency check of the surprisingly small lattice artifacts that we found , we compared the results of @xcite with those obtained in an older simulation with larger lattice spacing ( @xmath54 , @xmath55fm ) @xcite . the comparison ( fig . [ fig : scarr ] and [ fig : scafvv ] ) shows quite small scale breaking . larger scale breaking ( @xmath56 ) are observed for the ratio @xmath57 , which at fixed @xmath58 goes from @xmath59 ( at @xmath6 ) to @xmath60 ( at @xmath61 ) . to summarize : we showed that it is possible to simulate , with reasonable costs , very light dynamical quarks . compensating @xmath51 effects by introducing @xmath51 terms in the pqch - lagrangian is , in this case , a viable alternative to @xmath51 improvement of the action . the observed @xmath51 contributions in the light goldstone boson sector are surprisingly small , while nnlo are still important . the expected behavior predicted by pqchpt is already visible , although a quantitative determination of the lec s still needs further simulations at smaller masses and lattice spacing . most of the numerical calculations presented here have been done at the computers of nic - juelich and zeuthen . 9 s.r . sharpe , n. shoresh , phys . rev . * d62 * ( 2000 ) 094503 ; hep - lat/0006017 . s. drr , hep - lat/0208051 . qq+q collaboration , f. farchioni , i. montvay , e. scholz and l. scorzato , to appear in eur . j. ; hep - lat/0307002 . i. montvay , nucl . phys . * b466 * ( 1996 ) 259 ; hep - lat/9510042 . qq+q collaboration , f. farchioni , c. gebert , i. montvay and l. scorzato , eur . j. * c26 * ( 2002 ) 237 ; hep - lat/0206008 . bernard , m.f.l . golterman , phys . * d49 * ( 1994 ) 486 ; hep - lat/9306005 . j. gasser and h. leutwyler , annals phys . * 158 * ( 1984 ) 142 . g. rupak , n. shoresh , phys . rev . * d66 * ( 2002 ) 054503 ; hep - lat/0201019 . sharpe , r. van de water , these proceedings ; hep - lat/0308010 . o. baer , g. rupak and n. shoresh , hep - lat/0306021 . qq+q collaboration , f. farchioni , c. gebert , i. montvay , e. scholz and l. scorzato , phys . lett . * b561 * ( 2003 ) 102 ; hep - lat/0302011 .

the dependence of pseudo - scalar masses and decay constants on the sea and valence quark masses is investigated in the pseudo - goldstone boson sector of qcd with two light quark flavours . the sea quark masses are at present in the range @xmath0 whereas the valence quark masses satisfy @xmath1 . the values of the gasser - leutwyler low energy constants @xmath2 , @xmath3 , @xmath4 and @xmath5 are estimated . the computation is done with the wilson - quark lattice action at gauge coupling @xmath6 on @xmath7 lattices . @xmath8 effects are taken into account by applying chiral perturbation theory for the wilson lattice action as proposed by rupak and shoresh .

it is well known that in homogeneous magnetized two - fluid plasmas three electromagnetic modes with frequency less than the electron cyclotron frequency exist @xcite . these include the fast , alfvn ( or intermediate ) and slow modes , according to their different phase velocities @xcite . their dispersion relations can be obtained from a general one based on the hall - mhd model @xcite . recently , @xcite obtained the same relation from a simpler formulation involving a two - dimensional current density vector . the general dispersion relation for even lower frequency modes ( with wave frequency @xmath11 less than the ion cyclotron frequency @xmath12 ) have also been derived using different formulations @xcite . however , a comprehensive investigation of the wave polarizations is still lacking . in this paper we present analytical expressions of the dispersion relations and polarizations using an approach similar to that of ref . we shall consider the dense - plasma limit @xmath13 , where @xmath14 is the alfvn speed and @xmath15 is the light speed , so that the displacement current in the ampere s law can be ignored st63,fk69,sw89,is05,da09,kr94,be12 . the resulting analytical expressions are useful for analyzing the properties of low - frequency waves in different plasmas . in the next section we present the derivation of the dispersion relations and polarizations of the waves . in sec . iii the properties of the waves propagating at different angles and different @xmath1 regimes are discussed . the main results are summarized in sec . iv . the appendix gives the approximate dispersion relations in the near - perpendicular propagation , low-@xmath1 @xmath3 , and high-@xmath1 @xmath16 limits , where @xmath1 is the ratio of the plasma to magnetic pressures . linearized two - fluid and maxwell s equations are @xmath17where the subscripts @xmath18 denote ions and electrons , respectively , @xmath19 is the mass , @xmath20 is the charge , @xmath21 is the thermal pressure , @xmath22 * * * * is boltzmann constant , @xmath23 is the temperature , @xmath24 is the perturbed number density , @xmath25 is the perturbed velocity , @xmath26 is the perturbed current density , @xmath27 and @xmath28 are the perturbed electric and magnetic fields , respectively , @xmath29 is the ambient magnetic field , and @xmath30 is the ambient number density . as mentioned , the displacement current is neglected . the quasi - neutrality condition @xmath31 shall also be used . in the study an electron - proton plasma is considered , namely @xmath32 and @xmath33 . we shall consider plane waves , so that @xmath34exp@xmath35 , where @xmath11 is the wave frequency and @xmath36 is the wave vector . we can obtain from eq . ( [ eq : momentum ] ) the perpendicular and parallel ( to @xmath37 ) fluid velocities @xmath38and @xmath39the current density @xmath40 can then be expressed as @xmath41and @xmath42where @xmath43 , @xmath44 , @xmath45 , @xmath46 , @xmath47 and @xmath48 combining eqs . ( 3 ) and ( 4 ) leads to @xmath49from eqs.([eq : cur-1 ] ) ( [ eq : cur-3 ] ) , we get for the electric field and number density perturbation , @xmath50 \delta e_{x}+\left ( 1-q^{2}\right ) \frac{\omega ^{3}}{% \omega _ { ci}}\delta e_{y}-i\lambda _ { 0}\lambda _ { 2}v_{a}^{2}k_{\perp } k_{z}\delta e_{z } \notag \\ -\frac{\kappa t_{t}}{e}\left ( \lambda _ { 2}\widetilde{t}_{i}-q\lambda _ { 0}% \widetilde{t}_{e}\right ) \omega ^{2}k_{\perp } \frac{\delta n}{n_{0 } } = 0 , \label{eq : relation_x } \\ \left ( 1-q^{2}\right ) \frac{\omega ^{3}}{\omega _ { ci}}\delta e_{x}-i\left [ \lambda _ { 0}\lambda _ { 2}v_{a}^{2}k^{2}-\left ( 1+q\right ) \lambda _ { 1}\omega ^{2}\right ] \delta e_{y } \notag \\ -i\frac{\kappa t_{t}}{e}\left ( \lambda _ { 2}\widetilde{t}_{i}+\lambda _ { 0}% \widetilde{t}_{e}\right ) \omega \omega _ { ci}k_{\perp } \frac{\delta n}{n_{0 } } = 0 , \label{eq : relation_y } \\ i\lambda _ { e}^{2}k_{\perp } k_{z}\delta e_{x}-i\left ( 1+q+\lambda _ { e}^{2}k_{\perp } ^{2}\right ) \delta e_{z}-\frac{\kappa t_{t}}{e}\left ( q% \widetilde{t}_{i}-\widetilde{t}_{e}\right ) k_{z}\frac{\delta n}{n_{0 } } = 0 , \label{eq : relation_z}\end{aligned}\]]so that three electric field components can be written as @xmath51where @xmath52 and other definitions are : @xmath53 \lambda _ { 1}v_{a}^{2}k^{2}\omega ^{2 } \notag \\ & + & \left ( 1+q\right ) \lambda _ { 0}\lambda _ { 2}v_{a}^{4}k^{2}k_{z}^{2 } , \notag \\ \pi _ { \mathrm{ex } } & = & \left ( 1+q\right ) \left ( 1+q+\lambda _ { e}^{2}k_{\perp } ^{2}\right ) \left ( q\widetilde{t}_{i}-\widetilde{t}_{e}\right ) \omega ^{4 } \notag \\ & -&\left [ \begin{array}{c } \left ( 1+q+\lambda _ { e}^{2}k_{\perp } ^{2}\right ) \left ( \lambda _ { 2}% \widetilde{t}_{i}-q\lambda _ { 0}\widetilde{t}_{e}\right ) \\ + \left ( 1+q\right ) \left ( q\widetilde{t}_{i}-\widetilde{t}_{e}\right ) \lambda _ { 1}k_{z}^{2}/k^{2}% \end{array}% \right ] v_{a}^{2}k^{2}\omega ^{2 } \notag \\ & + & \lambda _ { 0}\lambda _ { 2}\left ( q\widetilde{t}_{i}-\widetilde{t}% _ { e}\right ) v_{a}^{4}k^{2}k_{z}^{2 } , \notag \\ \pi _ { \mathrm{ey } } & = & \left ( 1+q\right ) \left [ \left ( 1+q+\lambda _ { e}^{2}k_{\perp } ^{2}\right ) \omega ^{2}-\lambda _ { 1}v_{a}^{2}k_{z}^{2}% \right ] \omega \omega _ { ci } , \notag \\ \pi _ { \mathrm{ez } } & = & \pi _ { \mathrm{ex}}+\left ( 1-q^{2}\right ) v_{a}^{2}k^{2}\omega ^{2}. \notag\end{aligned}\ ] ] inserting above electric field components into the number density equation that is derived from eqs . ( [ eq : continuum ] ) , ( [ eq : vel - per ] ) and ( eq : vel - par ) , @xmath54 \frac{\delta n}{n_{0}}=-i\frac{% \omega ^{2}k_{\perp } } { b_{0}\omega _ { ci}}\delta e_{x}+\frac{\omega k_{\perp } % } { b_{0}}\delta e_{y}+ik_{z}\frac{e}{m_{i}}\lambda _ { 0}\delta e_{z } , \label{eq : ion - equation}\]]the general dispersion relation can be expressed as @xmath55with @xmath56 v_{a}^{2}k^{2 } , \notag \\ c & = & \left [ \left ( 1+q\right ) \left ( 1 + 2\beta \right ) + \left ( 1+q^{2}\right ) \rho ^{2}k^{2}\right ] v_{a}^{4}k^{2}k_{z}^{2 } , \notag \\ d & = & \beta v_{a}^{6}k^{2}k_{z}^{4}. \notag\end{aligned}\]]where @xmath57 , @xmath58 is the ion gyroradius , @xmath59 is the ion acoustic gyroradius , @xmath60 is the ion inertial length , @xmath61 is the electron inertial length , @xmath62 is the ion thermal speed , @xmath63 is the ion acoustic speed , @xmath64 is the sound speed , and @xmath65 . with respect to the existing ones st63,fk69,be12,is05,da09,ho99,ch11 , eq . ( [ eq : general dispersion equation ] ) represents a more general description of the low - frequency electromagnetic waves . three roots for @xmath66 correspond to the fast @xmath67 , alfvn @xmath68 , and slow @xmath69 modes , or @xcite,@xmath70with @xmath71 and @xmath72 . if we set @xmath73 , eq . ( [ eq : general dispersion equation ] ) yields two resonances @xmath74 : the ion cyclotron resonance @xmath75 and the electron cyclotron resonance @xmath76 . if we neglect the electron inertial terms @xmath77and terms of the order of @xmath78 , eq . ( [ eq : general dispersion equation ] ) recovers the hall - mhd dispersion relation @xcite , where only the ion cyclotron resonance exists . for the high oblique propagation , low-@xmath1 and high-@xmath1 limits , the approximate dispersion relations of the three modes are given in the appendix . ( [ eq : general dispersion equation ] ) can also be reduced to the well - known results in the cold two - fluid plasmas @xmath79 @xcite . once the electric field perturbation ( [ electric - field ] ) and the dispersion relation ( [ eq : roots ] ) are known , the magnetic field and velocity perturbations can be also expressed in terms of the number density perturbation , @xmath80@xmath81and@xmath82where @xmath83 , \notag \\ \pi _ { \mathrm{viy } } & = & \left [ q\left ( 1+q+\lambda _ { e}^{2}k^{2}\right ) \omega ^{2}-v_{a}^{2}k_{z}^{2}\right ] v_{a}^{2}k^{2 } , \notag \\ \pi _ { \mathrm{viz } } & = & \left ( 1+q+\lambda _ { e}^{2}k^{2}\right ) ^{2}\omega ^{4}-\left [ \left ( 1+q+\lambda _ { i}^{2}k^{2}\right ) v_{a}^{2}k_{z}^{2}+\left ( 1+q+q\lambda _ { e}^{2}k^{2}\right ) v_{a}^{2}k^{2}% \right ] \omega ^{2}+v_{a}^{4}k^{2}k_{z}^{2 } , \notag \\ \pi _ { \mathrm{vex } } & = & \omega \omega _ { ci}\left [ \left ( 1+q+\lambda _ { e}^{2}k^{2}\right ) ^{2}\omega ^{2}-\left ( 1+q+q\lambda _ { e}^{2}k^{2}\right ) v_{a}^{2}k_{z}^{2}\right ] , \notag \\ \pi _ { \mathrm{vey } } & = & \left [ \left ( 1+q+\lambda _ { e}^{2}k^{2}\right ) \omega ^{2}-qv_{a}^{2}k_{z}^{2}\right ] v_{a}^{2}k^{2 } , \notag \\ \pi _ { \mathrm{vez } } & = & \left ( 1+q+\lambda _ { e}^{2}k^{2}\right ) ^{2}\omega ^{4}-\left [ \left ( 1+q+q\lambda _ { e}^{2}k^{2}\right ) v_{a}^{2}k_{z}^{2}+\left ( 1+q+\lambda _ { i}^{2}k^{2}\right ) v_{a}^{2}k^{2}% \right ] \omega ^{2}+v_{a}^{2}k^{2}k_{z}^{2}. \notag\end{aligned}\]]note that we can explore the linear relation between arbitrary two variables through the eigenfunctions ( [ electric - field ] ) and ( [ magnetic - field])@xmath84([velocitye ] ) . for example , the polarizations of electromagnetic fields are @xmath85 and @xmath86 at parallel propagation , @xmath87 , eq . ( eq : general dispersion equation ) is written as @xmath88 \left [ \left ( 1+q\right ) \omega ^{2}-v_{t}^{2}k_{z}^{2}\right ] = 0 , \label{eq : parallel dispersion}\]]which describes the left - hand @xmath89 and right - hand @xmath90 circularly - polarized waves @xmath91 , \label{ion and electron cyclotron wave}\]]and ion acoustic wave @xmath92note that the dispersion relation ( [ ion and electron cyclotron wave ] ) can be directly derived from eqs . ( [ eq : relation_x ] ) and ( eq : relation_y ) ; ( [ ion acoustic wave ] ) can be derived by use of eqs . ( [ eq : relation_z ] ) and ( [ eq : ion - equation ] ) . the left- and right - hand waves have the perpendicular perturbations @xmath93whereas the ion acoustic wave has the parallel perturbations @xmath94 when the wave propagates at the perpendicular direction , @xmath95 , only one mode exists @xmath96its polarization properties are @xmath97 but different @xmath98 : @xmath99 , @xmath100 and @xmath101 . , width=604 ] at the parallel propagation , the ion acoustic wave can interact with the right / left circularly - polarized waves at interaction points where their @xmath102 and @xmath103 are equal as shown in fig . . a mode transition can occur at the interaction point . the mode transition can happen among three oblique waves in figs . ( 2)@xmath104(4 ) . also , figs . ( 2)@xmath104(4 ) include the wave electromagnetic polarizations as well as the magnetic helicity @xmath105 and the ion cross - helicity @xmath106 @xmath107and @xmath108where @xmath109 denotes the vector potential and @xmath110 is the magnetic field perturbation in the velocity unit . fig . ( 2 ) presents the dispersion relations and polarizations of the three oblique waves at different angles in the low-@xmath1 plasmas where @xmath111 and @xmath112 . it shows that approximate dispersion relations ( [ a31 ] ) and ( [ a33 ] ) can describe the exact one ( [ eq : roots ] ) well . the fast mode corresponds to the fast magnetosonic wave as @xmath113 and the whistler wave as @xmath114 . at the electron cyclotron frequency @xmath115 , the fast mode is the electron cyclotron wave . furthermore , the electron cyclotron wave can change to a ( quasi- ) electroacoustic wave extending to higher frequency @xmath116 @xcite . it is interesting to see that the fast magnetosonic wave has @xmath117 as @xmath118 , and @xmath119 as @xmath120 at the near - parallel propagation . @xmath121 and @xmath122 for the fast magnetosonic wave ; @xmath123 and @xmath124 for the whistler and electron cyclotron waves . in addition , @xmath125 for the fast magnetosonic wave @xcite , while @xmath126 for the whistler and ( quasi- ) electroacoustic waves . the alfvn mode is the shear alfvn wave at @xmath127 and the ( quasi- ) electroacoustic wave at @xmath128 until a transition into the electron cyclotron wave at @xmath115 . at near - parallel @xmath129 and oblique @xmath130 cases , the phase velocity of the ( quasi- ) electroacoustic wave is about the sound speed . it is the alfvn speed at the high oblique angle , therefore , ref . @xcite called the high oblique mode at @xmath131 as the shear alfvn wave . note that an ion cyclotron wave @xmath132 arises at near - parallel propagation ( panel ( a1 ) ) . electromagnetic polarizations are @xmath133 , @xmath134 and @xmath135 at @xmath127 ; at @xmath136 , @xmath137 becomes much larger than @xmath14 . at @xmath129 and @xmath138 , the magnetic - helicity @xmath105 decreases firstly from @xmath139 to @xmath140 at @xmath127 , and then increases with increasing @xmath141 at @xmath136 ; at @xmath142 , @xmath139 is nearly unchanged at @xmath127 , and it becomes increasing at @xmath143 until reaching @xmath144 corresponding to the electron cyclotron wave**. * * besides , the ion cross helicity @xmath145 depends on the wave scale , e.g. , @xmath146 as @xmath147 and @xmath148 as @xmath149 . the slow mode corresponds to the slow magnetosonic wave at @xmath113 , where @xmath150 , @xmath151 , @xmath152 and @xmath139 . it turns to the ion cyclotron wave at @xmath153 @xcite , where @xmath154 , @xmath155 , @xmath156 and @xmath157 . at @xmath129 , the electric polarization @xmath158has an increment at the transition position where the slow magnetosonic wave changes to the ion cyclotron wave ; however , there is no such increment at @xmath159 and @xmath142 cases . figs . ( 3 ) and ( 4 ) present the dispersion relations and polarizations in @xmath160 and high-@xmath1 @xmath161 plasmas . here the alfvn mode interacts with the fast mode only . although the validity condition for approximate dispersion relations ( [ a22 ] ) and ( [ a23 ] ) is @xmath162 or @xmath163 , these expressions can describe the wave dispersion relations at @xmath164 as shown in fig . several mode properties in fig . ( 3 ) are obviously different from that in the low-@xmath1 plasmas ( fig . for example , to the near - parallel waves at @xmath165 , two circularly polarized @xmath166 modes are the fast and slow modes in @xmath164 plasmas but the fast and alfvn modes in the low-@xmath1 plasmas . here the fast ( slow ) mode exhibits the right - hand ( left - hand ) electric polarization and positive ( negative ) helicity . it also finds @xmath167 for the alfvn mode in @xmath164 plasmas . moreover , when the wave tends to more oblique propagation , the ion cross - helicity of the slow magnetosonic wave @xmath168 . ( 4 ) shows that the alfvn and slow modes are circularly polarized @xmath169 waves at @xmath170 in the high-@xmath2 plasmas , where the alfvn ( slow ) mode exhibits the right - hand ( left - hand ) electric polarization and positive ( negative ) helicity . these two modes also have the same ion cross - helicity distribution . note that three modes have no interaction point at the high oblique propagation as shown in panel ( c1 ) . it needs to note that the electric field polarizations also strongly depend on the ratio of the electron to ion temperature @xmath8 ( eq . ( eq : electric polarization ) ) . the electric field polarizations with different @xmath8 are presented in fig . ( 5 ) , where @xmath171 and @xmath10 , @xmath100 , and @xmath172 . it shows that the parallel polarization @xmath173 decreases obviously with decreasing @xmath8 . the transverse polarization @xmath174 is slightly affected by @xmath175 for the long - wavelength @xmath176 waves , but not for the long - wavelength fast mode in the high-@xmath1 plasmas or for the long - wavelength slow mode in the low-@xmath1 plasmas . to understand qualitatively above results , the complete expressions eq . ( [ eq : electric polarization ] ) can reduce to @xmath177 } { \left ( \omega ^{2}-v_{a}^{2}k_{z}^{2}\right ) \omega \omega _ { ci } } , \notag \\ \frac{\delta e_{z}}{i\delta e_{y } } & = & -\frac{k_{z}}{k_{\perp } } \frac{\left ( \omega ^{2}+v_{a}^{2}k^{2}\right ) } { \omega \omega _ { ci}}\widetilde{t}_{e},\end{aligned}\ ] ] where the long - wavelength @xmath178 and very low - frequency @xmath179 conditions are used , and the smaller terms of the order of @xmath78 are neglected . since @xmath180 and @xmath181 , @xmath182 decreases with decreasing @xmath8 . when @xmath183 , @xmath184 is independent on @xmath8 . we can also find @xmath185 corresponding to the fast wave @xmath186 in @xmath163 plasmas and @xmath187corresponding to the slow wave @xmath188 in @xmath189 plasmas , which indicate @xmath190 decreasing with smaller @xmath8 . besides , in fig . ( 5 ) the electric polarizations @xmath191 and @xmath173 in @xmath9 plasmas increase continuously as the slow magnetosonic wave changes to the ion cyclotron wave , while both polarizations are nearly unchanged in @xmath10 plasmas . note that the main characters in fig . ( 5 ) still appear in the electric polarization distributions of the oblique waves with @xmath192 . when the phase relation of the electric polarizations changes , a peak or a valley can occur in @xmath193 and @xmath194 distributions . for the fast mode in fig . ( 5 ) , the phase relation of @xmath7changes in @xmath10 plasmas , while it is unchanged in @xmath195 plasmas . for the alfvn mode , two transition points arise in the phase relation of @xmath196 in @xmath195 plasmas ; however , in the cold electron @xmath197 plasmas , the transition at the smaller @xmath198 disappears in @xmath189 plasmas , or two transitions are both missing in @xmath199 plasmas . for near - perpendicular alfvn wave , fig . ( 6 ) shows that the parallel polarization @xmath173 increases ( decreases ) with @xmath8 as @xmath200 @xmath201 . at @xmath202 , the transverse polarization @xmath193 decreases with increasing @xmath8 for the kinetic - scale alfvn waves @xmath203 . these results indicate the important role of the electron temperature @xmath204 on the kinetic - scale alfvn waves le99,ya14 . moreover , there is no transition of the phase relation of @xmath205 at @xmath202 case . the reason is that the wave frequency @xmath11 is smaller than the ion cyclotron frequency @xmath12 at @xmath202 , which can not satisfy the frequency condition @xmath206 for the changing of the phase relation of transverse electric polarization @xcite . in this study ions and electrons are treated separately in comparison with one fluid element @xmath207 method adopted in previous studies @xcite . this method is helpful to obtain the linear eigenfunctions including the ion and electron velocities as well as the ion and electron cross - helicities . it found that the fast and alfvn modes are nearly linearly polarized at the very low - frequency @xmath208 , and circularly polarized at @xmath209 at the near - parallel propagation in the low-@xmath1 plasmas . two circularly polarized modes become the fast and slow modes in a narrow frequency regime @xmath210 in @xmath164 plasmas ; they are the alfvn and slow modes in @xmath211 plasmas . to the ion cross - helicity @xmath212 of the long - wavelength slow mode , @xmath126 in the low-@xmath1 plasmas , @xmath213 as @xmath214 in @xmath164 plasmas , and @xmath215 in the high-@xmath1 plasmas . it also found that the negative magnetic - helicity @xmath216 of the alfvn mode can occur at the small or moderate angles in the low-@xmath1 plasmas , while @xmath167 arises always at the high oblique angle in the low-@xmath1 plasmas or at the general angle in @xmath217 plasmas . our results exhibited the sensitivity of the electric polarizations on the temperature ratio @xmath8 . the parallel polarization @xmath218 decreases with @xmath8 as @xmath219 . the transverse polarizations @xmath220 also decreases with @xmath8 for the long - wavelength fast mode in the high-@xmath1 plasmas , or for the long - wavelength slow mode in the low-@xmath1 plasmas , while @xmath221 at other long - wavelength cases are slightly affected by @xmath8 . furthermore , the phase relation of @xmath205 of the alfvn mode will change in @xmath195 plasmas , but this change can disappear in the cold electron @xmath10 plasmas . for the fast mode , the phase relation of @xmath222 changes in @xmath10 plasmas , while the unchanged phase relation arises in @xmath9 plasmas . we have also presented the approximate dispersion relations in the near - perpendicular propagation , low-@xmath1 , and high-@xmath1 limits . these approximations can describe nicely the exact dispersion relations of the three modes given by eq . ( [ eq : roots ] ) . it notes that the condition of @xmath13 used in the study leads to the neglecting of the displacement current . however , this assumption is broken near the wave cutoff position which results in the validity condition of @xmath223 @xcite . also , the displacement current may be important in producing the parallel electric field of the low - frequency alfvn mode @xcite . therefore , a comprehensive study including the effect of the displacement current is needed . lastly , two - fluid model neglects the kinetic wave - particle interaction effects , such as landau damping and ion ( electron ) cyclotron resonance damping , which can only be captured by the kinetic model . these kinetic effects can strongly affect the wave dispersion relation and polarization properties . for example , the wave dispersion relation of the kinetic alfvn wave is depressed at ion scales in the high-@xmath1 plasmas where there can be the heavy landau damping @xcite . the two models also result in different phase relation between two electric components . however , since it is difficult to identify clearly all modes from the full kinetic theory , the two - fluid theory can be a useful guide to discard the modes in the kinetic theory . our complete expressions can be conveniently used to compare with the results of the kinetic model . in low-@xmath1 plasmas with @xmath189 , the frequency of the slow mode @xmath232 is much smaller than that of the fast mode @xmath233 and alfvn mode @xmath234 . so that the last two terms in eq . ( [ eq : general dispersion equation ] ) control the dispersion relation of the slow mode in high-@xmath1 plasmas with @xmath163 , the frequency of the fast mode @xmath242 is much larger than that of the alfvn and slow modes @xmath243 . the fast mode are dominant by the first two terms in eq . ( [ eq : general dispersion equation ] ) , whereas the alfvn and slow modes are dominant by the quadratic equation shown in eq . ( [ a21 ] ) . therefore , the wave dispersion relations of the three modes are the same as that given by eqs . ( [ a22 ] ) and ( [ a23 ] ) . the author thanks prof . m. y. yu for discussing and improving the paper . the author also thanks the anonymous referee for constructive comments and suggestions that improve the quality of the paper . this work was supported by the nnsfc 11303099 , the nsf of jiangsu province ( bk2012495 ) , and the key laboratory of solar activity at cas nao ( lsa201304 ) .

analytical expressions for the dispersion relations and polarizations of low - frequency waves in magnetized plasmas based on two - fluid model are obtained . the properties of waves propagating at different angles ( to the ambient magnetic field @xmath0 ) and @xmath1 ( the ratio of the plasma to magnetic pressures ) values are investigated . it is shown that two linearly polarized waves , namely the fast and alfvn modes in the low-@xmath2 @xmath3 plasmas , the fast and slow modes in the @xmath4 plasmas , and the alfvn and slow modes in the high-@xmath5 @xmath6 plasmas , become circularly polarized at the near - parallel ( to @xmath0 ) propagation . the negative magnetic - helicity of the alfvn mode occurs only at small or moderate angles in the low-@xmath1 plasmas , and the ion cross - helicity of the slow mode is nearly the same as that of the alfvn mode in the high-@xmath1 plasmas . it also shown the electric polarization @xmath7 decreases with the temperature ratio @xmath8 for the long - wavelength waves , and the transition between left- and right - hand polarizations of the alfvn mode in @xmath9 plasmas can disappear when @xmath10 . the approximate dispersion relations in the near - perpendicular propagation , low-@xmath1 , and high-@xmath1 limits can quite accurately describe the three modes .

recent progress in the study of qcd - like gauge theories has revealed that a confined phase can exist under certain conditions when one or more spatial directions are compactified and small @xcite . this is surprising , because a small compact direction in euclidean time gives rise to a deconfined phase for @xmath5 gauge theories at high temperatures @xcite . it is also intriguing , because one or more small compact directions give rise to a small effective coupling constant if the theory is asymptotically free . thus we now have four - dimensional field theories in which confinement holds , and holds under circumstances where semiclassical methods may be reliably applied . at present , there are two methods known for achieving this . the first method directly modifies the gauge action with terms nonlocal in the compact direction(s ) @xcite , while the second adds adjoint fermions with periodic boundary conditions in the compact direction(s ) @xcite , which is our subject here . confinement in @xmath5 gauge theories is associated with an unbroken global center symmetry , which is @xmath6 for @xmath5 . the order parameter for @xmath6 breaking in the compact direction is the polyakov loop , @xmath7 $ ] , which is the path - ordered exponential of the gauge field in the compact direction . the trace of @xmath3 in a representation @xmath8 represents the insertion of a heavy fermion in that representation into the system . unbroken @xmath6 symmetry implies @xmath9 in the confined phase , and correspondingly @xmath10 holds in the deconfined phase where @xmath6 symmetry is broken . in the case of adjoint fermions with periodic boundary conditions on @xmath0 , @xmath6 symmetry is restored if the circumference @xmath1 of @xmath2 is sufficiently small and the mass @xmath11 of the adjoint fermions is sufficiently light @xcite . if @xmath12 is sufficiently small , the effective potential has a global minimum when the polyakov loop eigenvalues are uniformly spaced around the unit circle . this is the unique @xmath6-symmetric solution for @xmath3 . experience with phenomenological models @xcite suggests that in fact it is the constituent mass which is relevant in determining the size of the fermionic contribution to the effective potential for @xmath3 . in order to explore the interrelationship of confinement and chiral symmetry breaking , we use a generalization of nambu - jona lasinio models known as polyakov - nambu - jona lasinio ( pnjl ) models @xcite . in njl models , a four - fermion interaction induces chiral symmetry breaking . there has been a great deal of work on njl models , both as phenomenological models for hadrons and as effective theories of qcd @xcite . njl models have been used to study hadronic physics at finite temperature , but they include only chiral symmetry restoration , and do not model deconfinement . this omission is rectified by the pnjl models , which include both chiral restoration and deconfinement . the earliest model of this type was derived from strong - coupling lattice gauge theory @xcite , but later work on continuum models have proven to be extremely powerful in describing the finite - temperature qcd phase transition @xcite . in pnjl models , fermions with njl couplings move in a nontrivial polyakov loop background , and the effects of gluons at finite temperature is modeled in a semiphenomenological way . we will develop a model of this type for both fundamental and adjoint fermions below . recent lattice simulations by cossu and delia @xcite have confirmed the existence of the small-@xmath1 confined region in @xmath13 lattice gauge theory with two flavors of adjoint fermions , and we will focus on this case in our analysis . even if the small-@xmath1 confined region exists and is accessible in lattice simulations , it is not necessarily the same phase as found for large @xmath1 . put slightly differently , we would like to know if the small-@xmath1 and large-@xmath1 confined regions are smoothly connected , and thus represent the same phase . our main result will be a phase diagram for adjoint periodic qcd for all values of @xmath1 , obtained using a pnjl model . on the way to this goal , we will use as tests of our model both standard qcd with fundamental fermions and adjoint qcd with the usual antiperiodic boundary conditions for fermions . our principal tool will be the effective potential for the chiral symmetry order parameter @xmath4 and the deconfinement order parameter @xmath3 . for a more detailed discussion , see reference @xcite . we take the fermionic part of the lagrangian of our pnjl model to be @xcite @xmath14+g_{d}\left[\det\bar{\psi}\left(1-\gamma_{5}\right)\psi+h.c.\right]\ ] ] where @xmath15 is associated with @xmath16 flavors of dirac fermions in the fundamental or adjoint representation of the gauge group @xmath5 . the @xmath17 s are the generators of the flavor symmetry group @xmath18 and @xmath19 ; @xmath20 represents the strength of the four - fermion scalar - pseudoscalar coupling and @xmath21 fixes the strength of an anomaly induced term . for simplicity , we take the lagrangian mass matrix @xmath22 to be diagonal : @xmath23 . the covariant derivative @xmath24 couples the fermions to a background polyakov loop via the component of the gauge field in the compact direction . it is generally convenient to use the language of finite temperature to describe both the case of finite temperature , @xmath25 , with antiperiodic boundary conditions , and the case of a periodic spatial direction , @xmath26 . the zero - temperature contribution to the fermionic effective potential is given by @xmath27 where @xmath28 , @xmath29 , @xmath30 is a constituent mass , and the constant @xmath31 is the dimensionality of the color representation , @xmath32 for the fundamental and @xmath33 for the adjoint . the last term , representing a sum of one - loop diagrams , is regularized by three - dimensional momentum space cutoff , @xmath34 @xcite . in pnjl models , the finite - temperature contribution from the fermion determinant depends on the background polyakov loop . it is convenient to work in a gauge where the temporal component of the background gauge field , @xmath35 , is constant and diagonal . the covariant derivative then becomes @xmath36 . the one - loop free energy of fermions in a representation @xmath8 of @xmath5 gauge theory with zero chemical potential can be expanded in terms of modified bessel functions @xmath37 which is rapidly convergent for all values of the mass @xcite . the plus sign is used for periodic boundary conditions and minus for antiperiodic . in what follows , we will take @xmath38 , and take the masses @xmath39 to be equal to a common mass which we also write as @xmath22 . in this case , the contribution to the effective action from @xmath20 and @xmath21 has the same form . it is convenient to take @xmath40 , and also to write the common constituent mass as @xmath41 @xcite . there is a possibility of directly modifying the strength of chiral symmetry breaking by adding additional couplings compatible with all symmetries have been added . in the case of adjoint fermions with periodic boundary conditions , the ability to freely vary @xmath42 allows a clear connection between the large-@xmath1 and small-@xmath1 confining regions of the phase diagram . the boundary conditions for the gauge bosons are periodic in all cases considered here , so @xmath1 and @xmath43 may be used equivalently in the gluonic sector . the one - loop finite - temperature free energy in a background polyakov loop is given by an expression similar to the one for fermions @xmath44\ ] ] where we have inserted a mass parameter in @xmath45 for purely phenomenological reasons explained below . the polyakov loop in the fundamental representation of @xmath13 can be diagonalized by a gauge transformation and written as @xmath46 with two independent angles . with the use of @xmath47 symmetry , it is sufficient to consider the case where @xmath48 is real . thus we consider only diagonal , special - unitary matrices with real trace , which may be parametrized by taking @xmath49 , @xmath50 , and @xmath51 , or @xmath52 $ ] with @xmath53 . the unique set of @xmath47-invariant eigenvalues are obtained for @xmath54 . for @xmath13 , we can write the gluonic effective potential in a high temperature expansion in terms of @xmath55 @xcite @xmath56 we will set the mass scale @xmath57 by requiring that @xmath58 yields the correct deconfinement temperature for the pure gauge theory , with a value of @xmath59 . this gives @xmath60 @xcite . we stress that the mass parameter @xmath57 should not be interpreted as a gauge boson mass , nor do we limit ourselves to @xmath61 . the crucial feature of this potential is that for sufficiently large values of the dimensionless parameter @xmath62 , the potential leads to a @xmath6-symmetric , confining minimum for @xmath3 @xcite . on the other hand , for small values of @xmath62 , the pure gauge theory will be in the deconfined phase . it will be important later that @xmath63 is a good representation of the gauge boson contribution for high temperatures ; in other pnjl models , the gauge boson contribution has sometimes been chosen so as to be valid over a more narrow range of temperatures . as a test of all the components of the effective potential we have assembled , we consider the case of two flavors of fundamental fermions at finite temperature . a very common choice of zero - temperature parameters for two degenerate light flavors is @xmath64 @xmath65 and @xmath66 @xcite . we will use these parameters , augmented by the gluonic sector parameter @xmath60 discussed in the previous section . in figure [ fig : tvsops_fundabc ] , we show the constituent mass @xmath11 and polyakov expectation value @xmath67 as a function of temperature , normalized by dividing by their values at @xmath68 and @xmath69 , respectively . the behavior in the crossover region is very similar to the results of fukushima @xcite , and shows the explanatory power of pnjl models . the constituent mass @xmath11 is heavy at low temperatures , due to chiral symmetry breaking . the larger the constituent mass , the smaller the @xmath47 breaking effect of the fermions . on the other hand , a small value for @xmath48 reduces the effectiveness of finite - temperature effects in restoring chiral symmetry . these synergistic effects combine in the case of fundamental representation fermions to give a single crossover temperature at which both order parameters are changing rapidly , in agreement with lattice simulations . and @xmath67 for two - flavor qcd with adjoint representation fermions with antiperiodic boundary conditions as a function of temperature.,width=288 ] and @xmath67 for two - flavor qcd with adjoint representation fermions with antiperiodic boundary conditions as a function of temperature.,width=288 ] adjoint @xmath13 fermions at finite temperature show a completely different behavior in lattice simulations from fundamental fermions . because the adjoint fermions respect the @xmath47 center symmetry , there is a true deconfinement transition where @xmath47 spontaneously breaks . lattice simulations have shown that chiral symmetry is restored at a substantially higher temperature than the deconfinement temperature @xcite . the @xmath68 parameters needed are @xmath20 and @xmath34 . rather than work directly with @xmath20 , we will consider the dimensionless coupling @xmath70 . a given ratio of @xmath71 determines the value of @xmath72 , and vice versa . the value of @xmath34 is determined by the requirement that @xmath73 is near @xmath74 @xcite . this in turn determines the value of the constituent mass for all @xmath75 . the ratio @xmath71 should be less than one in order for the cutoff theory to be meaningful . in the case of fundamental fermions , this ratio is relatively large , on the order of @xmath76 . we have generally found that for adjoint fermions a larger ratio of @xmath71 with @xmath73 fixed implies a larger value of @xmath77 . we will work with the representative case of @xmath78 and @xmath79 . this gives @xmath80 and thus @xmath81 , with @xmath82 . for comparison , the critical value of @xmath72 , @xmath83 , below which @xmath84 , is @xmath85 . in figure [ fig : tvsops_adjabc ] , we show the constituent mass @xmath11 and polyakov expectation value @xmath67 as a function of temperature , normalized by dividing by their values at @xmath68 and @xmath69 , respectively . we see that the deconfinement temperature @xmath86 is very close to its value in the pure gauge theory , due to the large adjoint fermion constituent mass . the transition is first order . the constituent mass @xmath11 has a slow decline to a second - order transition at a substantially higher temperature , as indicated by lattice simulations @xcite . we consider the behavior of @xmath11 and @xmath87 with periodic fermions using the same parameters we used for the antiperiodic case . figure [ fig : tvsops_adjpbc ] shows the behavior of @xmath11 and @xmath88 as a function of @xmath89 for the @xmath90 parameter set , with @xmath91 . we see that chiral symmetry breaking persists at @xmath92 , which is much higher than the chiral restoration temperature for antiperiodic fermions . the constituent mass @xmath11 does fall eventually as @xmath89 increases , and chiral symmetry is ultimately restored , but at a temperature on the order of @xmath34 . in figure [ fig : tvsops_adjpbc ] , @xmath87 shows three distinct phase transitions as a function of @xmath89 . as @xmath89 increases , the confined phase gives way to the deconfined phase in a first - order phase transition . because the constituent mass of the fermions is large , the critical value of @xmath89 for this transition is approximately equal to @xmath86 . as @xmath89 increases , there are two more first - order transitions , from the deconfined phase to the skewed phase , and then from the skewed phase to a small-@xmath1 confined phase we describe as reconfined . the ordering of the phases seen in the behavior of @xmath87 for @xmath78 persists as @xmath22 is increased @xcite . in figure [ fig : phasediagram_fit ] , we show the phase diagram in the @xmath93 plane , obtained by numerically minimizing @xmath94 . for most values of @xmath72 larger than @xmath95 , the confined large-@xmath1 phase and the reconfined phase at small @xmath1 are separated by three phase transitions as in figure [ fig : tvsops_adjpbc ] . all of these transitions are characterized by abrupt changes in @xmath87 , while the chiral order parameter shows only a slow decrease with increasing temperature . however , there is a narrow range of @xmath72 between approximately @xmath96 and @xmath97 where confinement holds at all temperatures , and chiral symmetry remains broken . in this extended phase diagram , the confined and reconfined regions are smoothly connected . although this connection appears only for small range of values , the corresponding range of constituent mass values is not necessarily small @xcite . our results bear directly on the recent work by cossu and delia @xcite , in which they performed lattice simulations of two - flavor @xmath13 gauge theory with periodic adjoint fermions . -@xmath72 plane . c , d , and s refer to the confined , deconfined , and skewed phase , respectively.,width=288 ] -@xmath72 plane . c , d , and s refer to the confined , deconfined , and skewed phase , respectively.,width=288 ] we have extended the pnjl treatment of @xmath13 gauge theories to the case of two adjoint fermions with periodic boundary conditions on @xmath0 . our simple model reproduces the known successes of pnjl models for fundamental fermions while at the same time reproducing the expected behavior at high temperatures needed with adjoint fermions . the large separation between the deconfinement transition and the chiral symmetry restoration transition for adjoint fermion theories with antiperiodic boundary conditions requires a pnjl model which reproduces the behavior of the pure gauge theory to much smaller values of @xmath1 than have been considered before . the results for our @xmath13 pnjl model with two flavors of periodic adjoint dirac fermions can be summarized in the phase diagram in figure [ fig : phasediagram_fit ] . they are completely compatible with the lattice simulations of cossu and delia @xcite . if @xmath22 is set to zero , there is a small region in the @xmath93 plane , lying above @xmath95 , that connects the large-@xmath1 and small-@xmath1 confined regions . because the largest contribution to the constituent mass @xmath11 is from chiral symmetry breaking , this behavior will persist for some small range of nonzero @xmath22 . thus there is a single confining region , accessible in principle in lattice simulations . 9 nishimura h and ogilvie m c 2010 _ phys . _ d * 81 * 014018 cossu g and delia m 2009 _ j. high energy phys . _ jhep07(2009)048 unsal m 2008 _ phys . rev . lett . _ * 100 * 032005 myers j c and ogilvie m c 2008 _ phys . _ d * 77 * 125030 gross d j , pisarski r d and yaffe l g 1981 _ rev . phys . _ * 53 * 43 weiss n 1981 _ phys . _ d * 24 * 475 myers j c and ogilvie m c 2009 _ j. high energy phys . _ jhep07(2009)095 gocksch a and ogilvie m 1985 _ phys . _ d * 31 * 877 fukushima k 2004 _ phys . _ b * 591 * 277 klevansky s p 1992 _ rev . phys . _ * 64 * 649 hatsuda t and kunihiro t 1994 _ phys . rept . _ * 247 * 221 meisinger p n and ogilvie m c 2002 _ phys . rev . _ d * 65 * 056013 meisinger p n , miller t r and ogilvie m c 2002 _ phys . _ d * 65 * 034009 meisinger p n and ogilvie m c 2010 _ phys . _ d * 81 * 025012 karsch f and lutgemeier m 1999 _ nucl . _ b * 550 * 449 engels j , holtmann s and schulze t 2005 _ nucl . _ b * 724 * 357

recent work on qcd - like theories has shown that the addition of adjoint fermions obeying periodic boundary conditions to gauge theories on @xmath0 can lead to a restoration of center symmetry and confinement for sufficiently small circumference @xmath1 of @xmath2 . at small @xmath1 , perturbation theory may be used reliably to compute the effective potential for the polyakov loop @xmath3 in the compact direction . periodic adjoint fermions act in opposition to the gauge fields , which by themselves would lead to a deconfined phase at small @xmath1 . in order for the fermionic effects to dominate gauge field effects in the effective potential , the fermion mass must be sufficiently small . this indicates that chiral symmetry breaking effects are potentially important . we develop a polyakov - nambu - jona lasinio ( pnjl ) model which combines the known perturbative behavior of adjoint qcd models at small @xmath1 with chiral symmetry breaking effects to produce an effective potential for the polyakov loop @xmath3 and the chiral order parameter @xmath4 . a rich phase structure emerges from the effective potential . our results @xcite are consistent with the recent lattice simulations of cossu and delia @xcite , which found no evidence for a direct connection between the small-@xmath1 and large-@xmath1 confining regions . nevertheless , the two confined regions are connected indirectly if an extended field theory model with an irrelevant four - fermion interaction is considered . thus the small-@xmath1 and large-@xmath1 regions are part of a single confined phase .

one of the mathematical challenges encountered in the study of systems exhibiting phase coexistence is an efficient description of microscopic phase boundaries . here various levels of detail are in general possible : the finest level is typically associated with a statistical - mechanical model ( e.g. , a lattice gas ) in which both the interface and the surrounding phases are represented microscopically ; at the coarsest level the interface is viewed as a macroscopic ( geometrical ) surface between two structureless bulk phases . an intermediate approach is based on effective ( and , often , solid - on - solid ) models , in which the interface is still microscopic represented by a stochastic field while the structural details of the bulk phases are neglected . a simple example of such an effective model is a _ gradient field_. to define this system , we consider a finite subset @xmath12 of the @xmath13-dimensional hypercubic lattice @xmath14 and , at each site of @xmath12 and its external boundary @xmath15 , we consider the real - valued variable @xmath16 representing the height of the interface at @xmath17 . the hamiltonian is then given by @xmath18 where the sum is over unordered nearest - neighbor pairs @xmath19 . a standard example is the quadratic potential @xmath20 with @xmath21 ; in general @xmath5 is assumed to be a smooth , even function with a sufficient ( say , quadratic ) growth at infinity . the gibbs measure takes the usual form @xmath22 where @xmath23 is the @xmath24-dimensional lebesgue measure ( the boundary values of @xmath25 remain fixed and implicit in the notation ) , @xmath6 is the inverse temperature and @xmath26 is a normalization constant . a natural question to ask is what are the possible limits of the gibbs measures @xmath27 as @xmath28 . unfortunately , in dimensions @xmath29 , the fields @xmath30 are very `` rough '' no matter how tempered the boundary conditions are assumed to be . as a consequence , the family of measures @xmath31 is not tight and no meaningful object is obtained by taking the limit @xmath28i.e . , the interface is _ delocalized_. on the other hand , in dimensions @xmath32 the fields are sufficiently smooth to permit a non - trivial thermodynamic limit the interface is _ localized_. these facts are established by combinations of brascamp - lieb inequality techniques and/or random walk representation ( see , e.g. , @xcite ) which , unfortunately , apply only for convex potentials with uniformly positive curvature . thus , somewhat surprisingly , even for @xmath33 the problem of localization in high - dimension is still open ( * ? ? ? * open problem 1 ) . as it turns out , the thermodynamic limit of the measures @xmath34 is significantly less singular once we restrict attention to the gradient variables @xmath0 . these are defined by @xmath35 where @xmath1 is the nearest - neighbor edge @xmath36 oriented in one of the positive lattice directions . indeed , the @xmath37-marginal of @xmath27 always has at least one ( weak ) limit `` point '' as @xmath38 . the limit measures satisfy a natural dlr condition and are therefore called _ gradient gibbs measures_. ( precise definitions will be stated below or can be found in @xcite . ) one non - standard aspect of the gradient variables is that they have to obey a host of constraints . namely , @xmath39 holds for each lattice plaquette @xmath9 , where the edges @xmath40 are listed counterclockwise and are assumed to be positively oriented . these constraints will be implemented at the level of _ a priori _ measure , see sect . [ s : model ] . it would be natural to expect that the character ( and number ) of gradient gibbs measures depends sensitively on the potential @xmath5 . however , this is not the case for the class of uniformly strictly - convex potentials ( i.e. , the @xmath5 s such that @xmath41 for all @xmath37 ) . indeed , funaki and spohn @xcite showed that , in these cases , the translation - invariant , ergodic , gradient gibbs measures are completely characterized by the _ tilt _ of the underlying interface . here the tilt is a vector @xmath42 such that @xmath43 for every edge @xmath1which we regard as a vector in @xmath44 . furthermore , the correspondence is one - to - one , i.e. , for each tilt there exists precisely one gradient gibbs measure with this tilt . alternative proofs permitting extensions to discrete gradient models have appeared in sheffield s thesis @xcite . it is natural to expect that a serious violation of the strict - convexity assumption on @xmath5 may invalidate the above results . actually , an example of a gradient model with multiple gradient gibbs states of the same tilt has recently been presented @xcite ; unfortunately , the example is not of the type considered above because of the lack of translation invariance and its reliance on the discreteness of the fields . the goal of this paper is to point out a general mechanism by which the model with a sufficiently non - convex potential @xmath5 fails the conclusions of funaki - spohn s theorems . the mechanism driving our example will be the occurrence of a structural surface phase transition . to motivate the forthcoming considerations , let us recall that phase transitions typically arise via one of two mechanisms : either due to the breakdown of an internal symmetry , or via an abrupt turnover between energetically and entropically favored states . the standard examples of systems with these kinds of phase transitions are the ising model and the @xmath45-state potts model with a sufficiently large @xmath45 , respectively . in the former , at sufficiently low temperatures , there is a spontaneous breaking of the symmetry between the plus and minus spin states ; in the latter , there is a first - order transition at intermediate temperatures between @xmath45 ordered , low - temperature states and a disordered , high - temperature state . our goal is to come up with a potential @xmath5 that would mimic one of the above situations . in the present context an analogue of the ising model appears to be a _ double - well potential _ of the form , e.g. , @xmath46 unfortunately , due to the underlying plaquette constraints , the symmetry between the wells can not be completely broken and , even at the level of ground states , the system appears to be disordered . on @xmath2 this can be demonstrated explicitly by making a link to the _ ice model _ , which is a special case of the six vertex model @xcite . a similar equivalence has been used @xcite to study a roughening transition in an sos interface . to see how the equivalence works exactly , note that the ground states of the system are such that all @xmath37 s equal @xmath47 . let us associate a unit flow with each _ dual _ bond whose sign is determined by the value of @xmath48 for its direct counterpart @xmath1 . the plaquette constraint then translates into a _ no - source - no - sink _ condition for this flow . if we mark the flow by arrows , the dual bonds at each plaquette are constrained to one of six zero - flux arrangements of the six vertex model ; cf fig . [ fig1 ] and its caption . the weights of all zero - flux arrangements are equal ; we thus have the special case corresponding to the ice model . the ice model can be `` exactly solved '' @xcite : the ground states have a non - vanishing residual entropy @xcite and are disordered with infinite correlation length ( * ? ? ? 8.10.iii ) . however , it is not clear how much of this picture survives to positive temperatures . the previous discussion shows that it will be probably quite hard to realize a symmetry - breaking transition in the context of the gradient model . it is the order - disorder mechanism for phase transitions that seems considerably more promising . there are two canonical examples of interest : a potential with _ two centered wells _ and a _ triple - well potential _ ; see fig . [ fig2 ] . both of these lead to a gradient model which features a phase transition , at some intermediate temperature , from states with the @xmath37 s lying ( mostly ) within the thinner well to states whose @xmath37 s fluctuate on the scale of the thicker well(s ) . our techniques apply equally to these as well as other similar cases provided the widths of the wells are sufficiently distinct . notwithstanding , the analysis becomes significantly cleaner if we abandon temperature as our principal parameter ( e.g. , we set @xmath49 ) and consider potentials @xmath5 that are simply _ defined _ by @xmath50 here @xmath51 and @xmath52 are positive numbers and @xmath53 is a parameter taking values in @xmath54 $ ] . for appropriate values of the constants , @xmath5 defined this way will have a graph as in fig . [ fig2](a ) . to get the graph in part ( b ) , we would need to consider @xmath5 s of the form @xmath55 where @xmath47 are the ( approximate ) locations of the off - center wells . the idea underlying the expressions and is similar to that of the fortuin - kasteleyn representation of the potts model @xcite . in the context of continuous - spin models similar to ours , such representation has fruitfully been used by zahradnk @xcite . focusing on , we can interpret the terms on the right - hand side of as two distinct states of each bond . ( we will soon exploit this interpretation in detail . ) the indexing of the coupling constants suggests the names : `` o '' for _ ordered _ and `` d '' for _ it is clear that the extreme values of @xmath53 ( near zero or near one ) will be dominated by one type of bonds ; what we intend to show is that , for @xmath51 and @xmath52 sufficiently distinct from each other , the transition between the `` ordered '' and `` disordered '' phases is ( strongly ) first order . similar conclusions and proofs albeit more complicated apply also to the potential . however , for clarity of exposition , we will focus on the potential for the rest of the paper ( see , however , sect . [ sec2.5 ] ) . in addition , we will also restrict ourselves to two dimensions , even though the majority of our results are valid for all @xmath56 . we commence with a precise definition of our model . most of the work in this paper will be confined to the lattice torus @xmath57 of @xmath58 sites in @xmath2 , so we will start with this particular geometry . choosing the natural positive direction for each lattice axis , let @xmath59 denote the corresponding set of positively oriented edges in @xmath57 . given a configuration @xmath60 , we introduce the gradient field @xmath61 by assigning the variable @xmath35 to each @xmath62 . the product lebesgue measure @xmath63 induces a ( @xmath64-finite ) measure @xmath65 on the space @xmath66 via @xmath67 where @xmath68 denotes the dirac point - mass at zero . we interpret the measure @xmath65 as an _ a priori _ measure on _ gradient _ configurations @xmath69 . since the @xmath37 s arise as the gradients of the @xmath25 s it is easy to check that @xmath65 is entirely supported on the linear subspace @xmath70 of configurations determined by the condition that the sum of signed @xmath37s with a positive or negative sign depending on whether the edge is traversed in the positive or negative direction , respectively vanishes around each closed circuit on @xmath57 . ( note that , in addition to , the condition includes also loops that wrap around the torus . ) we will refer to such configurations as _ curl - free_. next we will define gradient gibbs measures on @xmath57 . for later convenience we will proceed in some more generality than presently needed : let @xmath71 be a collection of measurable functions @xmath72 and consider the partition function @xmath73 clearly , @xmath74 and , under the condition that @xmath75 is integrable with respect to the lebesgue measure on @xmath76 , also @xmath77 . we may then define @xmath78 to be the probability measure on @xmath79 given by @xmath80 this is the _ gradient gibbs measure _ on @xmath57 corresponding to the potentials @xmath81 . in the situations when @xmath82 for all @xmath1which is the principal case of interest in this paper we will denote the corresponding gradient gibbs measure on @xmath57 by @xmath83 . it is not surprising that @xmath78 obeys appropriate dlr equations with respect to all connected @xmath84 containing no topologically non - trivial circuit . explicitly , if @xmath85 in @xmath86 is a curl - free boundary condition , then the conditional law of @xmath87 given @xmath85 is @xmath88 here @xmath89 is the conditional probability with respect to the ( tail ) @xmath64-algebra @xmath90 generated by the fields on @xmath86 , @xmath91 is the partition function in @xmath12 , and @xmath92 is the _ a priori _ measure induced by @xmath65 on @xmath87 given the boundary condition @xmath85 . as usual , this property remains valid even in the thermodynamic limit . we thus say that a measure on @xmath93 is an _ infinite - volume gradient gibbs measure _ if it satisfies the dlr equations with respect to the specification in any finite set @xmath94 . ( as is easy to check e.g . , by reinterpreting the @xmath37 s back in terms of the @xmath25s@xmath92 is independent of the values of @xmath85 outside any circuit winding around @xmath12 , and so it is immaterial that it originated from a measure on torus . ) an important aspect of our derivations will be the fact that our potential @xmath5 takes the specific form , which can be concisely written as @xmath95 where @xmath96 is the probability measure @xmath97 . it follows that the gibbs measure @xmath83 can be regarded as the projection of the _ extended gradient gibbs measure _ , @xmath98 to the @xmath64-algebra generated by the @xmath37 s . here @xmath99 is the product of measures @xmath96 , one for each bond in @xmath59 . as is easy to check , conditioning on @xmath100 yields the corresponding extension @xmath101 of the finite - volume specification the result is independent of the @xmath102 s outside @xmath12 because , once @xmath85 is fixed , these have no effect on the configurations in @xmath12 . the main point of introducing the extended measure is that , if conditioned on the @xmath102 s , the variables @xmath48 are distributed as gradients of a gaussian field albeit with a non - translation invariant covariance matrix . as we will see , the phase transition proved in this paper is manifested by a jump - discontinuity in the density of bonds with @xmath103 which at the level of @xmath37-marginal results in a jump in the characteristic scale of the fluctuations . notably , the extended measure @xmath104 plays the same role for @xmath83 as the so called edwards - sokal coupling measure @xcite does for the potts model . similarly as for the edwards - sokal measures @xcite , there is a one - to - one correspondence between the infinite - volume measures on @xmath37 s and the corresponding infinite - volume extended gradient gibbs measures on @xmath105 s . explicitly , if @xmath93 is an infinite - volume gradient gibbs measure for potential @xmath5 , then @xmath106 , defined by ( extending the consistent family of measures of the form ) @xmath107 is a gibbs measure with respect to the extended specifications @xmath108 . for the situations with only a few distinct values of @xmath109 , it may be of independent interest to study the properties of the @xmath102-marginal of the extended measure , e.g. , using the techniques of percolation theory . however , apart from some remarks in sect . [ sec - diskuse ] , we will not pursue these matters in the present paper . now we are ready to state our main results . throughout we will consider the potentials @xmath5 of the form with @xmath110 . as a moment s thought reveals , the model is invariant under the transformation @xmath111 for any fixed @xmath112 . in particular , without loss of generality , one could assume from the beginning that @xmath113 and regard @xmath114 as the sole parameter of the model . however , we prefer to treat the two terms in on an equal footing , and so we will keep the coupling strengths independent . given a shift - ergodic gradient gibbs measure , recall that its tilt is the vector @xmath115 such that holds for each bond . the principal result of the present paper is the following theorem : [ t : main ] for each @xmath116 there exists a constant @xmath117 and , if @xmath118 a number @xmath119 such that , for interaction @xmath5 with @xmath120 , there are two distinct , infinite - volume , shift - ergodic gradient gibbs measures @xmath121 and @xmath122 of zero tilt for which @xmath123 and @xmath124 here @xmath125 is a constant of order unity . an inspection of the proof actually reveals that the above bounds are valid for any @xmath126 satisfying @xmath127 , where @xmath128 is a constant of order unity . as already alluded to , this result is a consequence of the fact that the density of ordered bonds , i.e. , those with @xmath103 , undergoes a jump at @xmath120 . on the torus , we can make the following asymptotic statements : [ t : torus ] let @xmath129 denote the fraction of ordered bonds on @xmath57 , i.e. , @xmath130 for each @xmath116 there exists @xmath117 such that the following holds : under the condition , and for @xmath131 as in theorem [ t : main ] , @xmath132 and @xmath133 the present setting actually permits us to determine the value of @xmath131 via a duality argument . this is the only result in this paper which is intrinsically two - dimensional ( and intrinsically tied to the form of @xmath5 ) . all other conclusions can be extended to @xmath56 and to more general potentials . [ t : dual ] let @xmath134 . if @xmath135 , then @xmath131 is given by @xmath136 theorem [ t : torus ] is proved in sect . [ sec4.2 ] , theorem [ t : main ] is proved in sect . [ sec4.3 ] and theorem [ t : dual ] is proved in sect . [ sec5.3 ] . the phase transition described in the above theorems can be interpreted in several ways . first , in terms of the extended gradient gibbs measures on torus , it clearly corresponds to a transition between a state with nearly all bonds ordered ( @xmath103 ) to a state with nearly all bonds disordered ( @xmath137 ) . second , looking back at the inequalities ( [ e : ener - bd1][e : ener - bd2 ] ) , most of the @xmath37 s will be of order at most @xmath138 in the ordered state while most of them will be of order at least @xmath139 in the disordered state . hence , the corresponding ( effective ) interface is significantly rougher at @xmath140 than it is at @xmath141 ( both phases are rough according to the standard definition of this term ) and we may thus interpret the above as a kind of _ first - order roughening _ transition that the interface undergoes at @xmath131 . finally , since the gradient fields in the two states fluctuate on different characteristic scales , the entropy ( and hence the energy ) associated with these states is different ; we can thus view this as a standard energy - entropy transition . ( by the energy we mean the expectation of @xmath142 ; notably , the expectation of @xmath143 is the same in both measures ; cf . ) energy - entropy transitions for spin models have been studied in @xcite and , quite recently , in @xcite . next let us turn our attention to the conclusions of theorem [ t : torus ] . we actually believe that the dichotomy ( [ below - pt][above - pt ] ) applies ( in the sense of almost - sure limit of @xmath129 as @xmath144 ) to all translation - invariant extended gradient gibbs states with zero tilt . the reason is that , conditional on the @xmath102 s , the gradient fields are gaussian with uniformly positive stiffness . we rest assured that the techniques of @xcite and @xcite can be used to prove that the gradient gibbs measure with zero tilt is unique for almost every configuration of the @xmath102 s ; so the only reason for multiplicity of gradient gibbs measures with zero tilt is a phase transition in the @xmath102-marginal . however , a detailed write - up of this argument would require developing the precise and somewhat subtle correspondence between the gradient gibbs measures of a given tilt and the minimizers of the gibbs variational principle ( which we have , in full detail , only for convex periodic potentials @xcite ) . thus , to keep the paper at manageable length , we limit ourselves to a weaker result . the fact that the transition occurs at @xmath131 satisfying is a consequence of a _ duality _ between the @xmath102-marginals at @xmath53 and @xmath145 . more generally , the duality links the marginal law of the configuration @xmath146 with the law of @xmath147 ; see theorem [ t : dualita ] and remark [ remark : mira - dualita ] . [ at the level of gradient fields , the duality provides only a vague link between the flow of the weighted gradients @xmath148 along a given curve and its flux through this curve . unfortunately , this link does not seem to be particularly useful . ] the point @xmath120 is self - dual which makes it the most natural candidate for a transition point . it is interesting to ponder about what happens when @xmath114 decreases to one . presumably , the first - order transition ( for states at zero tilt ) disappears before @xmath114 reaches one and is replaced by some sort of critical behavior . here the first problem to tackle is to establish the _ absence _ of first - order phase transition for small @xmath149 . via a standard duality argument ( see @xcite ) this would yield a power - law lower bound for bond connectivities at @xmath131 . another interesting problem is to determine what happens with measures of non - zero tilt . we expect that , at least for moderate values of the tilt @xmath115 , the first - order transition persists but shifts to lower values of @xmath53 . thus , one could envision a whole phase diagram in the @xmath53-@xmath115 plane . unfortunately , we are unable to make any statements of this kind because the standard ways to induce a tilt on the torus ( cf @xcite ) lead to measures that are not reflection positive . we proceed by an outline of the principal steps of the proof to which the remainder of this paper is devoted . the arguments are close in spirit to those in @xcite ; the differences arise from the subtleties in the setup due to the gradient nature of the fields . the main line of reasoning is basically thermodynamical : consider the @xmath102-marginal of the extended torus state @xmath104 which we will regard as a measure on configurations of ordered and disordered bonds . let @xmath150 denote ( the @xmath144 limit of ) the expected fraction of ordered bonds in the torus state at parameter @xmath53 . clearly @xmath150 increases from zero to one as @xmath53 sweeps through @xmath54 $ ] . the principal observation is that , under the assumption @xmath135 , the quantity @xmath151 is small , uniformly in @xmath53 . hence , @xmath152 must undergo a jump from values near zero to values near one at some @xmath119 . by usual weak - limiting arguments we construct two distinct gradient gibbs measures at @xmath131 , one with high density of ordered bonds and the other with high density of disordered bonds . the crux of the matter is thus to justify the uniform smallness of @xmath151 . this will be a consequence of the fact that the simultaneous occurrence of ordered and disordered bonds at any two given locations is ( uniformly ) unlikely . for instance , let us estimate the probability that a particular plaquette has two ordered bonds emanating out of one corner and two disordered bonds emanating out of the other . here the technique of chessboard estimates @xcite allows us to disseminate this pattern all over the torus via successive reflections ( cf theorem [ t : chess ] in sect . [ s : chessboard ] ) . this bounds the quantity of interest by the @xmath153-power of the probability that every other horizontal ( and vertical ) line is entirely ordered and the remaining lines are disordered . the resulting `` spin - wave calculation''i.e . , diagonalization of a period-2 covariance matrix in the fourier basis and taking its determinant is performed ( for all needed patterns ) in sect . [ s : spin - wave ] . once the occurrence of a `` bad pattern '' is estimated by means of various spin - wave free energies , we need to prove that these `` bad - pattern '' spin - wave free energies are always worse off than those of the homogeneous patterns ( i.e. , all ordered or all disordered)this is the content of theorem [ t : min ] . then we run a standard peierls contour estimate whereby the smallness of @xmath151 follows . extracting two distinct , infinite - volume , ergodic gradient gibbs states @xmath121 and @xmath122 at @xmath120 , it remains to show that these are both of zero tilt . here we use the fact that , conditional on the @xmath102 s , the torus measure is symmetric gaussian with uniformly positive stiffness . hence , we can use standard gaussian inequalities to show exponential tightness of the tilt , uniformly in the @xmath102 s ; cf lemma [ l : tilt ] . duality calculations ( see sect . [ s : duality ] ) then yield @xmath120 . our proof of phase coexistence applies to any potential of the form shown in fig . [ fig2]even if we return to parametrization by @xmath10 . the difference with respect to the present setup is that in the general case we would have to approximate the potentials by a quadratic well at each local minimum and , before performing the requisite gaussian calculations , estimate the resulting errors . here is a sketch of the main ideas : we fix a scale @xmath154 and regard @xmath48 to be in a well if it is within @xmath154 of the corresponding local minimum . then the requisite quadratic approximation of @xmath10-times energy is good up to errors of order @xmath155 . the rest of the potential `` landscape '' lies at energies of at least order @xmath156 and so it will be only `` rarely visited '' by the @xmath37 s provided that @xmath157 . on the other hand , the same condition ensures that the spin - wave integrals are essentially not influenced by the restriction that @xmath48 be within @xmath154 of the local minimum . thus , to make all approximations work we need that @xmath158 which is achieved for @xmath159 by , e.g. , @xmath160 . this approach has recently been used to prove phase transitions in classical @xcite as well as quantum @xcite systems with highly degenerate ground states . we refer the reader to these references for further details . a somewhat more delicate issue is the proof that both coexisting states are of zero tilt . here the existing techniques require that we have some sort of uniform convexity . this more or less forces us to use the @xmath5 s of the form @xmath161 where the @xmath162 s are uniformly convex functions . clearly , our choice is the simplest potential of this type ; the question is how general the potentials obtained this way can be . we hope to return to this question in a future publication . as was just mentioned , the core of our proofs are estimates of the spin - wave free energy for various regular patterns of ordered and disordered bonds on the torus . these estimates are rather technical and so we prefer to clear them out of the way before we get to the main line of the proof . the readers wishing to follow the proof in linear order may consider skipping this section and returning to it only while reading the arguments in sect . [ sec4.2 ] . throughout this and the forthcoming sections we assume that @xmath163 is an even integer . we will consider six partition functions @xmath164 , @xmath165 , @xmath166 , @xmath167 , @xmath168 and @xmath169 on @xmath57 that correspond to six regular configurations each of which is obtained by reflecting one of six possible arrangements of `` ordered '' and `` disordered '' bonds around a lattice plaquette to the entire torus . these quantities will be the `` building blocks '' of our analysis in sect . [ sec : pt ] . the six plaquette configurations are depicted in fig . [ fig3 ] . we begin by considering the homogeneous configurations . here @xmath164 is the partition function @xmath170 for all edges of the `` ordered '' type : @xmath171 similarly , @xmath165 is the quantity @xmath170 for @xmath172 i.e. , with all edges `` disordered . '' next we will define the partition functions @xmath166 and @xmath167 which are obtained by reflecting a plaquette with three bonds of one type and the remaining bond of the other type . let us split @xmath59 into the even @xmath173 and odd @xmath174 horizontal and vertical edges with the even edges on the lines of sites in the @xmath17 direction with even @xmath175 coordinates and lines of sites in @xmath175 direction with even @xmath17 coordinates . similarly , we will also consider the decomposition of @xmath59 into the set of horizontal edges @xmath176 and vertical edges @xmath177 . letting @xmath178 the partition function @xmath166 then corresponds to the quantity @xmath170 . the partition function @xmath167 is obtained similarly ; with the roles of `` ordered '' and `` disordered '' interchanged . note that , since we are working on a square torus , the orientation of the pattern we choose does not matter . it remains to define the partition functions @xmath168 and @xmath169 corresponding to the patterns with two `` ordered '' and two `` disordered '' bonds . for the former , we simply take @xmath170 with the potential @xmath179 note that the two types of bonds are arranged in a `` mixed periodic '' pattern ; hence the index @xmath180 . as to the quantity @xmath169 , here we will consider a `` mixed aperiodic '' pattern . explicitly , we define @xmath181 the `` mixed aperiodic '' partition function @xmath169 is the quantity @xmath170 for this choice of @xmath81 . again , on a square torus it is immaterial for the values of @xmath168 and @xmath169 which orientation of the initial plaquette we start with . as usual , associated with these partition functions are the corresponding free energies . in finite volume , these quantities can be defined in all cases by the formula @xmath182 where the factor @xmath183 has been added for later convenience and where the @xmath53-dependence arises via the corresponding formulas for @xmath184 in each particular case . the goal of this section is to compute the thermodynamic limit of the @xmath185 s . for homogeneous and isotropic configurations , an important role will be played by the momentum representation of the lattice laplacian @xmath186 defined for all @xmath187 in the corresponding brillouin zone @xmath188\times[-\pi,\pi]$ ] . using this quantity , the `` ordered '' free energy will be simply @xmath189 ^ 2}\frac{{\text{\rm d}\mkern0.5mu}{\boldsymbol k}}{(2\pi)^2}\log\bigl\{{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } \widehat d({\boldsymbol k})\bigr\},\ ] ] while the disordered free energy boils down to @xmath190 ^ 2}\frac{{\text{\rm d}\mkern0.5mu}{\boldsymbol k}}{(2\pi)^2}\log\bigl\{{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } } } \widehat d({\boldsymbol k})\bigr\}.\ ] ] it is easy to check that , despite the logarithmic singularity at @xmath191 , both integrals converge . the bond pattern underlying the quantity @xmath168 lacks rotation invariance and so a different propagator appears inside the momentum integral : @xmath192\\ + \,\frac12\int_{[-\pi,\pi]^2}\frac{{\text{\rm d}\mkern0.5mu}{\boldsymbol k}}{(2\pi)^2}\log\bigl\{{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } |1- { \text{\rm e}\mkern0.7mu}^{{\text{\rm i}\mkern0.7mu}k_1}|^2+{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } } } |1-{\text{\rm e}\mkern0.7mu}^{{\text{\rm i}\mkern0.7mu}k_2}|^2\bigr\}.\end{gathered}\ ] ] again , the integral converges as long as ( at least ) one of @xmath51 and @xmath52 is strictly positive . the remaining partition functions come from configurations that lack translation invariance and are `` only '' periodic with period two . consequently , the fourier transform of the corresponding propagator is only block diagonal , with two or four different @xmath193 s `` mixed '' inside each block . in the @xmath194 cases we will get the function @xmath195 \,+\,\frac14\int_{[-\pi,\pi]^2}\frac{{\text{\rm d}\mkern0.5mu}{\boldsymbol k}}{(2\pi)^2}\log\bigl\{\det\pi_{{\text{\rm uo}}}({\boldsymbol k})\bigr\},\ ] ] where @xmath196 is the @xmath197-matrix @xmath198 \frac12({{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } -{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } } } ) & { { \mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } |a_+|^2+\frac12({{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } + { { \mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } } } ) \end{matrix } \,\right)\ ] ] with @xmath199 and @xmath200 defined by @xmath201 the extra factor @xmath202on top of the usual @xmath202in front of the integral arises because @xmath203 combines the contributions of two fourier models ; namely @xmath193 and @xmath204 . a calculation shows @xmath205 implying that the integral in converges . the free energy @xmath206 is obtained by interchanging the roles of @xmath51 and @xmath52 and of @xmath53 and @xmath207 . in the ma - cases we will assume that @xmath208otherwise there is no distinction between any of the six patterns . the corresponding free energy is then given by @xmath209 \\+\,\frac18\int_{[-\pi,\pi]^2}\frac{{\text{\rm d}\mkern0.5mu}{\boldsymbol k}}{(2\pi)^2}\log\biggl\{\bigl(\frac{{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } -{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } } } } 2\bigr)^4\det\pi_{{\text{\rm ma}}}({\boldsymbol k})\biggr\}.\end{gathered}\ ] ] here @xmath210 is the @xmath211-matrix @xmath212 \\*[1 mm ] \\*[1 mm ] 0 & |a_+|^2 & |b_+|^2 & \!\!\!{r}(|a_+|^2+|b_+|^2 ) \end{matrix } \,\right)\ ] ] with the abbreviation @xmath213 note that @xmath214 in the cases of our interest . observe that @xmath215 is a quadratic polynomial in @xmath216 , i.e. , @xmath217 . moreover , @xmath218 annihilates @xmath219 when @xmath220 , and so @xmath221 is a root of @xmath222 . hence @xmath223 , i.e. , @xmath224 setting @xmath220 inside the large braces yields @xmath225 implying that the integral in is well defined and finite . the fact that @xmath218 has zero eigenvalue at @xmath220 is not surprising . indeed , @xmath220 corresponds to @xmath226 in which case a quarter of all sites in the @xmath227-pattern get decoupled from the rest . this indicates that the partition function blows up ( at least ) as @xmath228 as @xmath229 implying that there should be a zero eigenvalue at @xmath220 per each @xmath211-block @xmath218 . a formal connection between the quantities in and those in ( [ e : fs][e : fma ] ) is guaranteed by the following result : [ t : fe ] for all @xmath230 and uniformly in @xmath231 , @xmath232 proof this is a result of standard calculations of gaussian integrals in momentum representation . we begin by noting that the lebesgue measure @xmath233 can be regarded as the product of @xmath65 , acting only on the gradients of @xmath25 , and @xmath234 for some fixed @xmath235 . neglecting temporarily the _ a priori _ bond weights @xmath53 and @xmath207 , the partition function @xmath236 , @xmath230 , is thus the integral of the gaussian weight @xmath237 against the measure @xmath233 , where the covariance matrix @xmath238 is defined by the quadratic form @xmath239 here @xmath240 are the bond weights of pattern @xmath241 . indeed , the integral over @xmath234 with the gradient variables fixed yields @xmath242 which cancels the term in front of the gaussian weight . the purpose of the above rewrite was to reinsert the `` zero mode '' @xmath243 into the partition function ; @xmath244 was not subject to integration due to the restriction to gradient variables . to compute the gaussian integral , we need to diagonalize @xmath238 . for that we will pass to the fourier components @xmath245 with the result @xmath246 where @xmath247 is the reciprocal torus , @xmath248 is the kronecker delta and @xmath249 now if the horizontal part of @xmath240 is translation invariant in the @xmath250-th direction , then @xmath251 whenever @xmath252 , while if it is `` only '' 2-periodic , then @xmath251 unless @xmath253 or @xmath254 . similar statements apply to the vertical part of @xmath240 and @xmath255 . since all of our partition functions come from 2-periodic configurations , the covariance matrix can be cast into a block - diagonal form , with @xmath211 blocks @xmath256 collecting all matrix elements that involve the momenta @xmath257 . due to the reinsertion of the `` zero mode''cf all of these blocks are non - singular ( see also the explicit calculations below ) . hence we get that , for all @xmath258 , @xmath259^{{\mathchoice { \myffrac{1}{8 } in \scriptstyle } { \myffrac{1}{8 } in \scriptstyle } { \myffrac{1}{8 } in \scriptscriptstyle } { \myffrac{1}{8 } in \scriptscriptstyle } } } , \ ] ] where @xmath260 and @xmath261 denote the numbers of ordered and disordered bonds in the underlying bond configuration and where the exponent @xmath262 takes care of the fact that in the product , each @xmath193 gets involved in _ four _ distinct terms . taking logarithms and dividing by @xmath263 , the sum over the reciprocal torus converges to a riemann integral over the brillouin zone @xmath264\times[-\pi,\pi]$ ] ( the integrand has only logarithmic singularities in all cases , which are harmless for this limit ) . it remains to justify the explicit form of the free energies in all cases under considerations . here the situations @xmath265 are fairly standard , so we will focus on @xmath266 and @xmath267 for which some non - trivial calculations are needed . in the former case we get that @xmath268 with @xmath269 for all values that are not of this type . plugging into we find that the @xmath270-subblock of @xmath271 reduces essentially to the @xmath197-matrix in . explicitly , @xmath272 since @xmath273 whenever @xmath274 , the block matrix @xmath271 will only be a function of moduli - squared of @xmath199 and @xmath275 . using in we get . as to the @xmath227-case the only non - zero elements of @xmath276 are @xmath277 so , again , @xmath273 whenever @xmath274 and so @xmath278 depends only on @xmath279 and @xmath280 . an explicit calculation shows that @xmath281 where @xmath218 is as in . plugging into , we get . next we establish the crucial fact that the spin - wave free energies corresponding to inhomogeneous patterns @xmath282 exceed the smaller of @xmath283 and @xmath284 by a quantity that is large , independent of @xmath53 , once @xmath110 . [ t : min ] there exists @xmath285 such that if @xmath286 with @xmath287 , then for all @xmath231 , @xmath288 let us use @xmath289 and @xmath290 to denote the integrals @xmath291 ^ 2}\frac{{\text{\rm d}\mkern0.5mu}{\boldsymbol k}}{(2\pi)^2}\log\bigl\{\widehat d({\boldsymbol k})\bigr\ } , \quad\text{and}\quad j=\int_{[-\pi,\pi]^2}\frac{{\text{\rm d}\mkern0.5mu}{\boldsymbol k}}{(2\pi)^2}\log\bigl| a_-\bigr|.\ ] ] we will prove with @xmath292 . first , we have @xmath293 and @xmath294 while an inspection of yields @xmath295 \\ + \frac18\int_{[-\pi,\pi]^2}\frac{{\text{\rm d}\mkern0.5mu}{\boldsymbol k}}{(2\pi)^2}\log\bigl\{{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } { { \mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } } } ( { { \mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } -{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } } } ) ^2 \bigl|a_+ a_- b_+ b_-\bigr|^2\bigr\ } \\\ge -\log\bigl[p(1-p)\bigr]+\frac38{\log { { \mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } } + \frac18\log{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } } } + \frac14\log(1-\xi ) + j.\end{gathered}\ ] ] using that @xmath296 we thus get @xmath297 which agrees with for our choice of @xmath125 . coming to the free energy @xmath298 , using we evaluate @xmath299 yielding @xmath300+\frac18\log\frac{{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } } { { { \mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } } } } + \frac38{\log { { \mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } } + \frac18{\log { { \mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont d } } } } } } + j.\ ] ] bounding @xmath301 we thus get @xmath302 in agreement with . the computation for @xmath206 is completely analogous , interchanging only the roles of @xmath51 and @xmath52 as well as @xmath53 and @xmath207 . from the lower bound @xmath303 and the inequality @xmath304 we get again @xmath305 which is identical to . finally , for the free energy @xmath306 , we first note that @xmath307 + \frac12\log{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } + j,\ ] ] which yields @xmath308 under the condition that @xmath309 , we again get . for the complementary values of @xmath53 , we will compare @xmath306 with @xmath283 : @xmath310 since we now have @xmath311 , this yields with the above choice of @xmath125 . in this section we will apply the calculations from the previous section to the proof of theorems [ t : main ] and [ t : torus ] . throughout this section we assume that @xmath312 and that @xmath163 is even . we begin with a review of the technique of chessboard estimates which , for later convenience , we formulate directly in terms of extended configurations @xmath313 . our principal tool will be chessboard estimates , based on reflection positivity . to define these concepts , let us consider the torus @xmath57 and let us split @xmath57 into two symmetric halves , @xmath314 and @xmath315 , sharing a `` plane of sites '' on their boundary . we will refer to the set @xmath316 as _ plane of reflection _ and denote it by @xmath317 . the half - tori @xmath318 inherit the nearest - neighbor structure from @xmath57 ; we will use @xmath319 to denote the corresponding sets of edges . on the extended configuration space , there is a canonical map @xmath320induced by the reflection of @xmath314 into @xmath315 through @xmath317which is defined as follows : if @xmath321 are related via @xmath322 , then we put @xmath323 and @xmath324 here @xmath325 denotes that @xmath1 is orthogonal to @xmath53 while @xmath326 indicates that @xmath1 is parallel to @xmath317 . the minus sign in the case when @xmath325 is fairly natural if we recall that @xmath48 represents the difference of @xmath16 between the endpoints of @xmath317 . this difference changes sign under reflection through @xmath317 if @xmath325 and does not if @xmath326 . let @xmath327 be the @xmath64-algebras of events that depend only on the portion of @xmath328-configuration on @xmath319 ; explicitly @xmath329 . reflection positivity is , in its essence , a bound on the correlation between events ( and random variables ) from @xmath330 and @xmath331 . the precise definition is as follows : [ d : rp ] let @xmath332 be a probability measure on configurations @xmath333 and let @xmath334 be the corresponding expectation . we say that @xmath332 is _ reflection positive _ if for any plane of reflection @xmath317 and any two bounded @xmath330-measurable random variables @xmath335 and @xmath336 the following inequalities hold : @xmath337 and @xmath338 here , @xmath339 denotes the random variable @xmath340 . next we will discuss how reflection positivity underlines our principal technical tool : chessboard estimates . consider an event @xmath341 that depends only on the @xmath328-configurations on the plaquette with the lower - left corner at the torus origin . we will call such an @xmath341 a _ plaquette event_. for each @xmath342 , we define @xmath343 to be the event depending only on the configuration on the plaquette with the lower - left corner at @xmath17 which is obtained from @xmath341 as follows : if both components of @xmath17 are even , then @xmath343 is simply the translate of @xmath341 by @xmath17 . in the remaining cases we first reflect @xmath341 along the side(s ) of the plaquette in the direction(s ) where the component of @xmath17 is odd , and then translate the resulting event appropriately . ( thus , there are four possible `` versions '' of @xmath343 , depending on the parity of @xmath17 . ) here is the desired consequence of reflection positivity : [ t : chess ] let @xmath332 be a reflection - positive measure on configurations @xmath333 . then for any plaquette events @xmath344 and any distinct sites @xmath345 , @xmath346 see ( * ? ? ? * theorem 2.2 ) . the moral of this result whose proof boils down to the cauchy - schwarz inequality for the inner product @xmath347is that the probability of any number of plaquette events factorizes , as a bound , into the product of probabilities . this is particularly useful for contour estimates ( of course , provided that the word contour refers to a collection of plaquettes on each of which some `` bad '' event occurs ) . indeed , by the probability of a contour will be suppressed exponentially in the number of constituting plaquettes . in light of , our estimates will require good bounds on probabilities of the so called _ disseminated events _ @xmath348 . unfortunately , the event @xmath341 is often a conglomerate of several , more elementary events which makes a direct estimate of @xmath348 complicated . here the following subadditivity property will turn out to be useful . [ l : sub ] suppose that @xmath332 is a reflection - positive measure and let @xmath349 and @xmath341 be plaquette events such that @xmath350 . then @xmath351 this is lemma 6.3 of @xcite . apart from the above reflections , which we will call _ direct _ , one estimate namely in the proof of theorem [ t : main ] requires the use of so called _ diagonal reflections_. assuming @xmath163 is even , these are reflections in the planes @xmath317 of sites of the form @xmath352 here @xmath17 is a site that the plane passes through and @xmath353 and @xmath354 are the unit vectors in the @xmath17 and @xmath175-coordinate directions . as before , the plane has two components one corresponding to @xmath355 and the other corresponding to @xmath356and it divides @xmath57 into two equal parts . this puts us into the setting assumed in definition [ d : rp ] . some care is needed in the definition of reflected configurations : if @xmath357 is the bond obtained by reflecting @xmath1 through @xmath317 , then @xmath358 this is different compared to because the reflection in @xmath359 preserves orientations of the edges , while that in @xmath360 reverses them . while we will only apply these reflections in @xmath134 , we note that the generalization to higher dimensions is straightforward ; just consider all planes as above with @xmath361 replaced by various pairs @xmath362 of distinct coordinate vectors . these reflections will of course preserve the orientations of all edges in directions distinct from @xmath363 and @xmath364 . here we will provide the proof of phase transition in the form stated in theorem [ t : torus ] . we follow pretty much the standard approach to proofs of order - disorder transitions which dates all the way back to @xcite . a somewhat different approach ( motivated by another perspective ) to this proof can be found in @xcite . in order to use the techniques decribed in the previous section , we have to determine when the extended gradient gibbs measure @xmath104 on @xmath57 obeys the conditions of reflection positivity . [ p : rp ] let @xmath5 be of the form with any probability measure @xmath96 for which @xmath365 . then @xmath104 is reflection positive for both direct and diagonal reflections . the proof is the same for both types of reflections so we we proceed fairly generally . pick a plane of reflection @xmath317 . let @xmath366 be a site on @xmath317 and let us reexpress the @xmath48 s back in terms of the @xmath25 s with the convention that @xmath367 . then @xmath368 next , let us introduce the quantity @xmath369 ( we note in passing that the removal of @xmath317 from the first sum is non - trivial even for diagonal reflections once @xmath32 . ) clearly , @xmath370 is @xmath330-measurable and the full @xmath105-interaction is simply @xmath371 . the gibbs measure @xmath104 can then be written @xmath372 now pick a bounded , @xmath330-measurable function @xmath373 and integrate the function @xmath374 with respect to the torus measure @xmath104 . if @xmath375 is the @xmath64-algebra generated by random variables @xmath16 and @xmath109 , with @xmath17 and @xmath1 `` on '' @xmath317 , we have @xmath376 where the values of @xmath377 on @xmath317 are implicit in the integral . this proves the property in ; the identity follows by the reflection symmetry of @xmath104 . let us consider two good plaquette events , @xmath378 and @xmath379 , that all edges on the plaquette are ordered and disordered , respectively . let @xmath380 denote the corresponding bad event . given a plaquette event @xmath341 , let @xmath381^{\frac1{|{\mathbb t}_l|}}\ ] ] abbreviate the quantity on the right - hand side of and define @xmath382 the calculations from sect . [ s : spin - wave ] then permit us to draw the following conclusion : [ l : bad ] for each @xmath383 there exists @xmath384 such that if @xmath385 , then @xmath386 moreover , there exist @xmath387 such that @xmath388 and @xmath389 proof the event @xmath390 can be decomposed into a disjoint union of events @xmath391 each of which admits exactly one arrangement of ordered and disordered bonds around the plaquette ; see fig . [ fig3 ] for the relevant patterns . if @xmath391 is an event of type @xmath392 , then @xmath393\bigr\}.\ ] ] by theorem [ t : min ] , the right - hand side is bounded by @xmath394 , uniformly in @xmath53 . applying lemma [ l : sub ] , we conclude that @xmath395 is small uniformly in @xmath396 $ ] once @xmath397 . ( the values @xmath398 are handled by a limiting argument . ) the bounds ( [ dis - bd][ord - bd ] ) follow by the fact that @xmath399 which is ( large ) negative for @xmath53 near one and ( large ) positive for @xmath53 near zero . from @xmath400 we immediately infer that the bad events occur with very low frequency . moreover , a standard argument shows that the two good events do not like to occur in the same configuration . an explicit form of this statement is as follows : [ l : good - good ] let @xmath401 be the random variable from . there exists a constant @xmath402 such that for all ( even ) @xmath403 and all @xmath396 $ ] , @xmath404 proof the claim follows from the fact that , for some constant @xmath405 , @xmath406 uniformly in @xmath407 . indeed , the expectation in is the average of the probabilities @xmath408 over all @xmath409 . if @xmath17 and @xmath175 denotes the plaquettes containing the bonds @xmath1 and @xmath410 , respectively , then this probability is bounded by @xmath411 . but @xmath412 and so by the latter probability is bounded by @xmath413 , where we used @xmath414 . it remains to prove . consider the event @xmath415 where , without loss of generality , @xmath416 . we claim that on this event , the good plaquettes at @xmath17 and @xmath175 are separated from each other by a @xmath417-connected circuit of bad plaquettes . to see this , consider the largest connected component of good plaquettes containing @xmath17 and note that no plaquette neighboring on this component can be good , because ( by definition ) the events @xmath378 and @xmath379 can not occur at neighboring plaquettes ( we are assuming that @xmath208 ) . by chessboard estimates , the probability in @xmath104 of any such ( given ) circuit is bounded by @xmath395 to its size ; a standard peierls argument in toroidal geometry ( cf the proof of ( * ? ? ? * lemma 3.2 ) ) now shows that the probability in is dominated by the probability of the shortest possible contour which is @xmath418 . ( the contour argument requires that @xmath395 be smaller than some constant , but this we may assume to be automatically satisfied because the left - hand side of is less than one . ) now we are in a position to prove our claims concerning the torus state : proof of theorem [ t : torus ] let @xmath129 be the fraction of ordered bonds on @xmath57 ( cf . ) and let @xmath419 be the expectation of @xmath401 in the extended torus state @xmath104 with parameter @xmath53 . since @xmath420 is log - convex in the variable @xmath421 , and @xmath422 we can conclude that the function @xmath423 is non - decreasing . moreover , as the thermodynamic limit of the torus free energy exists ( cf proposition [ prop - fe ] in sect . [ sec5.3 ] ) , the limit @xmath424 exists at all but perhaps a countable number of @xmath53s namely the set @xmath425 $ ] of points where the limiting free energy is not differentiable . next we claim that @xmath426 tends to zero as @xmath144 for all @xmath427 and all @xmath428 . indeed , if this probability stays uniformly positive along some subsequence of @xmath163 s for some @xmath427 , then the boundedness of @xmath401 ensures that for some @xmath429 and some @xmath116 we have @xmath430 _ and _ @xmath431 for all @xmath163 in this subsequence . vaguely speaking , this implies @xmath432 because one is then able to extract two infinite - volume gibbs states with distinct densities of ordered bonds . a formal proof goes as follows : consider the cumulant generating function @xmath433 and note that its thermodynamic limit , @xmath434 , is convex in @xmath435 and differentiable at @xmath436 whenever @xmath428 . but @xmath430 in conjunction with the exponential chebyshev inequality implies @xmath437 which by taking @xmath144 and @xmath438 yields a lower bound on the right derivative at origin , @xmath439 . by the same token @xmath431 implies an upper bound on the left derivative , @xmath440 . hence , both probabilities can be uniformly positive only if @xmath432 . to prove the desired claim it remains to show that @xmath441 jumps from values near zero to values near one at some @xmath119 . to this end we first observe that @xmath442 , \qquad p\not\in{\mathcal d}.\ ] ] this follows by the fact that on the event @xmath443whose probability tends to one as @xmath144the quantity @xmath444 is bounded between @xmath445(1-\chi(p)+\epsilon)$ ] and @xmath446(1-\chi(p)-\epsilon)$ ] provided @xmath447 . lemma [ l : good - good ] now implies @xmath448\le c{\fraktura z}({\mathcal b}),\ ] ] with @xmath449 defined in . by lemma [ l : bad ] , for each @xmath383 there is a constant @xmath384 such that @xmath450\cup[1-\delta,1],\qquad p\not\in{\mathcal d},\ ] ] once @xmath451 . but the bounds ( [ dis - bd][ord - bd ] ) ensure that @xmath452 $ ] for @xmath453 and @xmath454 $ ] for @xmath455 . hence , by the monotonicity of @xmath152 , there exists a unique value @xmath119 such that @xmath456 for @xmath140 while @xmath457 for @xmath141 . in light of our previous reasoning , this proves the bounds ( [ below - pt][above - pt ] ) . in order to prove theorem [ t : main ] , we will need to derive a concentration bound on the tilt of the torus states . this is the content of the following lemma : [ l : tilt ] let @xmath84 and let @xmath458 be the set of bonds with both ends in @xmath12 . given a configuration @xmath459 , we use @xmath460 to denote the vector @xmath461 of empirical tilt of the configuration @xmath48 in @xmath12 . suppose that @xmath462 . then @xmath463 for each @xmath383 , each @xmath84 and each @xmath163 . we will derive a bound on the exponential moment of @xmath464 . let us fix a collection of numbers @xmath465 and let @xmath466 be the conditional law of the @xmath37 s given a configuration of the @xmath102 s . let @xmath467 be the corresponding law when all @xmath468 . in view of the fact that @xmath466 and @xmath467 are gaussian measures and @xmath469 , we have @xmath470 ( note that both measures enforce the same loop conditions . ) the right - hand side is best calculated in terms of the gradients . the result is @xmath471 the fact that @xmath472 and the identity @xmath473 , valid for any gaussian random variable , now allow us to conclude @xmath474 choosing @xmath475 on @xmath458 and zero otherwise , we get @xmath476 noting that @xmath477 implies that at least one of the components of @xmath464 is larger ( in absolute value ) than @xmath478 , the desired bound follows by a standard exponetial - chebyshev estimate . [ rem4.9 ] we note that the symmetry of the law of the @xmath37 s in @xmath466 is crucial for the above argument . in particular , it is not clear how to control the tightness of the empirical tilt @xmath464 in the measure obtained by normalizing @xmath479 , where @xmath480 is a `` built - in '' tilt . in the strictly convex cases , these measures were used by funaki and spohn @xcite to construct an infinite - volume gradient gibbs state with a given value of the tilt . proof of theorem [ t : main ] with theorem [ t : torus ] at our disposal , the argument is fairly straightforward . consider a weak ( subsequential ) limit of the torus states at @xmath141 and then consider another weak limit of these states as @xmath481 . denote the result by @xmath482 . next let us perform a similar limit as @xmath483 and let us denote the resulting measure by @xmath484 . as is easy to check , both measures are extended gradient gibbs measures at parameter @xmath131 . next we will show that the two measures are distinct measures of zero tilt . to this end we recall that , by and the invariance of @xmath104 under rotations , @xmath485 when @xmath141 while implies that @xmath486 when @xmath140 . but @xmath487 is a local event and so @xmath488 while @xmath489 for all @xmath1 ; i.e. , @xmath490 . moreover , the bound being uniform in @xmath53 and @xmath163survives the above limits unscathed and so the tilt is exponentially tight in volume for both @xmath482 and @xmath484 . it follows that @xmath491 as @xmath492 almost surely with respect to both @xmath482 and @xmath484 ; i.e. , both measures are supported entirely on configurations with zero tilt . it remains to prove the inequalities ( [ e : ener - bd1][e : ener - bd2 ] ) and thereby ensure that the @xmath37-marginals @xmath121 and @xmath122 of @xmath482 and @xmath484 , respectively , are distinct as claimed in the statement of the theorem . the first bound is a consequence of the identity @xmath493 which extends via the aforementioned limits to @xmath482 ( as well as @xmath484 ) . indeed , using chebyshev s inequality and the fact that @xmath494 we get @xmath495 to prove , the translation and rotation invariance of @xmath104 gives us @xmath496 let @xmath497 denote the integral of @xmath498 with respect to @xmath65 . since we have @xmath499 , simple scaling of all fields yields @xmath500 . intepreting the inner expectation above as the ( negative ) @xmath10-derivative of @xmath501 at @xmath49 , we get @xmath502 from here follows by taking @xmath144 on the right - hand side . as to the inequality for the disordered state , here we first use that the diagonal reflection allows us to disseminate the event @xmath503 around any plaquette containing @xmath1 . explicitly , if @xmath9 is a plaquette , then @xmath504 ( we are using that the event in question is even in @xmath37 and so the changes of sign of @xmath48 are immaterial . ) direct reflections now permit us to disseminate the resulting plaquette event all over the torus : @xmath505 bounding the indicator of the giant intersection by @xmath506 for @xmath507 , and invoking the scaling of the partition function @xmath508 , we deduce @xmath509^{\frac1{4|{\mathbb t}_l|}}.\ ] ] choosing @xmath510 , letting @xmath144 and @xmath483 , we thus conclude @xmath511 noting that @xmath512 , the bound is also proved . the goal of this section is to prove theorem [ t : dual ] . for that we will establish an interesting duality that relates the model with parameter @xmath53 to the same model with parameter @xmath145 . the duality relation that our model satisfies boils down , more or less , to an algebraic fact that the plaquette condition , represented by the delta function @xmath513 , can formally be written as @xmath514 we interpret the variable @xmath515 as the _ dual field _ that is associated with the plaquette @xmath516 . as it turns out ( see theorem [ t : dualita ] ) , by integrating the @xmath37 s with the @xmath515 s fixed a gradient measure is produced whose interaction is the same as for the @xmath37 s , except that the @xmath109 s get replaced by @xmath517 s . this means that if we assume that @xmath518 which is permissible in light of the remarks at the beginning of section [ sec2.2 ] , then the duality simply exchanges @xmath51 and @xmath52 ! we will assume that holds throughout this entire section . the aforementioned transformation works nicely for the plaquette conditions which guarantee that the @xmath37 s can _ locally _ be integrated back to the @xmath25 s . however , in two - dimensional torus geometry , two additional global constraints are also required to ensure the _ global _ correspondence between the gradients @xmath37 and the @xmath25 s . these constraints , which are by definition built into the _ a priori _ measure @xmath65 from sect . [ s : model ] , do not transform as nicely as the local plaquette conditions . to capture these subtleties , we will now define another _ a priori _ measure that differs from @xmath65 in that it disregards these global constraints . consider the linear subspace @xmath519 of @xmath66 that is characterized by the equations @xmath520 for each plaquette @xmath9 . this space inherits the euclidean metric from @xmath66 ; we define @xmath521 as the corresponding lebesgue measure on @xmath522 scaled by a constant @xmath523 which will be determined momentarily . in order to make the link with @xmath65 , we define @xmath524 clearly , @xmath525 consider also the projection @xmath526 which is defined , for any configuration @xmath527 , by @xmath528 then we have : there exist constants @xmath523 such that , in the sense of distributions , @xmath529 moreover , we have @xmath530 and @xmath531 here , @xmath532 is a multiple of the lebesgue measure on the two - dimensional space @xmath533 , which can be formally identified with @xmath534 . proof we begin with . consider the orthogonal decomposition @xmath535 . clearly , @xmath536 . choosing an orthonormal basis @xmath537 in @xmath538 ( where @xmath539 ) the measure @xmath521 can be written as @xmath540 let @xmath541 denote the vectors in @xmath79 such that if @xmath542 then @xmath543 . then @xmath544 with all but one of these vectors linearly independent . this means that we can replace the linear functionals @xmath545 by the plaquette conditions . fixing a particular plaquette , @xmath546 , we find that @xmath547 provided that @xmath548 the expression is now easily checked to be equivalent to : applying the constraints from the plaquettes distinct from @xmath549 , we find that @xmath550 . the corresponding @xmath68-function becomes @xmath551 , and so we can set @xmath552 in the remaining @xmath68-functions . integration over @xmath553 yields an overall multiplier @xmath554 . in order to prove , pick a subtree @xmath555 of @xmath57 as follows : @xmath555 contains the horizontal bonds in @xmath556 and the vertical bonds in @xmath557 . as is easy to check , @xmath555 is a spanning tree . denoting by @xmath558 the measure on the right - hand side of pick a bounded , continuous function @xmath559 with bounded support and consider the integral @xmath560 . the complement of @xmath555 contains exactly @xmath561 edges and there are as many @xmath68-functions in and , in which all @xmath48 , @xmath562 , appear with coefficient @xmath563 . we may thus resolve these constraints and substitute for all @xmath564 into @xmath565call the result of this substitution @xmath566 . then we can integrate all of these variables which reduces our attention to the integral @xmath567 . as is easy to check , the transformation @xmath35 for @xmath568 with the convention @xmath569 turns the measure @xmath570 into @xmath571 and makes @xmath566 into @xmath572 . we have thus deduced @xmath573 from here we get by noting that the latter integral can also be written as @xmath574 . to derive @xmath575 , let us write @xmath576 . since @xmath521 is the @xmath523-multiple of the lebesgue measure on @xmath522 and since @xmath577 and @xmath578 represent orthogonal coordinates in @xmath579 , we have @xmath580 where @xmath581 is the lebesgue measure on @xmath582 . plugging into we find that @xmath583 which in turn implies . it is of some interest to note that the measure @xmath521 is also reflection positive for direct reflections . one proof of this fact goes by replacing the @xmath68-functions in by gaussian kernels and noting that the linear term in @xmath553 ( in the exponent ) exactly cancels . the status of reflection positivity for the diagonal reflections is unclear . now we can state the principal duality relation . for that let @xmath584 denote the dual torus which is simply a copy of @xmath57 shifted by half lattice spacing in each direction . let @xmath585 denote the set of dual edges . we will adopt the convention that if @xmath1 is a direct edge , then its dual i.e . , the unique edge in @xmath585 that cuts through @xmath1will be denoted by @xmath586 . then we have : [ t : dualita ] given two collections @xmath587 and @xmath588 of positive weights on @xmath59 , consider the partition functions @xmath589 and @xmath590 if @xmath587 and @xmath591 are dual in the sense that @xmath592 then @xmath593z_{l,(\kappa_b)}.\ ] ] proof we will cast the partition function @xmath594 into the form on the right - hand side of . let us regard this partition function as defined on the dual torus @xmath584 . the proof commences by rewriting the definition with the help of as @xmath595 where @xmath596 is the plaquette curl for the dual plaquette @xmath597 with the center at @xmath17 . rearranging terms and multiplying by the exponential ( gaussian ) weight from , we are thus supposed to integrate the function @xmath598 against the ( unconstrained ) lebesgue measure @xmath599 . here @xmath600 if @xmath568 is dual to the bond @xmath586 . completing the squares and integrating over the @xmath37 s produces the function @xmath601\int_{{\mathbb r}}{\text{\rm d}\mkern0.5mu}\theta \int_{{\mathbb r}^{{\mathbb t}_l}}\prod_{x\in{\mathbb t}_l}{\text{\rm d}\mkern0.5mu}\phi_x\ , \exp\biggl\{-\frac12\sum_{b\in{\mathbb b}_l}\frac1{\kappa^\star_{b^\star}}(\nabla_b\phi)^2-{\text{\rm i}\mkern0.7mu}\theta\sum_{x\in{\mathbb t}_l}\phi_x\biggr\}.\ ] ] invoking , we can replace all @xmath602 by @xmath109 . the integral over @xmath553 then yields @xmath603 times the @xmath68-function of @xmath604 which by the substitution @xmath605 that has no effect on the rest of the integral can be converted to @xmath606 . invoking the definition of @xmath65 , this leads to the partition function . [ remark : mira - dualita ] let @xmath607 be the extended gradient gibbs measure @xmath104 for @xmath97 with parameter @xmath53 and let @xmath608 be the corresponding measure with the _ a priori _ measure @xmath65 replaced by @xmath521 . then the above duality shows that the law of @xmath146 governed by @xmath607 is the same as the law of its dual @xmath609defined via in measure @xmath610 , once @xmath53 and @xmath611 are related by @xmath612 indeed , the probability in measure @xmath610 of seeing the configuration @xmath609 with @xmath613 ordered bonds and @xmath614 disordered bonds is proportional to @xmath615 . considering the dual configuration @xmath146 and letting @xmath616 denote the number of disordered bonds and @xmath617 the number of ordered bonds in @xmath146 , we thus have @xmath618 for @xmath53 and @xmath611 related as in , the right - hand side is proportional to the probability of @xmath146 in measure @xmath607 . we believe that the difference between the two measures disappears in the limit @xmath144 and so the @xmath102-marginals of the states @xmath482 and @xmath484 at @xmath131 can be considered to be dual to each other . however , we will not pursue this detail at any level of rigor . in order to use effectively the duality relation from theorem [ t : dualita ] , we have to show that the difference in the _ a priori _ measure can be neglected . we will do this by showing that both partition functions lead to the same free energy . this is somewhat subtle due to the presence ( and absence ) of various constraints , so we will carry out the proof in detail . [ prop - fe ] let @xmath619 $ ] and recall that @xmath620 denotes the integral of @xmath497 with respect to @xmath99 . similarly , let @xmath621 denote the integral of @xmath622 with respect to @xmath99 . then ( the following limits exist as @xmath144 and ) @xmath623 for all @xmath396 $ ] . before we commence with the proof , let us establish the following variance bounds for homogeneous gaussian measures relative to the _ a priori _ measure @xmath65 and @xmath624 : [ l : var - bd ] let @xmath625 be the ( standard ) gaussian gradient measure @xmath626 and @xmath627 be the measure obtained by replacing @xmath65 by @xmath624 . for @xmath628 , let @xmath629 there exists an absolute constant @xmath630 such that for all @xmath403 and all @xmath628 , @xmath631 proof in measure @xmath625 , we can reintroduce back the fields @xmath632 and @xmath633 then equals @xmath634 . discrete fourier transform implies that @xmath635 where @xmath636 is the reciprocal torus and @xmath186 is the discrete ( torus ) laplacian . simple estimates show that the sum is bounded by a constant times @xmath637 , uniformly in @xmath163 . hence , @xmath638 for some absolute constant @xmath639 . as for the other measure , we recall the definitions and use these to write @xmath640 if @xmath1 is horizontal ( and @xmath641 if @xmath1 is vertical ) . the fact that the gaussian field is homogeneous implies via that the fields @xmath632 and the variables @xmath578 and @xmath577 are independent with @xmath632 distributed according to @xmath625 and @xmath578 and @xmath577 gaussian with mean zero and variance @xmath642 . in this case @xmath643 and so we get @xmath644 but @xmath645 and so the correction is bounded for all @xmath163 . proof of proposition [ prop - fe ] the proof follows the expected line : to compensate for the lack of obvious subadditivity of the torus partition function , we will first relate the periodic boundary condition to a `` fixed '' boundary condition . then we will establish subadditivity and hence the existence of the free energy for the latter boundary condition . fix @xmath646 and consider the partition function @xmath647 defined as follows . let @xmath648 be a box of @xmath58 sites and consider the set @xmath649 of edges with _ both _ ends in @xmath648 . let @xmath650 be as in subject to the restriction that @xmath651 for all @xmath17 on the _ internal _ boundary of @xmath648 . let @xmath652 we will now provide upper and lower bounds between the partition functions @xmath620 ( resp . @xmath621 ) and @xmath647 , for a well defined range of values of @xmath653 . comparing explicit expressions for @xmath620 and @xmath647 and using @xmath654 , we get @xmath655 to derive an opposite inequality , note that for @xmath656 we get that @xmath657 , where @xmath625 is as in . invoking one more time the gaussian identity @xmath658 in conjunction with lemma [ l : var - bd ] , yields @xmath659 hence , if @xmath660 we have that with probability at least @xmath202 in measure @xmath104 , _ all _ variables @xmath16 are in the interval @xmath661 $ ] . since the interaction that wraps @xmath648 into the torus is of definite sign , it follows that @xmath662 for all @xmath163 and all @xmath660 . concerning the star - partition function , lemma [ l : var - bd ] makes the proof of exactly the same . as for the alternative of , we invoke and restrict all @xmath663 on the internal boundary of @xmath648 to values less than @xmath653 and @xmath664 and @xmath665 to values less than @xmath666 . since @xmath667 for every vertical bond that wraps @xmath648 into the torus ( and similarly for the horizontal bonds ) , we now get @xmath668 where the factor @xmath669 comes from the integration over @xmath578 and @xmath577 . we conclude that , for @xmath670 , the partition functions @xmath620 , @xmath621 and @xmath647 lead to the same free energy , provided at least one of these exists . it remains to establish that the partition function @xmath647 is ( approximately ) submultiplicative for some choice of @xmath671 . choose , e.g. , @xmath672 and let @xmath673 be an integer . if two neighbors have their @xmath25 s between @xmath674 and @xmath675 , the energy across the bond is at most @xmath676 . splitting @xmath677 into @xmath678 boxes of size @xmath163 , and restricting the @xmath25 s to @xmath679 $ ] on the internal boundaries of these boxes , we thus get @xmath680^{p^2}\exp\bigl\{-\tfrac12{{\mathchoice { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{6.5}{6}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } { \kappa_{\text{\rm\fontsize{5}{5}\selectfont o } } } } } ( 2m_l)^2\,2(p-1)l\bigr\}.\ ] ] the exponent can be bounded below by @xmath681 for @xmath163 sufficiently large which implies that @xmath682^{1/(pl)^2}$ ] is increasing for all @xmath673 and all @xmath397 . this proves the claim for limits along multiples of any fixed @xmath163 ; to get the values `` in - between '' we just need to realize that , as before , @xmath683 , for any fixed @xmath684 . now we finally prove our claim concerning the value of the transitional @xmath53 : proof of theorem [ t : dual ] let @xmath685 denote the integral of @xmath497 with respect to the _ a priori _ measure @xmath686 with parameter @xmath53 and let @xmath687 denote the analogous quantity for @xmath622 . the arguments leading up to then yield @xmath688 whenever @xmath611 is dual to @xmath53 in the sense of . thus , using @xmath689 to denote the limit in with the negative sign , we have @xmath690 now , as a glance at the proof of theorem [ t : torus ] reveals , the value @xmath131 is defined as the unique point where the derivative of @xmath689 , which at the continuity points of @xmath152 is simply @xmath691 , jumps from values near @xmath692 to values near @xmath693 . eq . then forces the jump to occur at the self - dual point @xmath694 . in light of , this proves . the research of m.b . was supported by the nsf grant dms-0505356 and that of r.k . by the grants gar 201/03/0478 , msm 0021620845 , and the max planck institute for mathematics in the sciences , leipzig . the authors are grateful to scott sheffield for discussions that ultimately led to the consideration of the model , and for valuable advice how to establish the zero - tilt property of the coexisting states . discussions with jean - dominique deuschel helped us understand the problems described in remark [ rem4.9 ] . j. frhlich , r. israel , e.h . lieb and b. simon , _ phase transitions and reflection positivity . ii . lattice systems with short - range and coulomb interations _ , j. statist . * 22 * ( 1980 ) , no . 3 , 297347 . r. koteck and s.b . shlosman , _ existence of first - order transitions for potts models _ , in : s. albeverio , ph . combe , m. sirigue - collins ( eds . ) , proc . of the international workshop stochastic processes in quantum theory and statistical physics , lecture notes in physics , vol . 173 , pp . 248253 , springer - verlag , berlin , 1982 . y. velenik , _ localization and delocalization of random interfaces _ , lecture notes for a minicourse at the meeting `` topics in random interfaces and directed polymers , '' leipzig 2005 ; arxiv : math.pr/0509695 .

we consider the ( scalar ) gradient fields @xmath0with @xmath1 denoting the nearest - neighbor edges in @xmath2that are distributed according to the gibbs measure proportional to @xmath3 . here @xmath4 is the hamiltonian , @xmath5 is a symmetric potential , @xmath6 is the inverse temperature , and @xmath7 is the lebesgue measure on the linear space defined by imposing the loop condition @xmath8 for each plaquette @xmath9 in @xmath2 . for convex @xmath5 , funaki and spohn have shown that ergodic infinite - volume gibbs measures are characterized by their tilt . we describe a mechanism by which the gradient gibbs measures with non - convex @xmath5 undergo a structural , order - disorder phase transition at some intermediate value of inverse temperature @xmath10 . at the transition point , there are at least two distinct gradient measures with zero tilt , i.e. , @xmath11 . = 1

let @xmath2 be a simple graph with @xmath0 vertices and @xmath3 the adjacency matrix of @xmath2 . the eigenvalues @xmath4 of @xmath3 are said to be the eigenvalues of the graph @xmath2 . the energy of @xmath2 is defined as @xmath5 the characteristic polynomial of @xmath3 is also called the characteristic polynomial of @xmath2 , denoted by @xmath6 . using these coefficients of @xmath7 , the energy of @xmath2 can be expressed as the coulson integral formula @xcite : @xmath8dx . \label{energy-1}\end{aligned}\ ] ] for convenience , write @xmath9 and @xmath10 for @xmath11 . since the energy of a graph can be used to approximate the total @xmath12-electron energy of the molecular , it has been intensively studied . for details on graph energy , we refer to the recent book @xcite and reviews @xcite . one of the fundamental question that is encountered in the study of graph energy is which graphs ( from a given class ) have minimal and maximal energies . a large of number of papers were published on such extremal problems , see chapter 7 in @xcite . a connected graph on @xmath0 vertices with @xmath13 edges is called an @xmath14-graph . we call an @xmath14-graph a unicyclic graph , a bicyclic graph , a tricyclic graph , and a tetracyclic graph if @xmath15 and @xmath16 , respectively . follow @xcite , let @xmath17 be the graph obtained by the star @xmath18 with @xmath19 additional edges all connected to the same vertex , and @xmath20 be the bipartite @xmath14-graph with two vertices on one side , one of which is connected to all vertices on the other side . in @xcite , caporossi et al . gave the following conjecture : @xcite[conjecture - minimal energy ] connected graphs @xmath2 with @xmath21 vertices , @xmath22 edges and minimum energy are @xmath17 for @xmath23 $ ] , and @xmath20 otherwise . this conjecture is true when @xmath24 , @xmath25 @xcite , and when @xmath26 for @xmath21 @xcite . @xcite showed that @xmath20 is the unique bipartite graph of order @xmath0 with minimal energy for @xmath27 . hou @xcite proved that for @xmath21 , @xmath28 has the minimal energy among all bicyclic graphs of order @xmath0 with at most one odd cycle . let @xmath29 be the set of connected graphs with @xmath0 vertices and @xmath13 edges . let @xmath30 be the subset of @xmath29 which contains no disjoint two odd cycles of length @xmath31 and @xmath32 with @xmath33 ( mod @xmath34 , and @xmath35 . zhang and zhou @xcite characterized the graphs with minimal , second - minimal and third - minimal energy in @xmath36 for @xmath37 . combining the results ( lemmas 5 - 9 ) in @xcite with the fact that @xmath38 for @xmath39 , we can deduce the following lemma . @xcite[bicyclic - minimal energy-1 ] the graph with minimal energy in @xmath40 is @xmath41 for @xmath42 or @xmath37 , and @xmath28 for @xmath43 , respectively . @xcite proved that @xmath44 has minimal energy in @xmath45 for @xmath46 , and for @xmath47 , they wanted to characterize the graphs with minimal and second - minimal energy in @xmath48 , but left four special graphs without determining their ordering . huo et al . solved this problem in @xcite , and the results on minimal energy can be restated as follows . [ tricyclic - minimal energy-1 ] the graph with minimal energy in @xmath45 is @xmath44 for @xmath46 @xcite , and @xmath49 for @xmath47 @xcite , respectively . in @xcite , the authors claimed that they gave a complete solution to conjecture [ conjecture - minimal energy ] for @xmath50 and @xmath51 by showing the following two results . ( theorem 1 , @xcite)[bicyclic - minimal energy-2 ] let @xmath2 be a connected graph with @xmath0 vertices and @xmath52 edges . then @xmath53 with equality if and only if @xmath54 . ( theorem 2 , @xcite)[tricyclic - minimal energy-2 ] let @xmath2 be a connected graph with @xmath0 vertices and @xmath55 edges . then @xmath56 with equality if and only if @xmath57 . note that @xmath38 for @xmath39 , and @xmath58 for @xmath59 . in addition , there is a little gap in the original proofs ( even for large @xmath0 ) of lemmas [ bicyclic - minimal energy-2 ] and [ tricyclic - minimal energy-2 ] in @xcite , respectively . for completeness , we will prove the following two results in section 2 . [ bicyclic - thm ] @xmath41 if @xmath42 or @xmath37 , @xmath28 if @xmath39 has minimal energy in @xmath60 . [ tricyclic - thm ] the complete graph @xmath61 if @xmath42 , @xmath49 if @xmath62 or @xmath47 , @xmath44 if @xmath63 has minimal energy in @xmath64 . furthermore , @xmath65 has second - minimal energy in @xmath66 . li and li @xcite discussed the graph with minimal energy in @xmath67 , and claimed that the graph with minimal energy in @xmath68 is @xmath69 for @xmath70 , and @xmath71 for @xmath72 , respectively . note that @xmath73 for @xmath74 . in section 3 , we will first illustrate the correct version of this result , and then we will show the following theorem . [ tetracyclic - thm ] the wheel graph @xmath75 if @xmath62 , the complete bipartite graph @xmath76 if @xmath77 , @xmath69 if @xmath78 , @xmath71 if @xmath74 has minimal energy in @xmath79 . furthermore , @xmath71 has second - minimal energy in @xmath79 for @xmath80 . @xcite[lemma sn , e and bn , e ] @xmath81 if @xmath82 ; @xmath83 if @xmath84 . from lemma [ lemma sn , e and bn , e ] , we know that the bound @xmath23 $ ] in conjecture [ conjecture - minimal energy ] should be understood that @xmath85 . with theorems [ bicyclic - thm ] , [ tricyclic - thm ] and [ tetracyclic - thm ] , we give a complete solution to conjecture [ conjecture - minimal energy ] for @xmath86 and @xmath16 . the following three lemmas are need in the sequel . @xcite[edge - cut ] if @xmath87 is an edge cut of a simple graph @xmath2 , then @xmath88 , where @xmath89 is the subgraph obtained from @xmath2 by deleting the edges in @xmath87 . @xcite[lemma in zhang ] ( 1 ) suppose that @xmath90 and @xmath91 . then @xmath92 with equality if and only if @xmath93 . \(2 ) @xmath94 for @xmath21 . \(3 ) @xmath95 for @xmath96 . \(4 ) @xmath97 for @xmath21 . [ unicyclic - minimal energy-1 ] ( 1 ) @xcite @xmath98 has minimal energy in @xmath99 for @xmath100 or @xmath21 . \(2 ) @xmath101 and @xmath98 have , respectively , minimal and second - minimal energy in @xmath99 for @xmath102 . in particular , @xmath98 is the unique non - bipartite graph in @xmath99 with minimal energy for @xmath102 . by table 1 of @xcite , there are two @xmath103-graphs and five @xmath104-graphs . by simple computation , we can obtain the result ( 2 ) . * proof of theorem [ bicyclic - thm ] : * by lemma [ bicyclic - minimal energy-1 ] , it suffices to prove that @xmath105 when @xmath42 or @xmath37 , and @xmath106 when @xmath39 for @xmath107 . suppose that @xmath107 . as there is nothing to prove for the case @xmath108 , we suppose that @xmath21 . then @xmath2 has a cut edge @xmath109 such that @xmath110 contains exactly two components , say @xmath111 and @xmath112 , which are non - bipartite unicyclic graphs . let @xmath113 , @xmath114 , and @xmath115 . by lemmas [ edge - cut ] , [ lemma in zhang ] and [ unicyclic - minimal energy-1 ] , we have @xmath116 in particular , @xmath117 for @xmath118 . the proof is thus complete . [ bicyclic - remark ] the proof of theorem [ bicyclic - thm ] ( for large @xmath0 ) is similar to that of lemma [ bicyclic - minimal energy-2 ] except that in @xcite , the authors did not point out that @xmath111 and @xmath112 are non - bipartite unicyclic graphs . without this assumption , we know that the inequality does not hold when @xmath119 or @xmath120 equals to @xmath121 or @xmath122 by lemma [ unicyclic - minimal energy-1 ] ( 2 ) . moreover , the inequality @xmath123 does not hold . for example : @xmath124 for @xmath125 , since @xmath126 and @xmath127 by lemma [ edge - cut ] . [ bicyclic - n-5,6,7 ] @xmath41 is the unique non - bipartite graph in @xmath60 with minimal energy for @xmath128 . furthermore , @xmath41 has second - minimal energy in @xmath60 for @xmath62 or @xmath129 , and @xmath130 has third - minimal energy in @xmath131 . by table 1 of @xcite , there are five @xmath132-graphs . by simple calculation , we can prove the theorem for @xmath62 . by table 1 of @xcite , there are 19 @xmath133-graphs . by direct computation , we can prove the theorem for @xmath77 . by the results ( lemmas 5 - 9 ) in @xcite , we can obtain that @xmath134 has second - minimal energy in @xmath135 . on the other hand , from the proof of theorem [ bicyclic - thm ] , @xmath136 for @xmath137 . therefore @xmath134 has second - minimal energy in @xmath138 , and so the theorem is true for @xmath139 . * proof of theorem [ tricyclic - thm ] : * since @xmath61 is the unique graph in @xmath140 , the theorem holds for @xmath42 . by table 1 of @xcite , there are four @xmath141-graphs . by simple calculation , we can prove the theorem for @xmath62 . by table 1 of @xcite , there are 22 @xmath142-graphs . by direct computation , we can prove the theorem for @xmath77 . now suppose that @xmath125 . by lemma [ tricyclic - minimal energy-1 ] , it suffices to prove that @xmath143 when @xmath47 , and @xmath144 when @xmath46 for @xmath145 . suppose that @xmath145 and @xmath146 , @xmath147 are two disjoint odd cycles with @xmath33 ( mod @xmath34 . then there are at most two edge disjoint paths in @xmath2 connecting @xmath146 and @xmath147 . * there exists exactly an edge disjoint path @xmath148 connecting @xmath146 and @xmath147 . then there exists an edge @xmath13 of @xmath148 such that @xmath149 , where @xmath111 is an non - bipartite bicyclic graph with @xmath150 vertices and @xmath112 is an non - bipartite unicyclic graph with @xmath151 vertices . by lemmas [ edge - cut ] , [ lemma in zhang ] , [ unicyclic - minimal energy-1 ] , [ bicyclic - n-5,6,7 ] and theorem [ bicyclic - thm ] , we have @xmath152 in particular , @xmath153 for @xmath46 . * there exist exactly two edge disjoint paths @xmath154 and @xmath155 connecting @xmath146 and @xmath147 . then there exist two edges @xmath156 and @xmath157 such that @xmath158 is an edge of @xmath159 for @xmath160 , and @xmath161 , where @xmath162 and @xmath163 are non - bipartite unicyclic graphs . let @xmath164 and @xmath165 . then by lemmas [ edge - cut ] , [ lemma in zhang ] and [ unicyclic - minimal energy-1 ] , we have @xmath166 in particular , @xmath153 for @xmath46 . the proof is thus complete . [ tricyclic - remark ] the proof of theorem [ tricyclic - thm ] ( for large @xmath0 ) is similar to that of lemma [ tricyclic - minimal energy-2 ] except that in @xcite , the authors did not point out that @xmath111 and @xmath112 are non - bipartite graphs . li and li @xcite discussed the graph with minimal energy in @xmath167 , and we first restate their results . by the results ( see the proofs of lemma 2.2 and proposition 2.3 ) of @xcite , all we need is to show that @xmath182 when @xmath2 contains exactly @xmath183 ( @xmath184 ) cycles ( see case 7 of lemma 2.2 ) . from @xcite , we have @xmath185 where @xmath186 is the number of quadrangles in @xmath2 . it is easy to check that in this case , @xmath2 has at most @xmath187 quadrangles . therefore @xmath188 the proof is thus complete . in @xcite , the authors failed to get the above result in that ( in the proof of proposition 2.5 of @xcite ) they used the wrong formula @xmath191 instead of the correct one @xmath192 . they also gave the following result . by lemmas [ tetracyclic - minimal energy-1 ] , [ tetracyclic - minimal energy-2 ] , [ tetracyclic - minimal energy-4],[tetracyclic - minimal energy-5 ] and corollary [ tetracyclic - minimal energy-3 ] , we can characterize the graph with minimal energy in @xmath68 . for @xmath218 , the result follows by direct computation . suppose that @xmath219 . by direct calculation , we have that @xmath220 . let @xmath221 . then we have that @xmath222 , @xmath223 , @xmath224 , @xmath225 and @xmath226 . hence @xmath227 on the other hand , we have @xmath228 @xcite , and so @xmath217 . * proof of theorem [ tetracyclic - thm ] : * by table 1 of @xcite , there are two @xmath229-graphs . by simple calculation , we can prove the theorem for @xmath62 . by table 1 of @xcite , there are 20 @xmath230-graphs . by direct computation , we can prove the theorem for @xmath77 . by @xcite , there are 132 @xmath231-graphs . by direct computing , we can prove the theorem for @xmath139 . now suppose that @xmath37 . by lemma [ tetracyclic - minimal energy-6 ] and corollary [ tetracyclic - minimal energy-3 ] , it suffices to prove that @xmath232 for @xmath233 . * there exists exactly an edge disjoint path @xmath154 connecting @xmath146 and @xmath147 . then there exists an edge @xmath156 of @xmath154 such that @xmath234 , where either both @xmath111 and @xmath112 are non - bipartite bicyclic graphs , or @xmath111 is an non - bipartite tricyclic graph and @xmath112 is an non - bipartite unicyclic graph . let @xmath113 and @xmath114 . * subcase 1.1 . * both @xmath111 and @xmath112 are non - bipartite bicyclic graphs . then by lemmas [ edge - cut ] , [ lemma in zhang ] , [ bicyclic - n-5,6,7 ] , [ lemma compare tetracyclic with unicyclic ] and theorem [ bicyclic - thm ] , we have @xmath235 * subcase 1.2 . * @xmath111 is an non - bipartite tricyclic graph and @xmath112 is an non - bipartite unicyclic graph . it follows from theorem [ tricyclic - thm ] and lemma [ lemma compare tricyclic with unicyclic ] that @xmath236 . therefore by lemmas [ edge - cut ] , [ lemma in zhang ] , [ unicyclic - minimal energy-1 ] and [ lemma compare tetracyclic with unicyclic ] , we have @xmath237 * case 2 . * there exist exactly two edge disjoint paths @xmath155 and @xmath238 connecting @xmath146 and @xmath147 . then there exist two edges @xmath157 and @xmath239 such that @xmath158 is an edge of @xmath159 for @xmath240 , and @xmath241 , where @xmath162 is an non - bipartite bicyclic graph with @xmath119 vertices and @xmath163 is an non - bipartite unicyclic graph with @xmath120 vertices . by lemmas [ edge - cut ] , [ lemma in zhang ] , [ unicyclic - minimal energy-1 ] , [ bicyclic - n-5,6,7 ] , [ lemma compare tetracyclic with unicyclic ] and theorem [ bicyclic - thm ] , we have @xmath242 * case 3 . * there exist exactly three edge disjoint paths @xmath243 , @xmath244 and @xmath245 connecting @xmath146 and @xmath147 . then there exist three edges @xmath246 , @xmath247 and @xmath248 such that @xmath158 is an edge of @xmath159 for @xmath249 , and @xmath250 , where @xmath169 and @xmath170 are non - bipartite unicyclic graphs . let @xmath251 and @xmath252 . then by lemmas [ edge - cut ] , [ lemma in zhang ] , [ unicyclic - minimal energy-1 ] and [ lemma compare tetracyclic with unicyclic ] , we have @xmath253 i. gutman , the energy of a graph : old and new results , in : a. betten , a. kohn- ert , r. laue , a. wassermann ( eds . ) , _ algebraic combinatorics and applications _ , springer - verlag , berlin , 2001 , pp . 196211 .

the energy of a graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix . in this paper , we characterize the tetracyclic graph of order @xmath0 with minimal energy . by this , the validity of a conjecture for the case @xmath1 proposed by caporossi et al . @xcite has been confirmed . + * keywords : * minimal energy ; tetracyclic graph ; characteristic polynomial + * ams subject classification 2000 : * 05c50 ; 15a18 ; 05c35 ; 05c90 = 0.30 in [ section ] [ lem]theorem [ lem]corollary [ lem]conjecture [ lem]remark [ lem]definition * on the minimal energy of tetracyclic graphs * = 0.20 in = 0.20 in = 0.20 in hongping ma , yongqiang bai + school of mathematics and statistics , jiangsu normal university , + xuzhou 221116 , china + = 0.245 in

while the study of galaxy evolution has made important strides in recent years by being able to weigh individual galaxies ( i.e. , determine a mass ) , the field of quasar research is grappling with the issue of how to measure accurately the masses of a supermassive black holes ( smbhs ) for the distant quasar population . here the challenge is greater due to the fact that the sphere of influence of a smbh can only be resolved for a limited sample of nearby galaxies whereas the dynamical mass of a galaxy can easily be measured due to its large spatial extent . a significant leap forward in our ability to both accurately and efficiently measure the masses of smbhs , @xmath13 , at all redshifts will likely lead to new insights on questions such as how are black holes fueled , what it is the connection with its host galaxy , and how do smbhs evolve within a cosmological framework . spectroscopy enables us to probe the kinematics of ionized gas within the vicinity of a smbh in distant active galactic nuclei ( agns ) and luminous quasars to infer their black hole masses . traditionally , emission lines ( e.g. , , , h@xmath14 , and h@xmath3 ) detected in the optical and velocity - broadened between @xmath15 km s@xmath6 are used to probe the gravitational potential well of a smbh . this lower limit on the velocity width has been set somewhat arbitrarily since there exists a well - known population of both type 1 agns having narrower line widths ( i.e. , nls1 ; * ? ? ? * ) and those with intermediate - mass black holes @xcite . there are methods to determine the luminosity - weighted radial distance between the broad - line region ( blr ) and central source , @xmath16 , for agn ( @xmath17 ) through reverberation - mapping campaigns ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) based on balmer lines . even with the complex nature of the blr , this characteristic radius is tightly correlated with its luminosity @xcite , thus providing a means to infer such a distance to the blr in large quasar samples based solely on luminosity . then coupled with velocity information provides a viral mass estimate based on a single - epoch spectrum . such techniques have been applied to large quasar samples most notably the sloan digital sky survey ( sdss ) . a number of studies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) clearly demonstrate that such samples effectively probe smbhs above @xmath18 at @xmath12 ( an epoch of maximal black hole activity ) due to the wide area coverage and shallow depth . it is important to keep in mind that these black hole masses are based on calibrations using lower luminosity agns at low redshift ; their application to luminous quasars at high redshift is not well solidified with reverberation mapping @xcite . deep surveys , such as cosmos , goods and aegis , are effective probes of black hole accretion at lower masses . given that the black hole mass function is steeply declining at @xmath19 @xcite , studies of the global population with sdss are susceptible to large uncertainties when extrapolated to lower masses @xcite . while noble attempts have been made to characterize the low - mass end ( @xmath20 ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , these studies have been based on luminosity and do not consider virial velocities . these deep survey fields have considerable x - ray coverage that can be utilized to select agns that mitigate biases incurred by host galaxy dilution and obscuration . such selection then has the potential to effectively probe the lower luminosity agn population that may be powered by lower mass black holes or those accreting at sub - eddington rates . followup optical spectroscopic observations are enabling single - epoch virial black hole mass estimates ( down to @xmath21 ) based on the properties of their broad emission lines and continuum luminosity @xcite . mass estimates for these higher redshift agns , which actually constitute the majority of the population in deep surveys such as cosmos , rely on or since the h@xmath14 line ( used to calibrate recipes based on local samples ) is no longer available in the optical window at @xmath22 . while the assumption that the line is produced from the same physical region as h@xmath14 may hold @xcite , there are studies that indicate that the physics of the broad - line region is not so simple @xcite , especially for the most luminous quasars that can have significant outflows possibly in response to a more intense radiation field . furthermore , it is non - trivial to disentangle the broad line from fe emission that sits at its base , especially for lower mass black holes and maybe even those at the high mass end . we present first results of a near - infrared spectroscopic survey of broad - line agns ( blagns ) primarily in the cosmos field using the fiber multi - object spectrograph ( fmos ) mounted on the subaru telescope . with fmos , we now have the capability to simultaneously acquire near - infrared spectra of @xmath23 targets over a field of view of 0.19 square degrees . in this study , we report on the comparison between the h@xmath3 emission line profile , detected in the near - infrared , with that of present in previously available optical spectra . we aim to establish how effective recipes ( established locally ) are to measure single - epoch black hole masses out to @xmath1 and for blagns of lower luminosity ( i.e. , lower black hole mass ) , as compared to those found in the sdss . our sample is supplemented with agns from the extended _ chandra _ deep field - south ( ecdf - s ) survey that reach fainter depths , in x - rays , than the _ chandra_/cosmos survey and improve our statistics at lower black hole masses . in section [ xxfmos ] , we fully describe the fmos observations including target selection , data reduction , and success rate with respect to detecting the h@xmath3 emission line . we describe our method for fitting broad emission lines in section [ xxxfit ] . our results are described in section [ result ] . throughout this work , we assume @xmath24 km s@xmath6 mpc@xmath6 , @xmath25 , @xmath26 , and ab magnitudes . the capability of fmos @xcite to simultaneously acquire near - infrared spectra for a large number of objects over a wide field offers great potential for studies of galaxies @xcite and agns @xcite at high redshift . over a circular region of 30@xmath27 in diameter , it is possible to place up to 400 fibers , each with a 12 aperture , across the field . to detect emission lines in agns over a wide range in redshift , we elect to use the low - resolution mode that effectively covers two wavelength intervals of @xmath28 ( j band ) and @xmath29 ( h band ) simultaneously . the spectral resolution is @xmath30 , thus a velocity resolution fwhm @xmath31 km s@xmath6 at @xmath32 , suitable for the study of broad emission lines of agn . unfortunately , this mode requires an additional optical element ( i.e. , vph grating ) that reduces the total throughput to @xmath33% at 1.3 and impacts the limiting depth reachable in a few hours of integration time . accurate removal of the bright sky background when observing from the ground with near - infrared detectors remains the primary challenge . with fmos , an oh - airglow suppression filter @xcite is built into the system that significantly reduces the intensity of strong atmospheric emission lines that usually plague the j band and h band . equally important , there is the capability ( cross - beam - switching ; cbs hereafter ) to dither targets between fiber pairs effectively measuring the sky background spatially close to individual objects and through the same fibers as the science targets . in this mode , two sequential observations are taken by offsetting the telescope by 60@xmath34 while keeping the target within one of the two fibers . the trade off is that only 200 fibers are available in this cbs mode . we refer the reader to @xcite for full details of the instrument and its performance . below , we briefly describe our observing program using fmos including the selection of type 1 agns , data reduction , and success rate with respect to the detection of broad emission lines . complete details of our program will be presented in silverman et al . ( in prep . ) along with the full catalog of emission - line properties of broad - line agns in the cosmos and ecdf - s . our primary selection of agns is based on their having x - ray emission detected by _ chandra _ @xcite within the central square degree of cosmos ( hereafter c - cosmos ) . the high surface density of agns ( @xmath35 per deg@xmath36 ) at the limiting depths ( @xmath37 ergs @xmath38 s@xmath6 ) of c - cosmos field ensures that we make adequate use of the multiplex capabilities of fmos . in addition , two fmos pointings were observed further out from the center of the cosmos field thus we relied upon the catalog of optical and near - infrared counterparts to _ xmm - newton _ sources @xcite for agn selection . we further require that optical spectroscopy @xcite is available for each source that yields a reliable redshift and detection of at least a single broad ( fwhm @xmath39 km s@xmath6 ) emission line , namely in many cases . we then specifically targeted those with spectroscopic redshifts that allows us to detect either h@xmath14 ( @xmath40 and @xmath41 ) , h@xmath3 ( @xmath42 and @xmath40 ) , or ( @xmath43 and @xmath44 ) in the observed fmos spectral windows in low - resolution mode , i.e. , @xmath28 and @xmath29 . fibers are assigned to blagns ( for which we can detect emission lines of interest ) with a limiting magnitude of @xmath45 . those at @xmath46 are given higher priority to ensure that this sample ( of lower density on the sky ) is well represented in the final catalog ; this also effectively improves upon our success rate of detection both continuum and line emission . due to the sensitivity of fmos and the low number density of agns at @xmath47 , our sample has very few detections of in the near infrared . in addition , we have acquired fmos observations of x - ray selected agns in the ecdf - s @xcite survey with the inclusion of those that are only detected in the deeper 2@xmath484 ms data @xcite in the central region that covers goods . this deeper x - ray field offers the potential to extend the dynamic range of our study in terms of black hole mass and eddington ratio . we specifically select agns , as mentioned above , based on their optical properties determined through deep spectroscopic campaigns @xcite . as with the cosmos sample , we place higher priority on the brighter agns ( @xmath49 ) while targeting the fainter cases with lower priority . we have acquired near - infrared spectra with subaru / fmos of broad - line agns from the cosmos and ecdf - s surveys . the majority of the data was obtained during open use time through naoj over three nights in december 2010 ( i d ; s10b-108 ) and two nights in december 2011 ( i d ; s11b-098 ) . additional targets were observed during other programs being carried out in the cosmos field through the university of hawaii in s10b@xmath48s11a . weather conditions were acceptable although clouds , mainly cirrus , reduced our observing efficiency . the typical seeing was @xmath50 with considerable variation across the nights . we elected to use the cbs mode while taking two sequential exposures each of 15 minutes for each position ( namely a and b positions hereafter ) . these pairs of exposure were repeated multiple times in order to reach an effective total integration time of @xmath51 hours on - source . some time is lost to refocusing and repositioning fibers at regular intervals during the full observation . in the early data , only one spectrograph ( irs1 ) was available thus @xmath52 fibers were available for science targets . we use the publicly available software fibre - pac ( fmos image - based reduction package ; * ? ? ? the reduction routines are based on iraf tasks although several steps are processed by additional tools written by the fmos instrument team . since our observation are carried out using an abab nodding pattern in cbs mode of the telescope , an effective sky subtraction ( a@xmath48b ) can be performed using the two different sky images : a@xmath53 b@xmath54 and a@xmath53 b@xmath55 taken before and after the @xmath56-th exposure . after the initial background subtraction , a cross talk signal is removed by subtracting 0.15% for irs1 and 1% for irs2 from each quadrant . the difference in the bias between the quadrants is corrected to make the average over each quadrant equal . we further apply a flat field correction using a dome lamp exposure . bad pixels are masked throughout the reduction procedure . additional steps include the distortion correction and the removal of residual airglow lines . this procedure is carried out for both positions a and b. individual frames are combined into an averaged image and an associated noise image . finally , the wavelength calibration is carried out based on a reference image of a th - ar emission spectrum . individual one - dimensional science and error spectra are extracted both to be used for the fitting of emission line profiles . we perform a first attempt at flux calibration by using spectra of bright stars , usually @xmath57 per spectrograph . a single stellar spectrum for each spectrograph is chosen to apply a correction based on the spectroscopic magnitude and photometry from 2mass . an improvement of the absolute flux level is required to account for aperture effects . therefore , we scale the flux of each fmos spectra of our agns to match the deep infrared photometry using total magnitudes available from the ultravista survey available over the cosmos field . while scale factors can reach as high as @xmath33 , the median value is 1.64 . we have observed over 100 type 1 agns in the combined cosmos and ecdf - s fields to date . in figure [ sample ] , we show the x - ray flux and nir magnitude distribution of the 108 agns ( originally identified as _ chandra _ x - ray sources ) in the cosmos field that have @xmath58 , a redshift interval where we are capable of detecting h@xmath3 in the fmos spectroscopic window . the distributions are shown for both the observed objects and the 56 having a significant detection of a broad h@xmath3 emission line ( at the expected wavelength ) that subsequently yielded a black hole mass estimate . prior to our observations , we had no empirical assessment of the performance of fmos thus agn were targeted to faint infrared magnitudes , now understood not to be feasible using the low - resolution mode for the faintest objects ( @xmath59 ) . as shown in figure [ sample](@xmath60 ) , we have a reasonable level of success ( 71% ) with the detection of broad emission lines at brighter magnitudes ( @xmath46 ) . unfortunately , the success rate is significantly lower at fainter magnitudes thus dropping to 50% for the entire sample with @xmath61 . it is worth highlighting that a survey depth of @xmath46 is about two magnitudes fainter than current near - infrared spectroscopic observations of sdss quasars @xcite at similar redshifts . in figure [ lbol_z ] , we demonstrate this by plotting the bolometric luminosity ( based on @xmath62 and a bolometric correction of 5.15 ; * ? ? ? * ) of our agn compared to those from sdss surveys . @xcite describe in detail the benefits of using h@xmath3 for black hole mass measurements . in particular , the h@xmath3 emission line is stronger ( @xmath63 ) thus more easily detected as compared to h@xmath14 . this is clearly evident from our observations . we successfully detect a broad h@xmath3 line in @xmath64% of the cases ( as mentioned above ) while the detection of h@xmath14 is very low ( 17% ) . even so , we do detect h@xmath14 in a fair number of cases up to @xmath65 that will be presented in the full emission - line catalog ( silverman et al . in prep . ) . any future fmos campaign designed to detect the h@xmath14 emission line ( both broad and narrow ) should increase the exposure time significantly and/or use the high - resolution mode that has roughly three times higher throughput in the h band as compared to low - resolution mode ( see figure 19 of * ? ? ? we measure the properties of broad emission lines present in optical and near - infrared spectra to derive single - epoch black hole mass estimates . for the emission lines of interest here ( i.e. , and h@xmath3 ) , we specifically measure the full width at half maximum ( fhwm ) , total luminosity of the emission line in the case of h@xmath3 , and continuum luminosity at 3000 . due to the moderate luminosities of the agn sample , there can be a non - negligible host galaxy contribution that impacts the estimate of the agn continuum at redder wavelengths ; therefore , we chose to use the h@xmath3 line flux rather than the continuum luminosity . fortunately , the multi - wavelength photometry of the cosmos , including the hst imaging , enables us to determine the level of such contamination that will be fully assessed in a future study . emission lines are fit using a procedure as outlined below that enables us to characterize the line shape for even those that have a considerable level of noise . our fitting procedure of the continuum and line emission utilizes mpfitfun , a levenberg - marquardt least squares minimization algorithm as available within the idl environment . even though this routine has well - known computational issues , this algorithm is widely used due to its ease of use and fast execution time . the routine returns best - fit parameters and their errors as well as measure of the goodness of the overall fit . we further describe the individual components required to successfully extract a parameterization of the broad component used in determining virial masses . it is worth recognizing that each line has its own advantages and disadvantages that need to be considered carefully especially when fitting data of moderate signal - to - noise ratio ( s / n ) . a final inspection of each fit by eye is performed to remove obvious cases where a broad component is not adequately determined almost exclusively due to spectra having low s / n . we perform a fit to the h@xmath3 emission line ( if detected within the fmos spectral window ) in order to measure line width and integrated emission - line luminosity . based on the spectroscopic redshift as determined from optical spectroscopy , we select the spectrum at rest wavelengths centered on the emission line and spanning a range that enables an accurate determination of the continuum characterized by a power law , @xmath66 . we employ multiple gaussian components to describe the line profile . it is common practice to make such an assumption on the intrinsic shape of individual components , even though it has been demonstrated that broad - emission lines in agn are not necessarily of such a shape . the h@xmath3 line is fit with two or three gaussians ( including a narrow component ) and the [ ] @xmath676548,6684 lines with a pair of gaussians . the ratio of the [ ] lines is fixed at the laboratory value of 2.96 . the narrow width of the [ ] lines is fixed to match the narrow component of h@xmath3 . the width of the narrow components is limited to @xmath68 km s@xmath6 ( a range not corrected for intrinsic dispersion ) . the velocity profile of the broad components is characterized by the fwhm , measured using either one gaussian or the sum of two gaussians . we then correct the velocity width for the effect of instrumental dispersion to achieve an intrinsic profile width . the h@xmath3 luminosity discussed throughout this work is the sum of the broad components . there are cases for which the fitting routine returns a solution with the width of the narrow component pegged at the upper bound of 800 km s@xmath6 . it is worth highlighting that this minimization routine stops the fitting procedure when a parameters hit a limit thus the returned values are not the true best - fit values . for these , we inspect all fits by eye and decide whether such an additional broad component is real . for many cases , we can use the [ ] @xmath675007 line profile , within the fmos spectral window , to determine whether such a fit to the narrow line complex is accurate . in addition , we can use the available optical spectra for such comparisons . when the level of significance of the narrow line is negligible , we rerun the fitting routine and fix the narrow line width to the spectral resolution of fmos , @xmath69 km s@xmath6 . in figure [ haexam ] , we show three examples of our fits to the h@xmath3 emission line that span a range of line properties . [ [ section ] ] we fit the emission line , as done in @xcite , observed in optical spectra primarily from zcosmos @xcite , magellan / imacs @xcite , and sdss @xcite . the emission line is modeled by a combination of one or two broad gaussian functions to best characterize the line shape . we first remove the continuum ( before attempting to deal with the emission lines ) by fitting the emission in a window surrounding the line . as with h@xmath3 , a power - law function is chosen to best characterize the featureless , non - stellar light attributed to an accretion disk . we further include a broad fe emission component based on an empirical template @xcite that is convolved by a gaussian of variable width and straddles the base of the emission line . a least square minimization is implemented to determine the best - fit parameters . when possible , we optimize residuals of the fits on a case - by - case basis by trying to minimize the number of components . absorption features are either masked out or interpolated across . the fit returns two parameters required for black hole mass estimates : fwhm and monochromatic luminosity at 3000 . examples of our fits to the line are presented in figure [ mgexam ] . we can determine how closely the parameters ( i.e. , luminosity and fwhm ) required to estimate single - epoch black hole masses agree between the h@xmath3 and emission lines . any systematic offset or inherent scatter may only add additional uncertainty to the derived masses . we essentially want to establish whether or not the kinematics of the blr is consistent with photoionized gas in virial motion around the smbh . we provide all measurements and derived masses in table [ catalg ] . our first concern is to determine whether the h@xmath3 emission - line luminosity scales appropriately with the uv continuum luminosity . for the following analysis , we do not correct for extinction due to dust and any contamination by the host galaxy ; the impact of these , thought to be small , will be quantified in a later study . in figure [ lumlum ] , the continuum luminosity , @xmath70 at 3000 , is plotted against the emission - line luminosity of h@xmath3 . our data ( as shown by the red points ) spans two decades in luminosity and exhibits a clear correspondence between continuum and line emission . based on our agn sample , we determine the best - fit linear relation to be @xmath71 . for the linear fitting , we adopt a fitexy method @xcite . the level of dispersion of the data about this fit is 0.20 dex that will contribute to the dispersion in the final mass estimates . rlrrlllllllll 178 & cosmos & 149.58521 & @xmath722.05114 & 1.350 & 43.76@xmath730.03 & 45.35@xmath730.03 & & 3.685@xmath730.013 & 3.811@xmath730.079 & & 8.68@xmath730.03 & 8.78@xmath730.16 + 5275 & cosmos & 149.59021 & @xmath722.77450 & 1.400 & 44.10@xmath730.13 & 45.50@xmath730.02 & & 3.835@xmath730.028 & 3.973@xmath730.072 & & 9.18@xmath730.09 & 9.18@xmath730.14 + 322 & cosmos & 149.62421 & @xmath722.18067 & 1.190 & 43.16@xmath730.14 & 45.02@xmath730.02 & & 3.440@xmath730.044 & 3.581@xmath730.081 & & 7.84@xmath730.12 & 8.17@xmath730.16 + 192 & cosmos & 149.66358 & @xmath722.08522 & 1.220 & 43.24@xmath730.14 & 45.05@xmath730.02 & & 3.339@xmath730.075 & 3.490@xmath730.035 & & 7.68@xmath730.17 & 8.00@xmath730.07 + 157 & cosmos & 149.67512 & @xmath721.98275 & 1.330 & 42.98@xmath730.13 & 44.84@xmath730.03 & & 3.701@xmath730.040 & 3.632@xmath730.017 & & 8.28@xmath730.11 & 8.19@xmath730.04 + we can compare our data set to more luminous quasars from the sdss . in particular , we identify 327 quasars from the sdss sample in @xcite with @xmath74 , that cover a similar luminosity range as our high redshift sample . these were selected from 1178 quasars at @xmath74 based on high - quality data determined by adopting the following criteria : error in @xmath75 , @xmath76 , and @xmath77 . in addition , we include data from a recent study by @xcite that provides the emission line properties including h@xmath3 line of high - luminosity quasars from the sdss with @xmath78 . these samples are added to our data shown figure [ lumlum ] . we clearly see that sdss quasars fall along the @xmath79-@xmath80 relation as established above . furthermore , the sdss quasars have similar dispersion at both low-@xmath81 and high-@xmath81 samples to our sample , @xmath82 and @xmath83 , respectively . we highlight that our agn sample nicely extends such comparisons between continuum luminosity and line emission at higher redshifts and to lower luminosities . we are able to effectively establish a wider dynamic range , not present in the high-@xmath81 sdss sample due to the limited luminosity range around @xmath84 . by merging all three samples , we find the following relation based on a linear fit : @xmath85 . while there is very good agreement between the uv continuum and emission line luminosity , there is a small difference that may impact , even slightly , our comparison of the masses . based on the fmos sample , the mean ratio @xmath86 is @xmath87 , slightly higher than that found for both the low-@xmath81 and high-@xmath81 sdss quasar samples mentioned above ; @xmath88 and @xmath89 , respectively . this can be seen in the inset histogram in figure [ lumlum ] . if this was due to the effect of dust extinction , one would find the opposite trend with reduced uv continuum relative to the h@xmath3 line emission . there may be other explanations such as an underlying sed that may be different for x - ray selected samples @xcite and has an impact on the response seen in photoionized gas , or the effect of the host galaxy on the aperture corrections that differs in each band . while this issue is of importance , we reserve a detailed investigation to subsequent work since it is beyond the scope of this paper to adequately demonstrate such effects . here , we are primarily concerned with the magnitude of a luminosity offset and whether it contributes to an offset between the masses . with black hole mass scaling with the square root of the luminosity ( see below ) , the offset in luminosity , as determined above , amounts to a very small offset in @xmath90 of 0.07 . a second pillar for the use of as a black hole mass indicator is that the emitting - line gas is located essentially within the same clouds that emit balmer emission . while there are claims that this is the case by comparing the velocity profile of with h@xmath14 ( e.g. , * ? * ; * ? ? ? * ) , there are reported differences and trends that are not well understood @xcite . for example , tends to be narrower than h@xmath14 with a difference significantly larger at higher velocity widths ( see figure 2 of * ? ? ? * ) . our aim here is to compare the fwhm of the and h@xmath3 emission lines using a sample not yet explored , namely the moderate - luminosity agns at high - redshifts in survey fields such as cosmos . in figure [ fwhfwh ] , we plot the emission - line velocity width between h@xmath3 and emission lines . based on the fmos sample only , a positive linear correlation is seen between the velocity width of the two emission lines with the mean ratio of @xmath91 ( @xmath92 ) . our data is in very good agreement with those from the sdss , i.e. , @xmath93 ( @xmath94 ) for the low-@xmath81 sample from @xcite , @xmath95 ( @xmath96 ) for the high-@xmath81 sample from @xcite , and recent fmos results from sxds ( see * ? ? ? based on a linear fit to the data shown in figure [ fwhfwh ] , we measure a slope that is consistent with unity ; @xmath97 for fmos only and @xmath98 for fmos@xmath72sdss , and that can not substantiate the claim by @xcite for a shallower value . these results are supportive of a scenario where the and the h@xmath3 emitting regions are essentially co - spatial with respect to the central ionizing source . there are a few noticeable outliers well outside the dispersion of the sample . these objects then appear as outliers when comparing their masses based on different lines in the next section . upon inspection , we find that these are the result of the fmos spectra having low s / n . in some cases , there may be a fit to the h@xmath3 emission line , based on different parameter constraints , that is equally acceptable to the original fit as assessed by a chi - square goodness of fit and has a velocity width in closer agreement with . although , we refrain from such selective fitting in order to present results that may be obtained from using similar fitting algorithms on larger data sets where such close inspection is not feasible . we now calculate black hole masses ( @xmath13 ) based on our single - epoch spectra using both ( i ) @xmath62 and fwhm@xmath99 , and ( ii ) @xmath80 and fwhm@xmath100 . this calculation can be expressed as follows : @xmath101 we explicitly use the recipes provided by @xcite and @xcite for the cases of h@xmath3 and lines , respectively : @xmath102 we note that the calibration of the relation for has been carried out by many studies ( e.g. , * ? ? ? * ; * ? ? ? * ) and there are known differences between them . in figure [ masmas ] , we show @xmath13 for our sample derived from h@xmath3 and . our sample spans a range of @xmath103 consistent with that reported by previous studies of type 1 agns in cosmos @xcite and is complementary to the higher-@xmath104 quasar sample at similar redshifts with @xmath105 @xcite . we find the average ( dispersion ) in the black hole mass ratio of @xmath106 is 0.17 ( @xmath11 ) for our fmos sample . these results are similar to that determined from the sdss sample ; the average ( dispersion ) in the black hole mass ratio is @xmath107 ( @xmath108 ) at low @xmath81 and @xmath109 ( @xmath82 ) at high @xmath81 . while an offset of 0.17 dex is seen in the fmos sample , we conclude that the recipes established using local relations give consistent results between - and h@xmath3-based estimates . we have investigated the emission line properties of agns in cosmos and ecdf - s to establish whether and h@xmath3 provide comparable estimates of their black hole mass . this study is the first attempt to do so for agns of moderate luminosity , hence lower black hole mass ( @xmath110 ) , at high redshift that complements studies of more luminous quasars . our results clearly show that the velocity profiles of and h@xmath3 are very similar when characterized by fwhm and the relation between continuum luminosity and line luminosity is tight . we then find that virial black hole masses based on and h@xmath3 have very similar values and a level of dispersion ( @xmath111 ) comparable to luminous quasars from sdss . it is important to keep in mind that these results pertain to specific calibrations @xcite for estimating black hole mass . the use of other recipes , such as provided by @xcite , will show a discrepancy larger than seen here . to conclude , the locally - calibrated recipes for black holes masses using and h@xmath3 are applicable for fainter agn samples at high redshift . these results further support a lack of evolution in the physical properties of the broad line region in terms of quantities such as @xmath112 ( e.g. , * ? ? ? * ; * ? ? ? * ) , emission - line strengths ( e.g. , * ? ? ? * ) , and the inferred metallicities . as a final word of caution , such estimates of black hole mass are likely to have inherent dispersion as discussed above and systematic uncertainties that are not yet well understood . for instance , recipes for estimating black hole mass depend on the assumption that the gas is purely in virial motion . this is unlikely to be true for all cases since both outflows and inflows are common in agns . even so , there is evidence that the virial product of mass and luminosity ( as a proxy for the radius to the blr ) is a useful probe of the central gravitational potential . in the very least , it is important to establish the level of dispersion in such relations since observed trends usually rely on offsets comparable to the dispersion such as the redshift evolution of the relation between black holes and their host galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? we thank kentaro aoki and naoyuki tamura for their invaluable assistance during our subaru / fmos observations . k.m . acknowledges financial support from the japan society for the promotion of science ( jsps ) . data analysis were in part carried out on common - use data analysis computer system at the astronomy data center , adc , of the national astronomical observatory of japan ( naoj ) . bahcall , j. n. , kozlovsky , b .- z . , & salpeter , e. e. 1972 , , 171 , 467 bentz , m. c. , walsh , j. l. , barth , a. j. , et al . 2009 , , 705 , 199 brusa , m. , civano , f. , comastri , a. , et al . 2010 , , 716 , 348 civano , f. , elvis , m. , brusa , m. , et al . 2012 , , 201 , 30 collin , s. , kawaguchi , t. , peterson , b. m. , et al . 2006 , , 456 , 75 elvis , m. , civano , f. , vignali , c. , et al . 2009 , , 184 , 158 elvis , m. , hao , h. , civano , f. , et al . 2012 , , 759 , 6 fan , x. 2006 , , 50 , 665 green , p. j. , aldcroft , t. l. , richards , g. t. , et al . 2009 , , 690 , 644 greene , j. e. , & ho , l. c. 2004 , , 610 , 722 greene , j. e. , & ho , l. c. 2005 , , 630 , 122 greene , j. e. , & ho , l. c. 2007 , , 670 , 92 hao , h. , elvis , m. , kelly , b. c. , et al . 2012 , arxiv:1210.3044 hopkins , p. f. , richards , g. t. , & hernquist , l. 2007 , , 654 , 731 iwamuro , f. , motohara , k. , maihara , t. , et al . 2001 , , 53 , 355 iwamuro , f. , moritani , y. , yabe , k. , et al . 2012 , , 64 , 59 kaspi , s. , smith , p. s. , netzer , h. , et al . 2000 , , 533 , 631 kaspi , s. , maoz , d. , netzer , h. , et al . 2005 , , 629 , 61 kaspi , s. , brandt , w. n. , maoz , d. , et al . 2007 , , 659 , 997 kelly , b. c. , & shen , y. 2012 , arxiv:1209.0477 kimura , m. , maihara , t. , iwamuro , f. , et al . 2010 , , 62 , 1135 lehmer , b. d. , brandt , w. n. , alexander , d. m. , et al . 2005 , , 161 , 21 lilly , s. j. , le brun , v. , maier , c. , et al . 2009 , , 184 , 218 luo , b. , brandt , w. n. , xue , y. q. , et al . 2010 , , 187 , 560 maihara , t. , iwamuro , f. , yamashita , t. , et al . 1993 , , 105 , 940 maiolino , r. , mannucci , f. , baffa , c. , et al . 2001 , , 372 , l5 marziani , p. , sulentic , j. w. , stirpe , g. m. , et al . 2009 , , 495 , 83 marziani , p. , sulentic , j. w. , plauchu - frayn , i. , & del olmo , a. 2013 , arxiv:1301.0520 matsuoka , k. , nagao , t. , marconi , a. , et al . 2011 , , 527 , a100 mccracken , h. j. , milvang - jensen , b. , dunlop , j. , et al . 2012 , , 544 , a156 mcgill , k. l. , woo , j .- h . , treu , t. , & malkan , m. a. 2008 , , 673 , 703 mclure , r. j. , & jarvis , m. j. 2002 , , 337 , 109 mclure , r. j. , & dunlop , j. s. 2004 , , 352 , 1390 merloni , a. , & heinz , s. 2008 , , 388 , 1011 merloni , a. , bongiorno , a. , bolzonella , m. , et al . 2010 , , 708 , 137 nagao , t. , marconi , a. , & maiolino , r. 2006 , , 447 , 157 nobuta , k. , akiyama , m. , ueda , y. , et al . 2012 , , 761 , 143 osterbrock , d. e. , & pogge , r. w. 1987 , , 323 , 108 park , d. , kelly , b. c. , woo , j .- h . , & treu , t. 2012 , , 203 , 6 peng , c. y. , impey , c. d. , ho , l. c. , et al . 2006 , , 640 , 114 peterson , b. m. 1993 , , 105 , 247 press , w. h. , teukolsky , s. a. , vetterling , w. t. , & flannery , b. p. 1992 , numerical recipes in c. the art of scientific computing , cambridge : university press , 2nd ed . roseboom , i. g. , bunker , a. , sumiyoshi , m. , et al . 2012 , , 426 , 1782 schramm , m. , & silverman , j. d. 2012 , arxiv:1212.2999 shen , y. , greene , j. e. , strauss , m. a. , et al . 2008 , , 680 , 169 shen , y. , richards , g. t. , strauss , m. a. , et al . 2011 , , 194 , 45 shen , y. , & kelly , b. c. 2012 , , 746 , 169 shen , y. , & liu , x. 2012 , , 753 , 125 silverman , j. d. , mainieri , v. , salvato , m. , et al . 2010 , , 191 , 124 steffen , a. t. , strateva , i. , brandt , w. n. , et al . 2006 , , 131 , 2826 steinhardt , c. l. , & elvis , m. 2010 , , 402 , 2637 steinhardt , c. l. , & silverman , j. d. 2011 , arxiv:1109.1554 szokoly , g. p. , bergeron , j. , hasinger , g. , et al . 2004 , , 155 , 271 tamura , n. , ohta , k. , & ueda , y. 2006 , , 365 , 134 trakhtenbrot , b. , & netzer , h. 2012 , , 427 , 3081 trump , j. r. , impey , c. d. , mccarthy , p. j. , et al . 2007 , , 172 , 383 trump , j. r. , impey , c. d. , kelly , b. c. , et al . 2009 , , 700 , 49 trump , j. r. , impey , c. d. , kelly , b. c. , et al . 2011 , , 733 , 60 vestergaard , m. , & wilkes , b. j. 2001 , , 134 , 1 vestergaard , m. , & peterson , b. m. 2006 , , 641 , 689 vestergaard , m. , & osmer , p. s. 2009 , , 699 , 800 wang , j .- g . , dong , x .- b . , wang , t .- , et al . 2009 , , 707 , 1334 xue , y. q. , luo , b. , brandt , w. n. , et al . 2011 , , 195 , 10 yabe , k. , ohta , k. , iwamuro , f. , et al . 2012 , , 64 , 60 york , d. g. , adelman , j. , anderson , j. e. , jr . , et al . 2000 , , 120 , 1579

we present an analysis of broad emission lines observed in moderate - luminosity active galactic nuclei ( agns ) , typical of those found in x - ray surveys of deep fields , with the aim to test the validity of single - epoch virial black hole mass estimates . we have acquired near - infrared ( nir ; @xmath0 ) spectra of agns up to @xmath1 in the cosmos and extended _ chandra _ deep field - south survey , with the fiber multi - object spectrograph ( fmos ) mounted on the subaru telescope . these low - resolution ( @xmath2 ) nir spectra provide a significant detection of the broad h@xmath3 emission line that has been shown to be a reliable probe of black hole mass at low redshift . our sample has existing optical spectroscopy ( through programs such as zcosmos ) which provides a detection of , a broad emission line typically used for black hole mass estimation at @xmath4 . we carry out a spectral - line fitting procedure using both h@xmath3 and to determine the virial velocity of gas in the broad line region , the monochromatic continuum luminosity at 3000 , and the total h@xmath3 line luminosity . with a sample of 43 agns spanning a range of two decades in luminosity ( i.e. , @xmath5 ergs s@xmath6 ) , we find a tight correlation between the rest - frame ultraviolet and emission - line luminosity with a distribution characterized by @xmath7 and a dispersion @xmath8 . there is also a close one - to - one relationship between the fwhm of h@xmath3 and of up to 10000 km s@xmath6 with a dispersion of 0.14 in the distribution of the logarithm of their ratios . both of these then lead to there being very good agreement between h@xmath3- and -based masses over a wide range in black hole mass ( i.e. , @xmath9 ) . we do find a small offset in -based masses , relative to those based on h@xmath3 , of @xmath10 dex and a dispersion @xmath11 . in general , these results demonstrate that local scaling relations , using or h@xmath3 , are applicable for agn at moderate luminosities and up to @xmath12 .

superstring theories have been , for now a number of years , the most promising candidates for physics beyond the standard model ( sm ) . two major problems , however , have impeded extracting definite phenomenological predictions from these constructions : the large vacuum degeneracy and the issue of supersymmetry ( susy ) breaking . a large amount of work has been devoted to studying both questions , which has led to the proposal of several mechanisms and models in order to solve them . among those , gaugino condensation @xcite is the most promising one , and it has been implemented , more or less successfully , in superstring inspired scenarios . this is a non - perturbative effect which provides the typically flat stringy fields , the dilaton and the moduli , with a non - trivial potential which could eventually lead to their stabilization at realistic values . it can also give rise to susy breaking at the so - called condensation scale ( @xmath0 where @xmath1 is the reduced planck mass and @xmath2 is the string coupling constant ) , relating therefore both major problems of superstring phenomenology . so far all this is happening in the hidden sector of the theory , governed by a strong - type interaction , and , in this simple picture , gravity is responsible for transmitting the breakdown of susy to the observable sector ( where the sm particles and susy partners live ) , parameterized in terms of the gravitino mass @xmath3 which sets the scale of the soft breaking terms . on the other hand , it is typical of many superstring constructions to have anomalous @xmath4 symmetries whose anomaly cancellation is implemented by a green - schwarz mechanism @xcite , where the dilaton plays a crucial role . this anomalous symmetry induces a fayet - iliopoulos ( f - i ) d - term in the scalar potential , and this will generate extra contributions to the soft terms . therefore one might expect an interesting interplay between susy breaking through gaugino condensation and the presence of the @xmath4 . moreover , there is now a new scale in the theory , that of the @xmath4 breakdown , @xmath5 . given that both hidden and observable sector fields are charged under this symmetry , the f - i term will act effectively as an extra source of transmission of the susy breaking between sectors . in fact , given that in general @xmath6 , one expects that the f - i will set the scale of the soft breaking terms . all these issues have been recently discussed in a number of papers @xcite , but always in a global susy limit . in ref . @xcite it was pointed out that the contribution from the d - terms to the soft breaking terms is the dominant one , which was claimed to have phenomenological merits . in particular , this could be useful to get naturally universal soft masses , thus avoiding dangerous flavour changing neutral current effects . on the other hand , in ref . @xcite it was shown that that discussion had ignored the crucial role played by the dilaton in the analysis . furthermore , it was claimed ( based on particular ansatzs for the khler potential ) that the situation is in fact the opposite , namely the contribution from the f - terms to the soft breaking terms is the dominant one . here , we present the generalization of the results obtained in ref.@xcite to the supergravity ( sugra ) case , which is the correct framework in which to deal with effective theories coming from strings . whereas we have checked that the sugra corrections do not affect significantly the minimization of the potential , and thus the vacuum structure of the theory , we have seen that they are crucial for a correct treatment of the soft breaking terms . this is due to the fact that the analysis of the soft terms relies heavily on the cancellation of the cosmological constant , an issue which can only be properly addressed in the context of sugra . this single extra requirement will introduce very powerful constraints on the structure of the soft terms , which entirely contradict previous results . the structure of the paper is as follows : in section ii we first extend the formulation of the f - i term made in ref.@xcite to sugra , as a previous stage to analyze the vacuum structure of a typical model first presented in @xcite . in section iii we study the cancellation of the cosmological constant , and we apply the constraints we get from it to the calculation of the soft breaking terms , obtaining a definite hierarchy between the different contributions to the scalar masses and the gaugino mass . in section iv we illustrate our general results with a particular example of a gaugino condensation mechanism that stabilizes the dilaton , namely one condensate with non - perturbative corrections to the khler potential . finally , in section v we present our conclusions . before studying the interplay between gaugino condensation and the presence of a fayet - iliopoulos d - term in string theories , let us briefly introduce the general sugra formulation of such a f - i term . as has been stressed by arkani - hamed et al . @xcite the dilaton field , @xmath7 plays a crucial role in this task . under an anomalous @xmath4 transformation with gauge parameter @xmath8 , @xmath7 transforms as @xmath9 , where the green - schwarz coefficient , @xmath10 , is proportional to the apparent @xmath4 anomaly _ gs=_i q_i , [ dgs ] with @xmath11 the @xmath4 charges of the matter fields , @xmath12 ( typically @xmath13 ) . in order to be gauge invariant the khler potential must be a function of @xmath14 , where x is the vector superfield of @xmath4 @xcite . therefore , the @xmath4 invariant @xmath15-part of the sugra action reads _ d = d^4 [ ld ] with k = k(^*_i e^2q_ix , _ i ; s+|s - _ gsx ) . [ kx ] extracting the piece proportional to @xmath16 in eq . ( [ ld ] ) , and eliminating @xmath16 through its equation of motion we find @xmath17 $ ] , where is as in ref . @xcite . ] @xmath18 is the gauge coupling , the prime indicates a derivative with respect to the dilaton @xmath7 and @xmath19 indicates derivatives of the khler potential with respect to @xmath20 . consequently , the contribution to the potential is v_d= d_x^2 = g_x^2 ^2 [ vd ] with ^2= - . [ xi ] in writing eq . ( [ vd ] ) we have absorbed the @xmath21 factor into the redefinition of the vierbein @xmath22 , as it is usually done in sugra theories to get a standard gravity action . ( [ vd],[xi ] ) were obtained in ref . @xcite in the global susy picture . the expression of @xmath23 remains as given in ref . @xcite . let us now turn to the gaugino condensation effects , and how are they affected by the presence of the f - i potential . in order to discuss this issue , let us consider in detail the model presented initially by bintruy and dudas @xcite and recently reanalyzed by arkani - hamed et al . @xcite . the starting point is a scenario with gaugino condensation in the hidden sector of the theory , originated by a @xmath24 strong - like interaction . following @xcite we take the number of flavours @xmath25 , which corresponds to chiral superfields @xmath26 , @xmath27 that transform under @xmath28 as ( @xmath29,@xmath30 ) and ( @xmath31,@xmath32 ) respectively . the spectrum in this sector is completed by a @xmath24 singlet , @xmath33 , which has charge @xmath34 under the @xmath4 . it is also assumed that @xmath23 has positive sign . the superpotential of this model is given by w = m ( ) ^q+|q + ( n_c-1 ) ( ) ^ , [ sup ] where @xmath35 is the meson superfield and @xmath36 is the condensation scale , which is related to the dilaton by [ wnp ] ( ) ^3n_c-1 = e^-2(q+|q)s/_gs . as for the khler potential , @xmath37 , it was assumed in @xcite that it consists of a dilaton dependent part plus canonical terms for @xmath38 and @xmath33 . the scalar potential for such a theory in the framework of sugra is given by v = e^k / m_p^2 \ { | w+k |^2 + | w _ + k _ |^2 + | w_t+k_t |^2 - 3 | |^2 } + d_x^2 , [ pot ] where , as before , a prime indicates a derivative with respect to @xmath7 , and the subindices indicate derivatives of the superpotential and khler potential with respect to the corresponding fields . the terms generated by the sugra corrections are indicated explicitly by the presence of inverse powers of the planck mass , so that in the limit @xmath39 we recover the global susy case studied by the authors of refs . @xcite and @xcite . the d - part of the potential reads d_x^2 = g_x^2 ^2 . [ vd2 ] this model leads to susy breaking by gaugino condensation ( provided that @xmath7 is stabilized at a non - trivial value ) . in order to study the phenomenological implications of the presence of the f - i term , we have to compute the soft breaking terms , separating the f - i contribution . the first step in this task is to minimize the potential . following ref . @xcite we define the parameters & = & m ( ) ^q+|q , + & & [ exp ] + & = & ( ) ^ ( ) ^ , with @xmath40 . we have checked that the expansion in @xmath41 presented by refs . @xcite , around @xmath42 and @xmath43 , is still correct in the sugra case . to be more precise , we will parameterize & = & ^2 [ 1 + ( q+|q ) + a ^2 + ... ] , + & & [ sols ] + < t^2 > & = & 2 ^2 . it is straightforward to check that the minimization of @xmath44 with respect to the @xmath33 , @xmath38 fields at lowest order in @xmath41 imposes the form of the lowest order terms in eq . ( [ sols ] ) . in order to evaluate the next to leading order coefficients in this expansion , @xmath45 and @xmath46 , we have to solve the minimization conditions to the next order in @xmath41 . for this matter , and future convenience , the following ( lowest order in @xmath41 ) expressions are useful w & = & ^2 n_c , [ w ] + d_x & = & g^2 ^2 ^2 ( a - ( ) b ) [ dx ] , + [ wt ] & = & ^3/2 , + & = & ( ) , + & = & - ^2 . now , from the minimization condition , @xmath47 , we get b & = & b_0 + ( + ) , where @xmath48 is the result obtained in @xcite in the global susy case b_0 & = & ( ) . from the second minimization condition , @xmath49 , we get a = ( ) b - ( 1 - ( 1- ) + _ sugra ) , where _ sugra = k ( _ gs - ) - ( - 2 ( ) k + ( n_c+ ) _ gs k ) - , [ delta ] to be compared to the global case a_0 = ( ) b_0 - ( 1 - ( 1- ) ) . from the previous expressions we can write the final form of the d - term [ dd ] d_x = e^k ^2 ^2 ( ) ^2 ( 1 - ( 1- ) + _ sugra ) , where @xmath50 was given in eq . ( [ delta ] ) . finally , the third minimization condition , @xmath51 translates into an equation for the value of the third derivative of the khler potential , [ k ] & = & - ( 1 - ) ^2 + 2 ( ) ^2 + & & - ( ) ^3 . for phenomenological consistency we will assume that @xmath52 satisfies ( [ k ] ) at @xmath53 . notice that , since @xmath54 is small , one expects [ k3k2 ] kk , as was already pointed out in ref . @xcite . in that reference it was also claimed that one expects @xmath55 , based on a particular ansatz for @xmath37 . this was crucial to get the result that the f - term contribution ( in particular the @xmath56 one ) to the soft terms dominates over the d - term one . however , as we shall shortly see , this is not a consequence of the minimization and , in fact , general arguments indicate that the most likely case is precisely the opposite . in order to get quantitative results for the soft terms we need , beside the above minimization conditions , an additional condition , which is provided by the requirement of a vanishing cosmological constant . notice from eqs . ( [ w][wt ] ) that , for this matter , the d - part of the potential and the @xmath57 term in eq . ( [ pot ] ) , being of order @xmath58 and @xmath59 respectively , are irrelevant . consequently , at order @xmath60 the cancellation of the cosmological constant reads v e^k \ { | w+kw |^2 + | w _ + k _ w |^2 - 3 noticed at first sight that @xmath61 can not be much larger than @xmath62 , otherwise the first term above can not be cancelled by the @xmath63 term . actually , from ( [ cosmo ] ) it is possible to get non - trivial bounds on the relative ( and absolute ) values of @xmath64 and thus on the relative size of the various contributions to the soft terms . it is important to keep in mind that @xmath65 and that both @xmath66 have positive sign ( the former by assumption , the latter from positivity of the kinetic energy ) . ( [ cosmo ] ) translates into f(^2 ) ( + ^2)^2 = 3 ^2 , [ fsffi ] where f(^2 ) 1 + . [ fxi ] the @xmath67 in the right hand side corresponds to the @xmath68 contribution while the other term comes from the @xmath69 contribution . note that @xmath70 , reflecting the fact that both contributions are positive definite . we can treat eq . ( [ fsffi ] ) as a quadratic equation in @xmath23 which has @xmath71-dependent solutions given by ^2 = ( 1 ) ^2 . [ fsffi2 ] the existence of solutions requires a positive square root . hence , @xmath71 is constrained to be within the range 1 f(^2 ) , [ bounds ] which in particular implies that [ qbound ] . actually , @xmath72 is also bounded due to phenomenological reasons . namely @xmath73 should be @xmath74(1 tev ) to guarantee reasonable soft terms . since the perturbative and non - perturbative contributions to @xmath75 ( see eq . ( [ sup ] ) ) are of the same size at the minimum , we may apply this condition to the non - perturbative piece , i.e. [ westim ] |w_np| = ( n_c-1 ) ( e^- 2 ( ) s/ ) ^1/(- 1 ) ~o(1 ) , where we have used eq . ( [ wnp ] ) and everything is expressed in planck units . using @xmath76 we get @xmath77 . since @xmath78 , we finally get [ estimate ] ~9 . now , a non - trivial result about the relative sizes of @xmath61 and @xmath62 can be derived from the right hand side of the eq . ( [ bounds ] ) . by writing @xmath71 explicitly in terms of these derivatives of the khler potential and using eq . ( [ estimate ] ) we get | | - , which results in the general bound | | . [ kpkpp ] this bound means in particular that the @xmath79 assumption made in ref . @xcite is clearly inconsistent with the cancellation of the cosmological constant . as mentioned at the end of section ii , this assumption was crucial for the results obtained in that paper concerning the relative size of the f and d contributions to the soft terms . this suggests that those results must be revised , as we are about to do . besides eq . ( [ kpkpp ] ) , eq . ( [ fsffi ] ) provides interesting separate constraints on @xmath64 . namely , from eq . ( [ fsffi ] ) we can write the inequality @xmath80 , which implies ( 1 - ) ^2 ^2 ( 1 + ) ^2 . [ ffi ] this translates into an allowed range for @xmath61 , since @xmath65 . on the other hand , eq . ( [ fsffi ] ) also implies the inequality @xmath81 and , thus k ( ) ^2 ~o(100 ) . [ kss ] let us notice that the bounds eq . ( [ ffi ] ) and eq . ( [ kss ] ) are a consequence of imposing that neither the @xmath82 nor the @xmath83 contributions ( both positive ) may be larger then the ( negative sign ) contribution @xmath84 in eq . ( [ cosmo ] ) . it is also interesting to discuss in which cases the @xmath83 contribution dominates over the @xmath85 one or vice - versa . the @xmath83 contribution is maximized when @xmath71 is as large as possible , i.e. when the upper bound in eq . ( [ bounds ] ) gets saturated . then @xmath86 and hence ( see eqs . ( [ xi ] , [ estimate ] ) ) [ kp18 ] |k| ~18 . if @xmath87 is smaller ( larger ) than 18 , @xmath85 becomes dominant and @xmath23 tends to left ( right ) hand limit of ( [ ffi ] ) . condition ( [ kp18 ] ) does not guarantee that the @xmath83 contribution _ dominates _ over the @xmath85 one . this would require @xmath88 , i.e. @xmath89 and , from eq . ( [ bounds ] ) , @xmath90 . the latter condition clearly shows that @xmath83 dominance can only happen in a very restricted region of parameter space , and therefore is unlikely to appear in explicit constructions . let us turn now to the important issue of the soft terms , and how are they constrained by the previous bounds . the soft mass of any matter field , @xmath91 , is given by [ mfi ] m_^2 = m_f^2 + m_d^2 , where [ mfmd ] m_f^2=e^k |w|^2 m_d^2=- q _ are the respective contributions from the f and d terms to @xmath92 . here @xmath93 is the @xmath91 anomalous charge and we have assumed a canonical kinetic term for @xmath91 in the khler potential ( as for @xmath38 and @xmath33 ) . from eq . ( [ dd ] ) [ dsoft ] < d_x>= o(1)e^k ^2 ^2 ( ) ^2 . so , using eq . ( [ w ] ) we get [ mdmf ] = o(1 ) q_()^2 ( ) ^4 o(1)()^2 . this means that for @xmath94 the d contribution to the soft masses will be the dominant one ( contrary to what was claimed in ref . the @xmath95 case can not be excluded , but we could not implement it with the explicit example we use in the next section . on the other hand , notice that , since @xmath96 ( see eq . ( [ xi ] ) ) , the f - i scale becomes in this case comparable to @xmath1 ( or larger ) . so it is not surprising that for large @xmath61 the role of gravity as the messenger of the ( f - type ) susy breaking is not overriden by the `` gauge mediated '' ( d - type ) susy breaking associated to the f - i term . concerning gaugino masses , these are given by [ gaugino ] m_^2 = ( k)^-2 e^k | w + kw|^2 . using @xmath97 ( to allow @xmath98 in eq . ( [ cosmo ] ) ) we get [ gaugino2 ] m_^2 ( k)^-1 e^k masses are strongly suppressed with respect to the scalar masses , which poses a problem of naturality . namely , since gaugino masses must be compatible with their experimental limits , the scalar masses must be much higher than 1 tev , leading to unnatural electroweak breaking . this conclusion seems inescapable in this context . summarizing , the hierarchy of masses we expect is [ hierarchy ] m_^2 m_f^2 < m_d^2 , although the last inequality might be reversed in special cases . in this section we want to illustrate the points we have just been discussing with a particular example . for that purpose we shall take a model of gaugino condensation in which the dilaton is stabilized by non perturbative corrections to the khler potential @xcite ; in particular the ansatz we shall use is k = -(2 re s ) + k_np , where @xcite k_np = ( 1+e^-b ( - ) ) . [ ours ] this function depends on three parameters , @xmath99 , @xmath15 , and @xmath100 , the first of which just determines the value of @xmath101 at the minimum . since @xmath78 , we shall fix @xmath102 from now on . therefore this description is effectively made in terms of only @xmath15 and @xmath100 , which are positive numbers . the ansatz eq . ( [ ours ] ) can naturally implement the hierarchy @xmath103 , which , as we have seen in previous sections ( cf . ( [ k3k2 ] , [ kpkpp ] ) ) , is required to stabilize the dilaton at zero cosmological constant . actually , there is a curve of values of @xmath15 vs. @xmath100 for which this happens . this is shown in fig . 1 for a few different values of @xmath104 ( @xmath54 being determined by imposing a correct phenomenology ) ; all of them below the upper bound of @xmath105 found in the previous section , see eq . ( [ qbound ] ) . values of the @xmath100 and @xmath15 parameters ( in logarithmic scales ) that give a zero cosmological constant with the ansatz of eq . ( [ ours ] ) . the hidden gauge group is su(5 ) , and @xmath106 for all the curves . the solid line corresponds to @xmath107 and @xmath108 , the dashed line to @xmath109 and @xmath110 and the dash - dotted line to @xmath111 and @xmath112 . the thin lines show the corresponding lower bound on @xmath100 obtained from eq . ( [ lowbbound ] ) . the sizes of @xmath61 and @xmath62 at the minimum are & & + eq . ( [ kpkpp ] ) , while the first one has implications on the size of the soft terms as we shall shortly see . concerning the size of the different terms in the potential , it is remarkable how , for small ( big ) values of the @xmath15 ( @xmath100 ) parameter it is the @xmath113 term the one that dominates over @xmath114 in the potential and therefore cancels the @xmath84 . given that @xmath115 that means that in this region of the parameter space ( see the discussion after eq . ( [ kp18 ] ) ) ^2 = ( 1 - ) ^2 . [ xim ] as @xmath100 decreases ( i.e. @xmath15 increases ) the size of both @xmath116 terms becomes comparable and , eventually , the value of @xmath23 tends to a constant value given by @xmath104 . this can be understood from eq . ( [ ours ] ) in the asymptotic regime of `` small '' @xmath100 ( see fig . 1 ) . defining @xmath117 we have in this regime k_np|_min & ~ & , + k_np|_min & ~ & , [ kpmin ] + k_np|_min & ~ & , where the subscript @xmath118 denotes values at the minimum . we can now evaluate @xmath71 , which is given by f(^2 ) = 1- ~1 + . imposing the limit of eq . ( [ qbound ] ) , @xmath119 $ ] , we get a lower bound on @xmath100 , which is the one shown in fig . 1 for each example , [ lowbbound ] b ~ . it can also be seen in fig . 1 that @xmath100 approaches asymptotically this bound , and therefore @xmath71 will tend to a constant value in this limit . using eq . ( [ fsffi2 ] ) we finally obtain that @xmath23 goes to an asymptotic value given by @xmath72 . using eq . ( [ mdmf ] ) this means that the ratio @xmath120 tends to @xmath67 . in fact , since @xmath94 in these particular models , this same equation tells us that @xmath120 approaches @xmath67 from below . in other words , we are in a scenario where the scalar soft masses are dominated by the @xmath15-term ( see the discussion at the end of section iii ) . all this is illustrated in fig . 2 for the same three cases presented in fig . 1 . plot of @xmath120 vs @xmath15 , with @xmath100 fixed to the values given by the previous figure . same cases as before . the thin lines show the corresponding lower bound on @xmath120 derived from eq . ( [ qbound ] ) . the left hand limit of each curve is determined by the corresponding value of @xmath23 given by eq . ( [ xim ] ) , that is corresponds to @xmath121 driven by @xmath122 , whereas as we move in the parameter space towards larger ( smaller ) values of @xmath15 ( @xmath100 ) the @xmath114 term contributes more to the cancellation of the cosmological constant until we reach the constant value defined by @xmath123 . moreover the ratio @xmath120 tends to its maximum value . therefore this is an ansatz for @xmath124 for which we get the different possibilities for cancelling the cosmological constant , generically with the soft scalar masses dominated by their @xmath15 contribution . four - dimensional superstring constructions frequently present an anomalous @xmath4 , which induces a fayet - iliopoulos ( f - i ) d - term . on the other hand susy breaking through gaugino condensation is a desirable feature of string scenarios . the interplay between these two facts is not trivial and has important phenomenological consequences , especially concerning the size and type of the soft breaking terms . previous analyses @xcite , based on a global susy picture , led to contradictory results regarding the relative contribution of the f and d terms to the soft terms . in this paper we have examined this issue , generalizing the work of ref . @xcite to sugra , which is the appropriate framework in which to deal with effective theories coming from strings . we have extended the formulation of the f - i term made in @xcite to the sugra case , as well as considered the complete sugra potential in the analysis . this allows to properly implement the cancellation of the cosmological constant ( something impossible in global susy ) , which is crucial for a correct treatment of the soft breaking terms . moreover this condition yields powerful constraints on the khler potential , which leads to definite predictions on the relative size of the contributions to the soft terms . in particular we obtain the following hierarchy of masses [ hierarchy2 ] m_^2 m_f^2 < m_d^2 , where @xmath125 , @xmath126 are the contributions from the f and d terms , respectively , to the soft scalar masses , and @xmath127 are the gaugino masses . these results amend those obtained in ref . @xcite . the last inequality could be reversed if @xmath61 , i.e. the derivative of the khler potential with respect to the dilaton at the minimum , is large ( @xmath128 ) . this is not surprising , since in this way the f - i scale ( which is proportional to @xmath61 ) can be made comparable to @xmath1 , so that the role of gravity as the messenger of the ( f - type ) susy breaking is not overriden by the `` gauge mediated '' ( d - type ) susy breaking associated to the f - i term . this situation can not be excluded , but in our opinion it is unlikely to happen . the first inequality in ( [ hierarchy2 ] ) is rather worrying since it poses a problem of naturality . namely , given that gaugino masses must be compatible with their experimental limits , the scalar masses must be much higher than 1 tev , leading to unnatural electroweak breaking . this conclusion seems inescapable in this context . finally , we have illustrated all our results with explicit examples , in which the dilaton is stabilized by a gaugino condensate and non - perturbative corrections to the khler potential , keeping a vanishing cosmological constant . there it is shown how the various contributions to the soft terms are in agreement to the hierarchy expressed in eq . ( [ hierarchy2 ] ) . jac and jmm thank michael dine and steve martin for useful discussions held at the university of santa cruz , and the nato project crg 971643 that made it possible . the work of tb was supported by jnict ( portugal ) , bdc was supported by pparc and jac and jmm were supported by cicyt of spain ( contract aen95 - 0195 ) . finally , the authors would like to thank the british council / acciones integradas program for the financial support received through the grant hb1997 - 0073 . derendinger , l.e . ibez and h.p . nilles , ; m. dine , r. rohm , n. seiberg and e. witten , . m. green and j. schwarz , . m. dine , n. seiberg and e. witten , . casas , e.k . katehou and c. muoz , . p. bintruy and e. dudas , . n. arkani - hamed , m. dine and s. martin , . g. dvali and a. pomarol , . p. bintruy and p. ramond , ; p. bintruy , s. lavignac , p. ramond , ; p. bintruy , n. irges , s. lavignac , p. ramond ; j.k . elwood , n. irges and p. ramond , ; n. irges , s. lavignac , p. ramond , ; z. lalak , ; s.f . king and a. riotto , hep - ph/9806281 . faraggi and j.c . pati , hep - ph/9712516 . shenker , proceedings of the cargese school on random surfaces , quantum gravity and strings , cargese ( france ) , 1990 . t. banks and m. dine , . casas , ; p. binetruy , m.k . gaillard and y .- y . t. barreiro , b. de carlos and e.j . copeland , .

the interplay between gaugino condensation and an anomalous fayet - iliopoulos term in string theories is not trivial and has important consequences concerning the size and type of the soft susy breaking terms . in this paper we examine this issue , generalizing previous work to the supergravity context . this allows , in particular , to properly implement the cancellation of the cosmological constant , which is crucial for a correct treatment of the soft breaking terms . we obtain that the d - term contribution to the soft masses is expected to be larger than the f - term one . moreover gaugino masses must be much smaller than scalar masses . we illustrate these results with explicit examples . all this has relevant phenomenological consequences , amending previous results in the literature . # 1#2#3_nucl . phys . _ * b#1 * ( 19#2 ) # 3 # 1#2#3_phys . lett . _ * b#1 * ( 19#2 ) # 3 # 1#2#3_phys . rev . _ * d#1 * ( 19#2 ) # 3 # 1#2#3_phys . rev . lett . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_z . phys . _ * c#1 * ( 19#2 ) # 3 # 1#2#3_prog . theor . phys . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_mod . phys . lett . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_phys . rep . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_ann . phys . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_rev . mod . phys . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_helv . phys . acta _ * # 1 * ( 19#2 ) # 3 # 1#2#3_phys . rev . _ * d#1 * ( 19#2 ) # 3 # 1#2#3_phys . rev . lett . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_z . phys . _ * c#1 * ( 19#2 ) # 3 # 1#2#3_prog . theor . phys . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_mod . phys . lett . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_phys . rep . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_ann . phys . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_rev . mod . phys . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_helv . phys . acta _ * # 1 * ( 19#2 ) # 3 # 1#2#3_jetp lett . _ * # 1 * ( 19#2 ) # 3 # 1#2#3_j . phys . g._*g#1*(19#2 ) # 3 # 1#2#3_int . j. mod . phys . _ * a#1 * ( 19#2 ) # 3

lebesgue ( @xcite , 1904 ) was probably the first to show an example of a real function on the reals satisfying the rather surprising property that it takes on each real value in any nonempty open set ( see also @xcite ) . the functions satisfying this property are called _ everywhere surjective _ ( functions with even more stringent properties can be found in @xcite ) . of course , such functions are nowhere continuous but , as we will see later , it is possible to construct a _ lebesgue measurable _ everywhere surjective function . entering a very different realm , in 1906 pompeiu @xcite was able to construct a nonconstant differentiable function on the reals whose derivative _ vanishes on a dense set . _ passing to several variables , the first problem one meets related to the `` minimal regularity '' of functions at a elementary level is that of whether separate continuity implies continuity , the answer being given in the negative . in this paper , we will consider the families consisting of each of these kinds of functions , as well as two special families of sequences , and analyze the existence of large algebraic structures inside all these families . nowadays the topic of lineability has had a major influence in many different areas on mathematics , from real and complex analysis @xcite , to set theory @xcite , operator theory @xcite , and even ( more recently ) in probability theory @xcite . our main goal here is to continue with this ongoing research . .15 cm let us now fix some notation . as usual , we denote by @xmath0 and @xmath1 the set of positive integers , the set of rational numbers and the set of all real numbers , respectively . the symbol @xmath2 will stand for the vector space of all real continuous functions defined on an interval @xmath3 . in the special case @xmath4 , the space @xmath5 will be endowed with the topology of the convergence in compacta . it is well known that @xmath5 under this topology is an @xmath6-space , that is , a complete metrizable topological vector space . .15 cm by @xmath7 it is denoted the family of lebesgue measurable everywhere surjective functions @xmath8 . a function @xmath9 is said to be a _ pompeiu function _ ( see figure [ pompeiu ] ) provided that it is differentiable and @xmath10 vanishes on a dense set in @xmath1 . the symbols @xmath11 and @xmath12 stand for the vector spaces of pompeiu functions and of the derivatives of pompeiu functions , respectively . additional notation will be rather usual and , when needed , definitions will be provided . .15 cm the organization of this paper is as follows . in section 2 , a number of concepts concerning the linear or algebraic structure of sets inside a vector space or a linear algebra , together with some examples related to everywhere surjectivity and special derivatives , will be recalled . sections 3 , 4 , and 5 will focus on diverse lineability properties of the families @xmath7 , @xmath13 , @xmath12 , and certain subsets of discontinuous functions , so completing or extending a number of known results about several strange classes of real functions . concerning sequence spaces , section 6 will deal with subsets of convergent and divergent series for which classical tests of convergence fail and , finally , in section 7 convergence in measure versus convergence almost everywhere will be analyzed in the space of sequences of measurable lebesgue functions on the unit interval . a number of concepts have been coined in order to describe the algebraic size of a given set ; see @xcite ( see also the survey paper @xcite and the forthcoming book @xcite for an account of lineability properties of specific subsets of vector spaces ) . namely , if @xmath14 is a vector space , @xmath15 is a cardinal number and @xmath16 , then @xmath17 is said to be : 1 . _ lineable _ if there is an infinite dimensional vector space @xmath18 such that @xmath19 , 2 . _ @xmath15-lineable _ if there exists a vector space @xmath18 with dim@xmath20 and @xmath19 ( hence lineability means @xmath21-lineability , where @xmath22 , the cardinality of @xmath23 ) , and 3 . _ maximal lineable _ in @xmath14 if @xmath17 is @xmath24-lineable . if , in addition , @xmath14 is a topological vector space , then @xmath17 is said to be : 1 . _ dense - lineable _ in @xmath14 whenever there is a dense vector subspace @xmath18 of @xmath14 satisfying @xmath19 ( hence dense - lineability implies lineability as soon as dim@xmath25 ) , and 2 . _ maximal dense - lineable _ in @xmath14 whenever there is a dense vector subspace @xmath18 of @xmath14 satisfying @xmath19 and dim@xmath26 dim@xmath27 . and , according to @xcite , when @xmath14 is a topological vector space contained in some ( linear ) algebra then @xmath17 is called : 1 . _ algebrable _ if there is an algebra @xmath18 so that @xmath19 and @xmath18 is infinitely generated , that is , the cardinality of any system of generators of @xmath18 is infinite . densely algebrable _ in @xmath14 if , in addition , @xmath18 can be taken dense in @xmath14 . @xmath15-algebrable _ if there is an @xmath15-generated algebra @xmath18 with @xmath19 . strongly @xmath15-algebrable _ if there exists an @xmath15-generated _ free _ algebra @xmath18 with @xmath19 ( for @xmath28 , we simply say _ strongly algebrable _ ) . _ densely strongly @xmath15-algebrable _ if , in addition , the free algebra @xmath18 can be taken dense in @xmath14 . .15 cm note that if @xmath14 is contained in a commutative algebra then a set @xmath29 is a generating set of some free algebra contained in @xmath17 if and only if for any @xmath30 , any nonzero polynomial @xmath31 in @xmath32 variables without constant term and any distinct @xmath33 , we have @xmath34 and @xmath35 . observe that strong @xmath15-algebrability @xmath36 @xmath15-algebrability @xmath36 @xmath15-lineability , and none of these implications can be reversed ; see @xcite . .15 cm in @xcite the authors proved that the set of _ everywhere surjective _ functions @xmath8 is @xmath37-lineable , which is the best possible result in terms of dimension ( we have denoted by @xmath38 the cardinality of the continuum ) . in other words , the last set is maximal lineable in the space of all real functions . other results establishing the degree of lineability of more stringent classes of functions can be found in @xcite and the references contained in it . .15 cm turning to the setting of more regular functions , in @xcite the following results are proved : the set of _ differentiable _ functions on @xmath1 whose derivatives are discontinuous almost everywhere is @xmath38-lineable ; given a non - void compact interval @xmath3 , the family of differentiable functions whose derivatives are discontinuous almost everywhere on @xmath39 is dense - lineable in the space @xmath2 , endowed with the supremum norm ; and the class of differentiable functions on @xmath1 that are monotone on no interval is @xmath38-lineable . .15 cm finally , recall that every bounded variation function on an interval @xmath3 ( that is , a function satisfying @xmath40 ) is _ differentiable almost everywhere . _ a continuous bounded variation function @xmath41 is called strongly singular whenever @xmath42 for almost every @xmath43 and , in addition , @xmath44 is nonconstant on any subinterval of @xmath39 . et al . _ @xcite showed that the set of strongly singular functions on @xmath45 $ ] is densely strongly @xmath38-algebrable in @xmath46)$ ] . .15 cm a number of results related to the above ones will be shown in the next two sections . our aim in this section is to study the lineability of the family of lebesgue measurable functions @xmath8 that are everywhere surjective , denoted @xmath7 . this result is quite surprising , since ( as we can see in @xcite ) , the class of everywhere surjective functions contains a @xmath47-lineable set of non - measurable ones ( called _ jones functions _ ) . [ thm - mes - c - lineable ] the set @xmath7 is @xmath38-lineable . firstly , we consider the everywhere surjective function furnished in @xcite*example 2.2 . for the sake of convenience , we reproduce here its construction . let @xmath48 be the collection of all open intervals with rational endpoints . the interval @xmath49 contains a cantor type set , call it @xmath50 . now , @xmath51 also contains a cantor type set , call it @xmath52 . next , @xmath53 contains , as well , a cantor type set , @xmath54 . inductively , we construct a family of pairwise disjoint cantor type sets , @xmath55 , such that for every @xmath56 , @xmath57 . now , for every @xmath58 , take any bijection @xmath59 , and define @xmath60 as @xmath61 then @xmath44 is clearly everywhere surjective . indeed , let @xmath39 be any interval in @xmath62 . there exists @xmath63 such that @xmath64 . thus @xmath65 . .15 cm but the novelty of the last function is that @xmath44 is , in addition , zero almost everywhere , and in particular , it is ( lebesgue ) _ measurable . _ that is , @xmath66 . .15 cm now , taking advantage of the approach of @xcite*proposition 4.2 , we are going to construct a vector space that shall be useful later on . let @xmath67 where @xmath68 . then @xmath18 is a @xmath69-dimensional vector space because the functions @xmath70 @xmath71 are linearly independent . indeed , assume that there are scalars @xmath72 ( not all @xmath73 ) as well as positive reals @xmath74 such that @xmath75 for all @xmath76 . without loss of generality , we may assume that @xmath77 , @xmath78 and @xmath79 . then @xmath80 or @xmath81 , which is clearly a contradiction . therefore @xmath82 and we are done . note that each nonzero member @xmath83 ( with the @xmath84 s and the @xmath85 s as before ) of @xmath86 is ( continuous and ) surjective because @xmath87 and @xmath88 if @xmath89 ( with the values of the limits interchanged if @xmath90 ) . .15 cm next , we define the vector space @xmath91 observe that , since the @xmath44 is measurable and the functions @xmath92 in @xmath86 are continuous , the members of @xmath18 are measurable . fix any @xmath93 . then , again , there are finitely many scalars @xmath72 with @xmath78 , and positive reals @xmath79 such that @xmath94 and @xmath95 . now , fix a non - degenerate interval @xmath96 . then @xmath97 , which shows that @xmath98 is everywhere surjective . hence @xmath99 . .15 cm finally , by using the linear independence of the functions @xmath70 and the fact that @xmath44 is surjective , it is easy to see that the functions @xmath100 @xmath71 are linearly independent , which entails that @xmath18 has dimension @xmath69 , as required . in ( * example 2.34 ) it is exhibited one sequence of measurable everywhere surjective functions tending pointwise to zero . with theorem [ thm - mes - c - lineable ] in hand , we now get a plethora of such sequences , and even in a much easier way than @xcite . the family of sequences @xmath101 of lebesgue measurable functions @xmath8 such that @xmath102 converges pointwise to zero and such that @xmath103 for any positive integer @xmath104 and each non - degenerate interval @xmath39 , is @xmath38-lineable . consider the family @xmath105 consisting of all sequences @xmath106 given by @xmath107 where the functions @xmath98 run over the vector space @xmath18 constructed in the last theorem . it is easy to see that @xmath105 is a @xmath38-dimensional vector subspace of @xmath108 , that each @xmath109 is measurable , that @xmath110 @xmath111 for every @xmath76 , and that every @xmath109 is everywhere surjective if @xmath98 is not the zero function . it would be interesting to know whether @xmath7 is likewise the set of everywhere surjective functions maximal lineable in @xmath112 ( that is , @xmath37-lineable ) . in this section , we analyze the lineability of the set of pompeiu functions that are not constant on any interval . of course , this set is not a vector space . .15 cm firstly , the following version of the well - known stone weierstrass density theorem ( see e.g. @xcite ) for the space @xmath113 will be relevant to the proof of our main result . its proof is a simple application of the original stone weierstrass theorem for @xmath114 ( the banach space of continuous functions @xmath115 , endowed with the uniform distance , where @xmath116 is a compact topological space ) together with the fact that convergence in @xmath113 means convergence on each compact subset of @xmath1 . so we omit the proof . [ swforc(r ) ] suppose that @xmath117 is a subalgebra of @xmath5 satisfying the following properties : 1 . given @xmath118 there is @xmath119 with @xmath120 . given a pair of distinct points @xmath121 , there exists @xmath119 such that @xmath122 . then @xmath117 is dense in @xmath5 . in ( * proposition 7 ) , balcerzak , bartoszewicz and filipczak established a nice algebrability result by using the so - called _ exponential - like functions , _ that is , the functions @xmath123 of the form @xmath124 for some @xmath125 , some @xmath126 and some distinct @xmath127 . by @xmath128 we denote the class of exponential - like functions . the following lemma ( see @xcite or ( * ? ? ? * chapter 7 ) ) is a slight variant of the mentioned proposition 7 of @xcite . [ lemma - algebrabilitycriterium ] let @xmath129 be a nonempty set and @xmath130 be a family of functions @xmath131 . assume that there exists a function @xmath132 such that @xmath133 is uncountable and @xmath134 for every @xmath135 . then @xmath130 is strongly @xmath38-algebrable . more precisely , if @xmath136 is a set with card@xmath137 and linearly independent over the field @xmath138 , then @xmath139 is a free system of generators of an algebra contained in @xmath140 . lemma [ lemma - denselystralgebrable ] below is an adaptation of a result that is implicitly contained in ( * ? ? ? * section 6 ) . we sketch the proof for the sake of completeness . [ lemma - denselystralgebrable ] let @xmath141 be a family of functions in @xmath113 . assume that there exists a strictly monotone function @xmath142 such that @xmath143 for every exponential - like function @xmath144 . then @xmath141 is densely strongly @xmath38-algebrable in @xmath113 . if @xmath145 then @xmath133 is a non - degenerate interval , so it is an uncountable set . then , it is sufficient to show that the algebra @xmath117 generated by the system @xmath146 given in lemma [ lemma - algebrabilitycriterium ] is dense . for this , we invoke lemma [ swforc(r ) ] . take any @xmath147 . given @xmath118 , the function @xmath148 belongs to @xmath117 and satisfies @xmath120 . moreover , for prescribed distinct points @xmath121 , the same function @xmath6 fulfills @xmath122 , because both functions @xmath44 and @xmath149 are one - to - one . as a conclusion , @xmath117 is dense in @xmath5 . now we state and prove the main result of this section . [ thm - pnonconstant - algebrable ] the set of functions in @xmath150 that are nonconstant on any non - degenerated interval of @xmath1 is densely strongly @xmath38-algebrable in @xmath5 . from ( * ? * example 3.11 ) ( see also ( * ? ? ? * example 13.3 ) ) we know that there exists a derivable _ strictly increasing _ real - valued function @xmath151 ( with @xmath152 ) whose derivative vanishes on a dense set and yet does not vanish everywhere . by composition with the function @xmath153 , we get a strictly monotone function @xmath154 satisfying that @xmath155 is dense in @xmath1 but @xmath156 . observe that , in particular , @xmath44 is a pompeiu function that is nonconstant on any interval . .15 cm according to lemma [ lemma - denselystralgebrable ] , our only task is to prove that , given a prescribed function @xmath135 , the function @xmath157 belongs to @xmath158 , where @xmath159 by the chain rule , @xmath160 is a differentiable function and @xmath161 @xmath162 . hence @xmath163 vanishes at least on @xmath164 , so this derivative vanishes on a dense set . it remains to prove that @xmath160 is nonconstant on any open interval of @xmath1 . .15 cm in order to see this , fix one such interval @xmath165 . clearly , the function @xmath166 also belongs to @xmath167 . then @xmath166 is a nonzero entire function . therefore the set @xmath168 is discrete in @xmath1 . in particular , it is closed in @xmath1 and countable , so @xmath169 is open and dense in @xmath1 . of course , @xmath170 is discrete in @xmath171 . since @xmath172 is a homeomorphism , the set @xmath173 is discrete in @xmath1 . hence @xmath174 is a nonempty open set of @xmath165 . on the other hand , since @xmath164 is dense in @xmath1 , it follows that the set @xmath175 of all interior points of @xmath164 is @xmath176 . indeed , if this were not true , there would exist an interval @xmath177 . then @xmath178 on @xmath179 , so @xmath44 would be constant on @xmath179 , which is not possible because @xmath44 is strictly increasing . therefore @xmath180 is dense in @xmath1 , from which one derives that @xmath181 is dense in @xmath165 . thus @xmath182 . finally , pick any point @xmath183 in the last set . this means that @xmath184 , @xmath185 ( so @xmath186 ) and @xmath187 ( so @xmath188 ) . thus @xmath189 which implies that @xmath160 is nonconstant on @xmath165 , as required . .1 cm \1 . in view of the last theorem one might believe that the expression `` @xmath178 on a dense set '' ( see the definition of @xmath150 ) could be replaced by the stronger one `` @xmath178 almost everywhere '' . but this is not possible because every differentiable function is an n - function that is , it sends sets of null measure into sets of null measure ( see ( * ? ? ? * theorem 21.9 ) ) and every continuous n - function on an interval whose derivative vanishes almost everywhere must be a constant ( see ( * ? ? ? * theorem 21.10 ) ) . .9pt if a real function @xmath44 is a derivative then @xmath190 may be not a derivative ( see @xcite ) . this leads us to conjecture that the set @xmath12 of pompeiu derivatives ( and of course , any subset of it ) is not algebrable . .9pt \3 . nevertheless , from theorem 3.6 ( and also from theorem 4.1 ) of @xcite it follows that the family @xmath191 of bounded pompeiu derivatives is @xmath69-lineable . a quicker way to see this is by invoking the fact that @xmath191 is a vector space that becomes a banach space under the supremum norm @xcite . since it is not finite dimensional , a simple application of baire s category theorem yields dim@xmath192 . now , on one hand , we have that , trivially , @xmath191 is dense - lineable in itself . on the other hand , it is known that the set of derivatives that are positive on a dense set and negative on another is a dense @xmath193 set in the banach space @xmath191 @xcite . then , as the authors of @xcite suggest , it would be interesting to see whether this set is also dense - lineable . let @xmath194 and consider the function @xmath195 given by @xmath196 observe that @xmath44 is discontinuous at the origin since arbitrarily near of @xmath197 there exist points of the form @xmath198 at which @xmath44 has the value @xmath199 . on the other hand , fixed @xmath200 , the real - valued function of a real variable given by @xmath201 is everywhere a continuous function of @xmath202 . indeed , this is trivial if all @xmath203 s @xmath204 are not @xmath73 , while @xmath205 if some @xmath206 . of course , @xmath44 is continuous at any point of @xmath207 . .15 cm given @xmath208 , we denote by @xmath209 the vector space of all _ separately continuous _ functions @xmath210 that are _ continuous on _ @xmath211 . since card@xmath212 , it is easy to see that the cardinality ( so the dimension ) of @xmath209 equals @xmath38 . theorem [ thm - dsc(n)-c - algebrable ] below will show the algebrability of the family @xmath213 in a maximal sense . [ thm - dsc(n)-c - algebrable ] let @xmath58 with @xmath194 , and let @xmath208 . then the set @xmath214 is strongly @xmath38-algebrable . we can suppose without loss of generality that @xmath215 . consider the function @xmath216 given by . for each @xmath217 , we set @xmath218 it is easy to see that these functions generate a free algebra . indeed , if @xmath219 is a nonzero polynomial in @xmath220 variables with @xmath221 and @xmath222 are distinct positive real numbers , let @xmath223 the variable @xmath224 appears explicitly in the expression of @xmath225 , and @xmath226 . then one derives that the function @xmath227 has the form @xmath228 , where @xmath229 , @xmath125 , @xmath92 is a finite sum of the form @xmath230 with @xmath231 integers and @xmath232 , and @xmath98 is a finite linear combination of functions of the form @xmath233 where , in turn , each @xmath234 is a finite sum of the form @xmath235 , with each @xmath236 satisfying that either @xmath237 , or @xmath238 and @xmath239 simultaneously . then @xmath240 and , in particular , @xmath227 is not @xmath73 identically . this shows that the algebra @xmath86 generated by the @xmath241 s is free . .15 cm now , define the set @xmath117 as @xmath242 plainly , @xmath117 is an algebra of functions @xmath210 each of them being continuous on @xmath207 . but , in addition , this algebra is freely generated by the functions @xmath243 @xmath244 . to see this , assume that @xmath245 where @xmath246 are as above . suppose that @xmath247 . evidently , the function @xmath44 is onto ( note that , for example , @xmath248 , @xmath249 @xmath250 and @xmath251 ) . therefore @xmath252 for all @xmath76 , so @xmath253 , which is absurd because @xmath254 becomes large as @xmath255 . .15 cm hence our only task is to prove that every function @xmath256 as in the last paragraph belongs to @xmath257 . firstly , the continuity of each @xmath241 implies that @xmath258 . finally , the function @xmath259 is discontinuous at the origin . indeed , we have for all @xmath260 that @xmath261 as @xmath262 , due to . this is inconsistent with continuity at @xmath73 . the proof is finished . every real sequence @xmath263 generates a real series @xmath264 . in order to make the notation of this section consistent , we adopt the convention @xmath265 for every real number @xmath266 , and @xmath267 . and a series @xmath264 will be called divergent just whenever it does not converge . as it is commonly known , given a series @xmath268 , a refinement of the classical _ ratio test _ states that * if @xmath269 then @xmath268 converges , and * if @xmath270 then @xmath268 diverges . however , we can have convergent ( positive ) series for which @xmath271 and @xmath272 simultaneously . for instance , consider the series @xmath273 making @xmath274 , we have @xmath275 now , the series @xmath276 diverges with the same corresponding limsup and liminf . .15 cm analogously , a refinement of the classical _ root test _ asserts that * if @xmath277 then @xmath268 converges , and * if @xmath278 then @xmath268 diverges . but no of these conditions is sufficient because , for instance , the positive series @xmath279 converges , the series @xmath280 diverges but @xmath281 for both of them . .15 cm our goal in this section is to show that the set of convergent series for which the ratio test or the root test fails that is , the refinements of both tests provide no information whatsoever is lineable in a rather strong sense ; see theorem [ thm - test - series - lineability ] below . the same result will be shown to happen for divergent series . .15 cm in order to put these properties into an appropriate context , we are going to consider the space @xmath282 of all real sequences and its subset @xmath283 , the space of all absolutely summable real sequences . recall that @xmath284 becomes a frchet space under the product topology , while @xmath283 becomes a banach space ( so a frchet space as well ) if it is endowed with the @xmath285-norm @xmath286 . moreover , the set @xmath287 such that @xmath288 is a dense vector subspace of both @xmath284 and @xmath283 . a standard application of baire s category theorem together with the separability of these spaces yields that their dimension equals @xmath38 . .15 cm we need an auxiliary , general result about lineability . let @xmath14 be a vector space and @xmath289 be two subsets of @xmath14 . according to @xcite , we say that _ @xmath17 is stronger than @xmath290 _ whenever @xmath291 . the following assertion of which many variants have been proved can be found in @xcite and the references contained in them . [ maxdenslineable - criterium ] assume that @xmath14 is a metrizable topological vector space . let @xmath16 be a maximal lineable . suppose that there exists a dense - lineable subset @xmath29 such that @xmath17 is stronger than @xmath290 and @xmath292 . then @xmath17 is maximal dense - lineable in @xmath14 . [ thm - test - series - lineability ] the following four sets are maximal dense - lineable in @xmath283 , @xmath283 , @xmath284 and @xmath284 , respectively : 1 . the set of sequences in @xmath283 for whose generated series the ratio test fails . the set of sequences in @xmath283 for whose generated series the root test fails . 3 . the set of sequences in @xmath284 whose generated series diverges and the ratio test fails . 4 . the set of sequences in @xmath284 whose generated series diverges and the root test fails . we shall only show the first item , even in a very strong form . namely , our aim is to prove that the set @xmath293 is maximal dense - lineable . the remaining items can be done in a similar manner and are left to the reader : as a hint , suffice it to say that , instead of the collection of sequences @xmath294 used for ( a ) , one may use @xmath295 , @xmath296 and @xmath297 , respectively , to prove ( b ) , ( c ) and ( d ) . .15 cm let us prove ( a ) . consider , for every real number @xmath298 , the positive sequence @xmath299 for @xmath56 . since @xmath300 for all @xmath301 , the comparison test yields @xmath302 . next , take @xmath303 which is a vector subspace of @xmath283 . it can be easily seen that dim(@xmath304)@xmath305 . indeed , suppose that a linear combination of the type @xmath306 is identically @xmath73 . then , supposing without loss of generality that @xmath307 and @xmath308 , and dividing the previous expression by @xmath309 we obtain @xmath310 taking limits in the previous expression , as @xmath104 goes to @xmath311 , we have @xmath312 . inductively we can obtain that all @xmath313 s are @xmath73 , having that the set of sequences @xmath314 is linearly independent , thus dim(@xmath304)@xmath315 . .15 cm next , let us show that , given any sequence @xmath316 as in ( [ linearcombination ] ) ( with @xmath317 and @xmath318 ) , the ratio test does not provide any information on the convergence of @xmath319 . dividing numerators and denominators by @xmath320 , we get @xmath321 where @xmath322 @xmath323 , @xmath324 if @xmath104 is even , and @xmath325 if @xmath104 is odd ( @xmath326 ) . note that @xmath327 for all @xmath328 , @xmath329 for all @xmath330 , and @xmath331 . then @xmath332 consequently , @xmath333 belongs to @xmath117 , as we wished . this shows that @xmath117 is maximal lineable in @xmath283 . .15 cm finally , an application of lemma [ maxdenslineable - criterium ] with @xmath334 , @xmath335 and @xmath336 proves the maximal dense - lineability of @xmath117 . concerning parts ( c ) and ( d ) of the last theorem , one might believe that they happen because root and ratio test are specially non - sharp criteria . to be more precise , given a divergent series @xmath337 with positive terms ( notice that we may have @xmath338 , for instance with @xmath339 ) , one might believe that there are not many sequences @xmath340 essentially lower that @xmath341 such that @xmath342 still diverges . the following theorem will show that this is far from being true . in order to formulate it properly , a piece of notation is again needed . for a given sequence @xmath343 , we denote by @xmath344 the vector space of all sequences @xmath345 satisfying @xmath346 . it is a standard exercise to prove that , when endowed with the norm @xmath347 the set @xmath344 becomes a separable banach space , such that @xmath348 is a dense subspace of it . [ thmc_0((c_n))maxdenslineable ] assume that @xmath341 is a sequence of positive real numbers such that the series @xmath337 diverges . then the family of sequences @xmath349 such that the series @xmath342 diverges is maximal dense - lineable in @xmath344 . by baire s theorem , dim@xmath350 . we denote @xmath351 obviously , @xmath352 and @xmath353 . let us apply lemma [ maxdenslineable - criterium ] with @xmath354 then it is enough to show that @xmath17 is maximal lineable , that is , @xmath38-lineable . .15 cm to this end , we use the divergence of @xmath337 and the fact @xmath355 @xmath356 . letting @xmath357 , we can obtain inductively a sequence @xmath358 satisfying @xmath359 now , define the collection of sequences @xmath360 by @xmath361 since @xmath362 as @xmath363 , each sequence @xmath364 belongs to @xmath344 . we set @xmath365 this vector space is @xmath38-dimensional . indeed , if this were not the case , then there would exist @xmath366 , @xmath367 with @xmath368 and @xmath369 such that the sequence @xmath370 is identically zero . from the triangle inequality , we obtain for each @xmath371 that @xmath372 which is absurd . hence , the sequences @xmath373 @xmath374 are linear independent and dim@xmath375 . finally , we prove that each @xmath376 belongs to @xmath17 . note that such a sequence @xmath259 has the shape given in ( 6.2 ) , with @xmath368 and @xmath369 . but the fact @xmath377 @xmath378 as shown above entails that the cauchy convergence criterium for series does not hold for @xmath342 . consequently , this series diverges , as required . the property given in theorem [ thmc_0((c_n))maxdenslineable ] is topologically generic too , that is , the set @xmath17 above is _ residual _ in @xmath344 . indeed , we have that @xmath379 , where @xmath380 and each set @xmath381 is open and dense in @xmath344 . to prove this , fix @xmath382 with @xmath383 and observe that @xmath384 , where @xmath385 is given by @xmath386 . the continuity of the projections @xmath387 @xmath388 entails the continuity of @xmath389 , so @xmath390 is open in @xmath284 . the inclusion @xmath391 being continuous , we get that @xmath392 is open in @xmath344 . therefore @xmath393 is also open in @xmath344 . as for the density of @xmath393 , note that , due to the density of @xmath348 in @xmath344 , it is enough to show that , given @xmath394 and @xmath395 , there exist @xmath383 and @xmath396 with @xmath397 and @xmath398 . from the divergence of the positive series @xmath399 , it follows the existence of @xmath400 such that @xmath401 and @xmath402 . define the sequence @xmath403 by @xmath404 by construction , @xmath397 . finally , @xmath405 , as required . let @xmath406 be the lebesgue measure on @xmath1 . in this section we will restrict ourselves to the interval @xmath45 $ ] , which of course has finite measure @xmath407 ) = 1 $ ] . denote by @xmath408 the vector space of all lebesgue measurable functions @xmath45 \to { \mathbb{r}}$ ] , where two functions are identified whenever they are equal almost everywhere ( a.e . ) in @xmath45 $ ] . two natural kinds of convergence of functions of @xmath408 are a.e .- convergence and convergence in measure . recall that a sequence @xmath409 of measurable functions is said to converge in measure to a measurable function @xmath410 \to { \mathbb{r}}$ ] provided that @xmath411 : \ , |f_n(x ) - f(x)| > \alpha \ } ) = 0 \hbox { \ for all \ } \alpha > 0.\ ] ] convergence in measure is specially pleasant because it can be described by a natural metric on @xmath408 ; see e.g. @xcite . namely , the distance @xmath412 } { |f(x ) - g(x)| \over 1 + |f(x ) - g(x)| } \,dx \quad ( f , g \in l_0)\ ] ] satisfies that @xmath413 if and only if @xmath414 in measure ( the finiteness of the measure of @xmath45 $ ] is crucial ) . under the topology generated by @xmath415 , the space @xmath408 becomes a complete metrizable topological vector space for which the set @xmath416 of simple ( i.e. of finite image ) measurable functions forms a dense vector subspace . actually , @xmath408 is separable because the set @xmath417 of finite linear combinations with rational coefficients of functions of the form @xmath418}$ ] ( @xmath419 rational numbers ) is countable and dense in @xmath408 . here @xmath420 denotes the indicator function of the set @xmath17 . convergence in measure of a sequence @xmath409 to @xmath44 implies a.e .- convergence to @xmath44 of some subsequence @xmath421 ( see ( * ? ? ? * theorem 21.9 ) ) . but , generally , this convergence can not be obtained for the whole sequence @xmath409 . for instance , the so - called `` typewriter sequence '' given by @xmath422}\ ] ] ( where , for each @xmath104 , the non - negative integers @xmath423 and @xmath424 are uniquely determined by @xmath425 and @xmath426 ) satisfies that @xmath427 in measure but , for every point @xmath428 $ ] , the sequence @xmath429 does not converge . in order to face the lineability of this phenomenon , we need , once more , to put the problem in an adequate framework . let @xmath430 be the space of all sequences of measurable functions @xmath45 \to { \mathbb{r}}$ ] , endowed with the product topology . since @xmath408 is metrizable and separable , the space @xmath430 is also a complete metrizable separable topological vector space . again , by baire s theorem , this implies dim@xmath431 . moreover , the set @xmath432 is dense in the product space . now , we are ready to state our next theorem , with which we finish this paper . let @xmath435 be the typewriter sequence defined above , and let @xmath17 be the family described in the statement of the theorem , so that @xmath436 . extend each @xmath437 to the whole @xmath1 by defining @xmath438 for all @xmath439 $ ] . it is readily seen that , for each @xmath440 , the translated - dilated sequence @xmath441 @xmath356 also tends to @xmath73 in measure . consider the vector space @xmath442 the sequences @xmath443 @xmath444 are linearly independent . indeed , if it were not the case , there would be @xmath445 as well as real numbers @xmath446 with @xmath447 such that @xmath448 for all @xmath58 . in particular , @xmath449 for almost all @xmath76 . but @xmath450}$ ] , so @xmath451}$ ] for all @xmath452 . therefore @xmath453}(x ) + \cdots + c_s \chi_{[t_s , t_s+{1\over2}]}(x ) = 0 \hbox { \ for almost all \ , } x \in [ 0,1].\ ] ] but , for every @xmath454 $ ] , the left - hand side of the last expression equals @xmath455 , which is absurd . this shows the required linear independence . then @xmath456 . moreover , since @xmath408 is a topological vector space carrying the topology of convergence in measure , we get that every member of @xmath457 is a sequence tending to @xmath73 in measure . next , fix any @xmath458 as above , with @xmath459 and @xmath447 . for all @xmath460 $ ] , we have @xmath461 since @xmath447 and @xmath462 does not converge for every @xmath463 \,\ , ( \subset [ 0,1])$ ] , we derive that , for each @xmath460 $ ] , the sequence @xmath464 does not converge . this shows that @xmath19 . thus , @xmath17 is @xmath38-lineable . finally , an application of lemma [ maxdenslineable - criterium ] with @xmath465 puts an end on the proof .

it is proved the existence of large algebraic structures including large vector subspaces or infinitely generated free algebras inside , among others , the family of lebesgue measurable functions that are surjective in a strong sense , the family of nonconstant differentiable real functions vanishing on dense sets , and the family of non - continuous separately continuous real functions . lineability in special spaces of sequences is also investigated . some of our findings complete or extend a number of results by several authors .

teasing out signatures of interactions buried in overwhelming volumes of information is one of the most basic challenges in scientific research . understanding how information is organized can help us discover its fundamental underlying properties . researchers do this when they investigate the relationships between diseases , cell functions , chemicals , or particles , and we all learn new concepts and solve problems by understanding the relationships between the various entities present in our everyday lives . these entities can be represented as networks , or graphs , in which local behaviors can be easily understood , but whose global view is highly complex . these networks exhibit a long and varied list of global properties , including heavy - tailed degree distributions @xcite and interesting growth characteristics @xcite , among others . recent work has found that these global properties are merely products of a graph s local properties , in particular , graphlet distributions @xcite . these small , local substructures often reveal the degree distributions , diameter and other global properties of a graph @xcite , and have been shown to be a more complete way to measure the similarity between two or more graphs @xcite . our overall goal , and the goal of structural inference algorithms in general , is to learn the local structures that , in aggregate , help describe the observed interactions and generalize to explain further phenomena . for example , physicists and chemists have found that many chemical interactions are the result of underlying structural properties of the individual elements . similarly , biologists have agreed that simple tree structures are useful when organizing the evolutionary history of life , and sociologists find that clique - formation , _ e.g. _ , triadic closure , underlies community development @xcite . in other instances , the structural organization of the entities may resemble a ring , a clique , a star , or any number of complex configurations . in this work , we describe a general framework that can discover , from any large network , simple structural forms in order to make predictions about the topological properties of a network . in addition , this framework is able to extract mechanisms of network generation from small samples of the graph in order to generate networks that satisfy these properties . our major insight is that a network s _ clique tree _ encodes simple information about the structure of the network . we use the closely - related formalism of _ hyperedge replacement grammars _ ( hrgs ) as a way to describe the organization of real world networks . unlike previous models that manually define the space of possible structures @xcite or define the grammar by extracting frequent subgraphs @xcite , our framework can automatically discover the necessary forms and use them to recreate the original graph _ exactly _ as well as infer generalizations of the original network . our approach can handle any type of graph and does not make any assumption about the topology of the data . after reviewing some of the theoretical foundations of clique trees and hrgs , we show how to extract an hrg from a graph and use it to reconstruct the original graph . we then show how to use the extracted grammar to stochastically generate generalizations of the original graph . finally , we present experimental results that compare the stochastically generated graphs with the original graphs . we show that these generated graphs exhibit a wide range of properties that are very similar to the properties of the original graphs , and significantly outperform existing graph models at generating subgraph distributions similar to those found in the original graph . before we describe our method , some background definitions are needed . we begin with an arbitrary input _ hypergraph _ @xmath0 , where a _ hyperedge _ @xmath1 can connect multiple vertices @xmath2 . common _ graphs _ ( _ e.g. _ , social networks , web graphs , information networks ) are a particular case of hypergraphs where each edge connects exactly two vertices . for convenience , all of the graphs in this paper will be _ simple _ , _ connected _ and _ undirected _ , although these restrictions are not vital . in the remainder of this section we refer mainly to previous developments in clique trees and their relationship to hyperedge replacement grammars in order to support the claims made in sections 3 and 4 . all graphs can be decomposed ( though not uniquely ) into a _ clique tree _ , also known as a tree decomposition , junction tree , join tree , intersection tree , or cluster graph . within the data mining community , clique trees are best known for their role in exact inference in probabilistic graphical models , so we introduce the preliminary work from a graphical modelling perspective ; for an expanded introduction , we refer the reader to chapters 9 and 10 of koller and friedman s textbook @xcite . [ defn : cliquetree ] a _ clique tree _ of a graph @xmath0 is a tree @xmath3 , each of whose nodes @xmath4 is labeled with a @xmath5 and @xmath6 , such that the following properties hold : 1 . vertex cover : for each @xmath7 , there is a vertex @xmath8 such that @xmath9 . edge cover : for each hyperedge @xmath10 there is exactly one node @xmath8 such that @xmath11 . moreover , @xmath12 . running intersection : for each @xmath7 , the set @xmath13 is connected . the _ width _ of a clique tree is @xmath14 , and the _ treewidth _ of a graph @xmath15 is the minimal width of any clique tree of @xmath15 . unfortunately , finding the optimal elimination ordering and corresponding minimal - width clique tree is np - complete @xcite . fortunately , many reasonable approximations exist for general graphs : in this paper we employ the commonly used maximum cardinality search ( mcs ) heuristic introduced by tarjan and yannikakis @xcite in order to compute a clique tree with a reasonably - low , but not necessarily minimal , width . simply put , a clique tree of any graph ( or any hypergraph ) is a tree , each of whose nodes is labeled with nodes and edges from the original graph , such that _ vertex cover _ , _ edge cover _ and the _ running intersection _ properties hold , and the `` width '' of the clique tree measures how tree - like the graph is . the reason for the interest in finding the clique tree of a graph is because many computationally difficult problems can be solved efficiently when the data is constrained to be a tree . figure [ fig : expdtree ] shows a graph and a minimal - width clique tree of the same graph ( showing @xmath16 for each node @xmath4 ) . nodes are labeled with lowercase latin letters . we will refer back to this graph and clique tree as a running example throughout this paper . ; they are shown only for explanatory purposes . ] the key insight for this task is that a network s clique tree encodes robust and precise information about the network . an hrg , which is extracted from the clique tree , contains graphical rewriting rules that can match and replace graph fragments similar to how a context free grammar ( cfg ) rewrites characters in a string . these graph fragments represent a succinct , yet complete description of the building blocks of the network , and the rewriting rules of the hrg represent the instructions on how the graph is pieced together . for a thorough examination of hrgs , we refer the reader to the survey by drewes _ et al . _ @xcite . [ defn : tuple ] a _ hyperedge replacement grammar _ is a tuple @xmath17 , where 1 . @xmath18 is a finite set of nonterminal symbols . each nonterminal @xmath19 has a nonnegative integer _ rank _ , which we write @xmath20 . 2 . @xmath3 is a finite set of terminal symbols . @xmath21 is a distinguished starting nonterminal , and @xmath22 . 4 . @xmath23 is a finite set of production rules @xmath24 , where * @xmath19 , the left hand side ( lhs ) , is a nonterminal symbol . * @xmath25 , the right hand side ( rhs ) , is a hypergraph whose edges are labeled by symbols from @xmath26 . if an edge @xmath27 is labeled by a nonterminal @xmath28 , we must have @xmath29 . * exactly @xmath20 vertices of @xmath25 are designated _ external vertices_. the other vertices in @xmath25 are called _ internal _ vertices . when drawing hrg rules , we draw the lhs @xmath19 as a hyperedge labeled @xmath19 with arity @xmath20 . we draw the rhs as a hypergraph , with the external vertices drawn as solid black circles and the internal vertices as open white circles . if an hrg rule has no nonterminal symbols in its rhs , we call it a _ terminal rule_. [ defn : hrg ] let @xmath30 be an hrg and @xmath31 be a production rule of @xmath30 . we define the relation @xmath32 ( @xmath33 is derived in one step from @xmath34 ) as follows . @xmath34 must have a hyperedge @xmath27 labeled @xmath19 ; let @xmath35 be the vertices it connects . let @xmath36 be the external vertices of @xmath25 . then @xmath37 is the graph formed by removing @xmath27 from @xmath34 , making an isomorphic copy of @xmath25 , and identifying @xmath38 with the copies of @xmath39 for each @xmath40 . let @xmath41 be the reflexive , transitive closure of @xmath42 . then we say that @xmath30 generates a graph @xmath15 if there is a production @xmath43 and @xmath44 and @xmath15 has no edges labeled with nonterminal symbols . in other words , a derivation starts with the symbol @xmath45 , and we repeatedly choose a nonterminal @xmath19 and rewrite it using a production @xmath46 . the replacement hypergraph fragments @xmath25 can itself have other nonterminal hyperedges , so this process is repeated until there are no more nonterminal hyperedges . these definitions will be clearly illustrated in the following sections . clique trees and hyperedge replacement graph grammars have been studied for some time in discrete mathematics and graph theory literature . hrgs are conventionally used to generate graphs with very specific structures , _ e.g. _ , rings , trees , stars . a drawback of many current applications of hrgs is that their production rules must be hand drawn to generate some specific graph or class of graphs . very recently , kemp and tenenbaum developed an inference algorithm that learned probabilities from real world graphs , but still relied on a handful of rather basic hand - drawn production rules ( of a related formalism called vertex replacement grammar ) to which the learned probabilities were assigned @xcite . the main contribution of this paper is to combine prior theoretical work on clique trees , tree decomposition and treewidth to automatically learn an hrg for real world graphs . existing graph generators , like exponential random graphs , small world graphs , kronecker graphs , and so on , learn parameters from some input graph to generate new graphs stochastically . unlike these previous approaches , our model has the ability to reproduce the exact same graph topology where the new graph is guaranteed to be isomorphic to the original graph . our model is also able to stochastically generate different - sized graphs that share similar properties to the original graph . the first step in learning an hrg from a graph is to compute a clique tree from the original graph . then , this clique tree induces an hrg in a natural way , which we demonstrate in this section . let @xmath4 be an interior node of the clique tree @xmath3 , let @xmath47 be its parent , and let @xmath48 be its children . node @xmath4 corresponds to an hrg production rule @xmath46 as follows . first , @xmath49 . then , @xmath25 is formed by : * adding an isomorphic copy of the vertices in @xmath16 and the edges in @xmath50 * marking the ( copies of ) vertices in @xmath51 as external vertices * adding , for each @xmath52 , a nonterminal hyperedge connecting the ( copies of ) vertices in @xmath53 . figure [ fig : creation ] shows an example of the creation of an hrg rule . in this example , we focus on the middle clique - tree node @xmath54 , outlined in bold . we choose nonterminal symbol n for the lhs , which must have rank 2 because @xmath4 has 2 vertices in common with its parent . the rhs is a graph whose vertices are ( copies of ) @xmath55 . vertices d and e are marked external ( and numbered 1 and 2 , arbitrarily ) because they also appear in the parent node . the terminal edges are @xmath56 . there is only one child of @xmath4 , and the nodes they have in common are e and f , so there is one nonterminal hyperedge connecting e and f. next we deal with the special cases of the root and leaves . * root node . * if @xmath4 is the root node , then it does not have any parent cliques , but may still have one or more children . because @xmath4 has no parent , the corresponding rule has a lhs with rank 0 and a rhs with no external vertices . in this case , we use the start nonterminal @xmath45 as the lhs , as shown in figure [ fig : creation_root ] . the rhs is computed in the same way as the interior node case . for the example in fig . [ fig : creation_root ] , the rhs has vertices that are copies of c , d , and e. in addition , the rhs has two terminal hyperedges , @xmath57 . the root node has two children , so there are two nonterminal hyperedges on the rhs . the right child has two vertices in common with @xmath4 , namely , d and e ; so the corresponding vertices in the rhs are attached by a 2-ary nonterminal hyperedge . the left child has three vertices in common with @xmath4 , namely , c , d , and e , so the corresponding vertices in the rhs are attached by a 3-ary nonterminal hyperedge . * leaf node . * if @xmath4 is a leaf node , then the lhs is calculated the same as in the interior node case . again we return to the running example in fig . [ fig : creation_leaf ] ( on the next page ) . here , we focus on the leaf node @xmath58 , outlined in bold . the lhs has rank 2 , because @xmath4 has two vertices in common with its parent . the rhs is computed in the same way as the interior node case , except no new nonterminal hyperedges are added to the rhs . the vertices of the rhs are ( copies of ) the nodes in @xmath4 , namely , a , b , and e. vertices b and e are external because they also appear in the parent clique . this rhs has two terminal hyperedges , @xmath59 . because the leaf clique has no children , it can not produce any nonterminal hyperedges on the rhs ; therefore this rule is a terminal rule . we induce production rules from the clique tree by applying the above extraction method top down . because trees are acyclic , the traversal order does not matter , yet there are some interesting observations we can make about traversals of moderately sized graphs . first , exactly one hrg rule will have the special starting nonterminal @xmath45 on its lhs ; no mention of @xmath45 will ever appear in any rhs . similarly , the number of terminal rules is equal to the number of leaf nodes in the clique tree . larger graphs will typically produce larger clique trees , especially sparse graphs because they are more likely to have a larger number of small maximal cliques . these larger clique trees will produce a large number of hrg rules , one for each clique in the clique tree . although it is possible to keep track of each rule and its traversal order , we find , and will later show in the experiments section , that the same rules are often repeated many times . figure [ fig : production_rules ] shows the 6 rules that are induced from the clique trees illustrated in fig . [ fig : expdtree ] and used in the running example throughout this section . the hrg rule induction steps described in this section can be broken into two steps : ( i ) creating a clique tree and ( ii ) the hrg rule extraction process . unfortunately , finding a clique tree with minimal width _ i.e. _ , the treewidth @xmath60 , is np - complete . let @xmath61 and @xmath62 be the number of vertices and edges respectively in @xmath15 . tarjan and yannikakis maximum cardinality search ( mcs ) algorithm finds a usable clique tree @xcite in linear time @xmath63 , but is not guaranteed to be minimal . the running time of the hrg rule extraction process is determined exclusively by the size of the clique tree as well as the number of vertices in each clique tree node . from defn . [ defn : cliquetree ] we have that the number of nodes in the clique tree is @xmath62 . when minimal , the number of vertices in an the largest clique tree node @xmath64 ( minus 1 ) is defined as the treewidth @xmath60 , however , clique trees generated by mcs have @xmath64 bounded by the maximum degree of @xmath15 , denoted as @xmath65 @xcite . therefore , given an elimination ordering from mcs , the computational complexity of the extraction process is in @xmath66 . in this section we show how to use the hrg extracted from the original graph @xmath15 ( as described in the previous section ) to generate a new graph @xmath37 . ideally , @xmath37 will be similar to , or have features that are similar to the original graph @xmath15 . we present two generation algorithms . the first generation algorithm is _ exact generation _ , which , as the name implies , creates an isomorphic copy of the original graph @xmath67 . the second generation algorithm is a fast _ stochastic generation _ technique that generates random graphs with similar characteristics to the original graph . each generation algorithm starts with @xmath68 containing only the starting nonterminal @xmath45 . exact generation operates by reversing the hrg extraction process . in order to do this , we must store the hrg rules @xmath23 as well as the clique tree @xmath3 ( or at least the order that the rules were created ) . the first hrg rule considered is always the rule with the nonterminal labelled @xmath45 as the lhs . this is because the clique tree traversal starts at the root , and because the root is the only case that results in @xmath45 on the lhs . the previous section defined an hrg @xmath30 that is constructed from a clique tree @xmath3 of some given hypergraph @xmath15 , and defn . [ defn : hrg ] defines the application of a production rule @xmath69 that transforms some hypergraph @xmath70 into a new hypergraph @xmath37 . by applying the rules created from the clique tree in order , we will create an @xmath37 that is isomorphic to the original hypergraph @xmath15 . in the remainder of this section , we provide a more intuitive look at the exact generation property of the hrg by recreating the graph decomposed in the running example . with the rhs to create a new graph @xmath37 . ] using the running example from the previous section , the application of rule 1 illustrated in fig . [ fig : rule1 ] shows how we transform the starting nonterminal into a new hypergraph , @xmath37 . this hypergraph now has two nonterminal hyperedges corresponding to the two children that the root clique had in fig . [ fig : expdtree ] . the next step is to replace @xmath68 with @xmath37 and then pick a nonterminal corresponding to the leftmost unvisited node of the clique tree . with the rhs to create a new graph @xmath37 . ] we proceed down the left hand side of the clique tree , applying rule 2 to @xmath68 as shown in fig . [ fig : rule2 ] . the lhs of rule 2 matches the 3-ary hyperedge and replaces it with the rhs , which introduces a new internal vertex , two new terminal edges and a new nonterminal hyperedge . again we set @xmath68 to be @xmath37 and continue to the leftmost leaf in the example clique tree . with the rhs to create a new graph @xmath37 . ] the leftmost leaf in fig . [ fig : expdtree ] corresponds to the application of rule 3 ; it is the next to be applied to the new nonterminal in @xmath37 and replaced by the rhs as illustrated in figure [ fig : rule3 ] . the lhs of rule 3 matches the 2-ary hyperedge shown and replaces it with the rhs , which creates a new internal vertex along with two terminal edges . because rule 3 comes from a leaf node , it is a terminal rule and therefore does not add any nonterminal hyperedges . this concludes the left subtree traversal from fig . [ fig : expdtree ] . that is isomorphic to the original graph @xmath15 . ] continuing the example , the right subtree in the clique tree illustrated in fig . [ fig : expdtree ] has three further applications of the rules in @xmath23 . as illustrated in fig . [ fig : rule456 ] , rule 4 adds the final vertex , two terminal edges and one nonterminal hyperedge to @xmath37 . rule 5 and rule 6 do not create any more terminal edges or internal vertices in @xmath37 , but are still processed because of the way the clique tree is constructed . after all 6 rules are applied in order , we are guaranteed that @xmath15 and @xmath37 are isomorphic . there are many cases in which we prefer to create very large graphs in an efficient manner that still exhibit the local and global properties of some given example graph _ without storing the large clique tree _ as required in exact graph generation . here we describe a simple stochastic hypergraph generator that applies rules from the extracted hrg in order to efficiently create graphs of arbitrary size . in larger hrgs we usually find many @xmath24 production rules that are identical . we can merge these duplicates by matching rule - signatures in a dictionary , and keep a count of the number of times that each distinct rule has been seen . for example , if there were some additional rule 7 in fig . [ fig : production_rules ] that was identical to , say , rule 3 , then we would simply note that we saw rule 3 two times . to generate random graphs from a probabilistic hrg , we start with the special starting nonterminal @xmath71 . from this point , @xmath37 can be generated as follows : ( 1 ) pick any nonterminal @xmath19 in @xmath68 ; ( 2 ) find the set of rules @xmath72 associated with lhs @xmath19 ; ( 3 ) randomly choose one of these rules with probability proportional to its count ; ( 4 ) replace @xmath19 in @xmath68 with @xmath25 to create @xmath37 ; ( 5 ) replace @xmath68 with @xmath37 and repeat until there are no more nonterminal edges . however , we find that although the sampled graphs have the same mean size as the original graph , the variance is much too high to be useful . so we want to sample only graphs whose size is the same as the original graph s , or some other user - specified size . naively , we can do this using rejection sampling : sample a graph , and if the size is not right , reject the sample and try again . however , this would be quite slow . our implementation uses a dynamic programming approach to do this exactly while using quadratic time and linear space , or approximately while using linear time and space . we omit the details of this algorithm here , but the source code is available online at https://github.com / nddsg / hrg/. hrgs contain rules that succinctly represent the global and local structure of the original graph . in this section , we compare our approach against some of the state - of - the - art graph generators . we consider the properties that underlie a number of real - world networks and compare the distribution of graphs generated using generators for kronecker graphs , the exponential random graph , chung - lu graphs , and the graphs produced by the stochastic hyperedge replacement graph grammar . in a manner similar to hrgs , the kronecker and exponential random graph models learn parameters that can be used to approximately recreate the original graph @xmath15 or a graph of some other size such that the stochastically generated graph holds many of the same properties as the original graph . the chung - lu graph model relies on node degree sequences to yield graphs that maintain this distribution . except in the case of exact hrg generation described above , the stochastically generated graphs are likely not isomorphic to the original graph . we can , however , still judge how closely the stochastically generated graph resembles the original graph by comparing several of their properties . in order to get a holistic and varied view of the strengths and weaknesses of hrgs in comparison to the other leading graph generation models , we consider real - world networks that exhibit properties that are both common to many networks across different fields , but also have certain distinctive properties . .real networks [ cols=">,^,^",options="header " , ] [ tab : realnets ] the four real world networks considered in this paper are described in table [ tab : realnets ] . the networks vary in their number of vertices and edges as indicated , but also vary in clustering coefficient , diameter , degree distribution and many other graph properties . specifically , the enron graph is the email correspondence graph of the now defunct enron corporation ; the arxiv gr - qc graph is the co - authorship graph extracted from the general relativity and quantum cosmology section of arxiv ; the internet router graph is created from traffic flows through internet peers ; and , finally , dblp is the co - authorship graph from the dblp dataset . datasets were downloaded from the snap and konect dataset repositories . we compare several different graph properties from the 4 classes of graph generators ( hrg , kronecker , chung - lu and exponential random graph ( ergm ) models ) to the original graph @xmath15 . other models , such as the erds - rnyi random graph model , the watts - strogatz small world model , the barabsi - albert generator , etc . are not compared here due to limited space and because kronecker , chung - lu and ergm have been shown to outperform these earlier models when matching network properties in empirical networks . kronecker graphs operate by learning an initiator matrix and then performing a recursive multiplication of that initiator matrix in order to create an adjacency matrix of the approximate graph . in our case , we use kronfit @xcite with default parameters to learn a @xmath73 initiator matrix and then use the recursive kronecker product to generate the graph . unfortunately , the kronecker product only creates graphs where the number of nodes is a power of 2 , _ i.e. _ , @xmath74 , where we chose @xmath75 , @xmath76 , @xmath77 , and @xmath78 for enron , arxiv , routers and dblp graphs respectively to match the number of nodes as closely as possible . the chung - lu graph model ( cl ) takes , as input , a degree distribution and generates a new graph of the similar degree distribution and size @xcite . exponential random graph models ( ergms ) are a class of probabilistic models used to directly describe several structural features of a graph @xcite . we used default parameters in r s ergm package @xcite to generate graph models for comparison . in addition to the problem of model degeneracy , ergms do not scale well to large graphs . as a result , dblp and enron could not be modelled due to their size , and the arxiv graph always resulted in a degenerate model . therefore ergm results are omitted from this section . the main strength of hrg is to learn the patterns and rules that generate a large graph from only a few small subgraph - samples of the original graph . so , in all experiments , we make @xmath79 random samples of size @xmath80 node - induced subgraphs by a breadth first traversal starting from a random node in the graph @xcite . by default we set @xmath81 and @xmath82 empirically . we then compute tree decompositions from the @xmath79 samples , learn hrgs @xmath83 , and combine them to create a single grammar @xmath84 . for evaluation purposes , we generate 20 approximate graphs for the hrg , chung - lu , and kronecker models and plot the mean values in the results section . we did compute the confidence internals for each of the models , but omitted them from the graphs for clarity . in general , the confidence intervals were very small for hrg , kronecker and cl ( indicating good consistency ) , but very big in the few ergm graphs that we were able to generate because of the model degeneracy problem we encountered . here we compare and contrast the results of approximate graphs generated from hrg , kronecker product , and chung - lu . before the results are presented , we briefly introduce the graph properties that we use to compare the similarity between the real networks and their approximate counterparts . although many properties have been discovered and detailed in related literature , we focus on three of the principal properties from which most others can be derived . * degree distribution . * the degree distribution of a graph is the distribution of the number of edges connecting to a particular vertex . barabsi and albert initially discovered that the degree distribution of many real world graphs follows a power law distribution such that the number of nodes @xmath85 where @xmath86 and @xmath87 is typically between 2 and 3 @xcite . figure [ fig : real_degree ] shows the results of the degree distribution property on the four real world graphs ( @xmath88 or @xmath89 as a function of degree @xmath79 ) . recall that the graph results plotted here and throughout the results section are the mean averages of 20 generated graphs . each of the generated graphs is slightly different from the original graphs in their own way . as expected , we find that the power law degree distribution is captured by existing graph generators as well as the hrg model . * eigenvector centrality . * the principal eigenvector is often associated with the centrality or `` value '' of each vertex in the network , where high values indicate an important or central vertex and lower values indicate the opposite . a skewed distribution points to a relatively few `` celebrity '' vertices and many common nodes . the principal eigenvector value for each vertex is also closely associated with the pagerank and degree value for each node . figure [ fig : real_eig ] shows the eigenvector scores for each node ranked highest to lowest in each of the four real world graphs . because the x - axis represents individual nodes , fig . [ fig : real_eig ] also shows the size difference among the generated graphs . hrg performs consistently well across all four types of graphs , but the log scaling on the y - axis makes this plot difficult to discern . to more concretely compare the eigenvectors , the pairwise cosine distance between eigenvector centrality of @xmath15 and the mean eigenvector centrality of each model s generated graphs appear at the top of each plot in order . hrg consistently has the lowest cosine distance followed by chung - lu and kronecker . * hop plot . * the hop plot of a graph shows the number of vertex - pairs that are reachable within @xmath90 hops . the hop plot , therefore , is another way to view how quickly a vertex s neighborhood grows as the number of hops increases . as in related work @xcite we generate a hop plot by picking 50 random nodes and perform a complete breadth first traversal over each graph . figure [ fig : real_hopplot ] demonstrates that hrg graphs produce hop plots that are remarkably similar to the original graph . chung - lu performs rather well in most cases ; kronecker has poor performance on arxiv and dblp graphs , but still shows the correct hop plot shape . the previous network properties primarily focus on statistics of the global network . however , there is mounting evidence which argues that the graphlet comparisons are the most complete way measure the similarity between two graphs @xcite . the graphlet distribution succinctly describes the number of small , local substructures that compose the overall graph and therefore more completely represents the details of what a graph `` looks like . '' furthermore , it is possible for two very dissimilar graphs to have the same degree distributions , hop plots , etc . , but it is difficult for two dissimilar graphs to fool a comparison with the graphlet distribution . * graphlet correlation distance * recent work from systems biology has identified a new metric called the graphlet correlation distance ( gcd ) . the gcd computes the distance between two graphlet correlation matrices one matrix for each graph @xcite . it measures the frequency of the various graphlets present in each graph , _ i.e. _ , the number of edges , wedges , triangles , squares , 4-cliques , etc . , and compares the graphlet frequencies between two graphs . because the gcd is a distance metric , lower values are better . the gcd can range from @xmath91 $ ] , where the gcd is 0 if the two graphs are isomorphic . we computed the gcd between the original graph and each generated graph . figure [ fig : gcd_real ] shows the gcd results . although they are difficult to see due to their small size , fig . [ fig : gcd_real ] includes error bars for the 95% confidence interval . the results here are clear : hrg significantly outperforms the chung - lu and kronecker models . the gcd opens a whole new line of network comparison methods that stress the graph generators in various ways . recall that hrg learns the grammar from @xmath81 subgraph - samples from the original graph . in essence , hrg is extrapolating the learned subgraphs into a full size graph . this raises the question : if we only had access to a small subset of some larger network , could we use our models to infer a larger ( or smaller ) network with the same local and global properties ? for example , given the 34-node karate club graph , could we infer what a hypothetical karate franchise might look like ? using two smaller graphs , zachary s karate club ( 34 nodes , 78 edges ) and the protein - protein interaction network of _ s. cerevisiae _ yeast ( 1,870 nodes , 2,240 edges ) , we learned an hrg model with @xmath92 and @xmath93 , _ i.e. _ , no sampling , and generated networks of size-@xmath94 = 2x , 3x , , 32x . for the protein graph we also sampled down to @xmath95 . powers of 2 were used because the standard kronecker model can only generate graphs of that size . the chung - lu model requires a size-@xmath94 degree distribution as input . to create the proper degree distribution we fitted a poisson distribution ( @xmath96 ) and a geometric distribution ( @xmath97 ) to karate and protein graphs respectively and drew @xmath94 degree - samples from their respective distributions . in all cases , we generated 20 graphs at each size - point . + rather than comparing raw numbers of graphlets , the gcd metric compares the _ correlation _ of the resulting graphlet distributions . as a result , gcd is largely immune to changes in graph size . thus , gcd is a good metric for this extrapolation task . figure [ fig : xtrapol ] shows the mean gcd score and 95% confidence intervals for each graph model . not only does hrg generate good results at @xmath98x , the gcd scores remain mostly level as @xmath94 grows . we have shown that hrg can generate graphs that match the original graph from @xmath81 samples of @xmath82-node subgraphs . if we adjust the size of the subgraph , then the size of the clique tree will change causing the grammar to change in size and complexity . a large clique tree ought to create more rules and a more complex grammar , resulting in a larger model size and better performance ; while a small clique tree ought to create fewer rules and a less complex grammar , resulting in a smaller model size and a lower performance . to test this hypothesis we generated graphs by varying the number of subgraph samples @xmath79 from 1 to 32 , while also varying the size of the sampled subgraph @xmath80 from 100 to 600 nodes . again , we generated 20 graphs for each parameter setting . figure [ fig : grammarsize ] shows how the model size grows as the sampling procedure changes on the internet routers graph . plots for other graphs show a similar growth rate and shape , but are omitted due to space constraints . to test the statistical correlation we calculated pearson s correlation coefficient between the model size and sampling parameters . we find that the @xmath79 is slightly correlated with the model size on routers ( @xmath99 , @xmath100 ) , enron ( @xmath101 ) , arxiv ( @xmath102 ) , and dblp ( @xmath103 , @xmath104 ) . furthermore , the choice of @xmath80 affects the size of the clique tree from which the grammars are inferred . so its not surprising that @xmath80 is highly correlated with the model size on routers ( @xmath105r=0.71 ) , arxiv ( @xmath106 ) , and dblp ( @xmath107 ) all with @xmath108 . because we merge identical rules when possible , we suspect that the overall growth of the hrg model follows heaps law @xcite , _ i.e. _ , that the model size of a graph can be predicted from its rules ; although we save a more thorough examination of the grammar rules as a matter for future work . one of the disadvantages of the hrg model , as indicated in fig . [ fig : grammarsize ] , is that the model size can grow to be very large . but this again begs the question : do larger and more complex hrg models result in improved performance ? to answer this question we computed the gcd distance between the original graph and graphs generated by varying @xmath79 and @xmath80 . figure [ fig : size_score ] illustrates the relationship between model size and the gcd . we use the router and dblp graphs to shows the largest and smallest of our dataset ; other graphs show similar results , but their plots are omitted due to of space . surprisingly , we find that the performance of models with only 100 rules is similar to the performance of the largest models . in the router results , two very small models with poor performance had only 18 and 20 rules each . best fit lines are drawn to illustrate the axes relationship where negative slope indicates that larger models generally perform better . outliers can dramatically affect the outcome of best fit lines , so the faint line in the routers graph shows the best fit line if we remove the two square outlier points . without removing outliers , we find only a slightly negative slope on the best fit line indicating only a slight performance improvement between hrg models with 100 rules and hrg models with 1,000 rules . pearson s correlation coefficient comparing gcd and model size similarly show slightly negative correlations on routers ( @xmath109 , @xmath110 ) , enron ( @xmath111 ) , arxiv ( @xmath112 ) , and dblp ( @xmath113 , @xmath114 ) the overall execution time of the hrg model is best viewed in two parts : ( 1 ) rule extraction , and ( 2 ) graph generation . we previously identified the runtime complexity of the rule extraction process to be @xmath115 . however , this did not include @xmath79 samples of size-@xmath80 subgraphs . so , when sampling with @xmath79 and @xmath80 , we amend the runtime complexity to be @xmath116 where @xmath62 is bounded by the number of hyperedges in the size-@xmath80 subgraph sample and @xmath117 . graph generation requires a straightforward application of rules and is linear in the number of edges in the output graph . all experiments were performed on a modern consumer - grade laptop in an unoptimized , unthreaded python implementation . we recorded the extraction time while generating graphs for the size - to - gcd comparison in the previous section . although the runtime analysis gives theoretical upper bounds to the rule extraction process , fig . [ fig : size_time ] shows that the extraction runtime is highly correlated to the size of the model in routers ( @xmath106 ) , arxiv ( @xmath118 ) , enron ( @xmath119 ) , and dblp ( @xmath120 ) all with @xmath108 . simply put , more rules require more time , but there are diminishing returns . so it may not be necessary to learn complex models when smaller hrg models tend to perform reasonably well . lastly , we characterize the robustness of graph generators by introducing a new kind of test we call the _ infinity mirror _ @xcite . one of the motivating questions behind this idea was to see if hrg holds sufficient information to be used as a reference itself . in this test , we repeatedly learn a model from a graph generated by the an earlier version of the same model . for hrg , this means that we learn a set of production rules from the original graph @xmath15 and generate a new graph @xmath37 ; then we set @xmath121 and repeat thereby learning a new model from the generated graph recursively . we repeat this process ten times , and compare the output of the tenth recurrence with the original graph using gcd . we expect to see that all models degenerate over 10 recurrences because graph generators , like all machine learning models , are lossy compressors of information . the question is , how quickly do the models degenerate and how bad do the graphs become ? figure [ fig : inf_mir_gcd ] shows the gcd scores for the hrg , chung - lu and kronecker models at each recurrence . surprisingly , we find that hrg stays steady , and even improves its performance while the kronecker and chung - lu models steadily decrease their performance as expected . we do not yet know why hrg improves performance in some cases . because gcd measures the graphlet correlations between two graphs , the improvement in gcd may be because hrg is implicitly honing in on rules that generate the necessary graph patterns . yet again , further work is needed to study this important phenomenon . in this paper we have shown how to use clique trees ( also known as junction trees , tree decomposition , intersection trees ) constructed from a simple , general graph to learn a hyperedge replacement grammar ( hrg ) for the original graph . we have shown that the extracted hrg can be used to reconstruct a new graph that is isomorphic to the original graph if the clique tree is traversed during reconstruction . more practically , we show that a stochastic application of the grammar rules creates new graphs that have very similar properties to the original graph . the results of graphlet correlation distance experiments , extrapolation and the infinity mirror are particularly exciting because our results show a stark improvement in performance over existing graph generators . in the future , we plan to investigate differences between the grammars extracted from different types of graphs ; we are also interested in exploring the implications of finding two graphs which have a large overlap in their extracted grammars . among the many areas for future work that this study opens , we are particularly interested in learning a grammar from the actual growth of some dynamic or evolving graph . within the computational theory community there has been a renewed interest in quickly finding clique trees of large real world graphs that are closer to optimal . because of the close relationship of hrg and clique trees shown in this paper , any advancement in clique tree algorithms could directly improve the speed and accuracy of graph generation . perhaps the most important finding that comes from this work is the ability to interrogate the generation of substructures and subgraphs within the grammar rules that combine to create a holistic graph . forward applications of the technology described in this work may allow us to identify novel patterns analogous to the previously discovered triadic closure and bridge patterns found in real world social networks . thus , an investigation in to the nature of the extracted rules and their meaning ( if any ) is a top priority . we encourage the community to explore further work bringing hrgs to attributed graphs , heterogeneous graphs and developing practical applications of the extracted rules . given the current limitation related to the growth in the number of extracted rules as well as the encouraging results from small models , we are also looking for sparsification techniques that might limit the model s size while still maintaining performance . this work is supported by the templeton foundation under grant fp053369-m / o . s. aguinaga and t. weninger . the infinity mirror test for analyzing the robustness of graph generators . in _ acm sigkdd workshop on mining and learning with graphs _ , mlg 16 , new york , ny , usa , 2016 .

discovering the underlying structures present in large real world graphs is a fundamental scientific problem . in this paper we show that a graph s clique tree can be used to extract a hyperedge replacement grammar . if we store an ordering from the extraction process , the extracted graph grammar is guaranteed to generate an isomorphic copy of the original graph . or , a stochastic application of the graph grammar rules can be used to quickly create random graphs . in experiments on large real world networks , we show that random graphs , generated from extracted graph grammars , exhibit a wide range of properties that are very similar to the original graphs . in addition to graph properties like degree or eigenvector centrality , what a graph `` looks like '' ultimately depends on small details in local graph substructures that are difficult to define at a global level . we show that our generative graph model is able to preserve these local substructures when generating new graphs and performs well on new and difficult tests of model robustness . = [ circle , minimum width=10 , draw , fill = black!5 , inner sep=1.5 ] = [ circle , minimum width=10 , draw = black!40 , fill = black!05 , inner sep=1.5 , text = black!40 ] = [ rounded corners=3pt , draw , minimum height=14pt , inner sep=0 ] = [ itxset , ultra thick ] = [ textnode , circle , draw , fill , text = white ] = [ textnode , circle , draw ] = [ textnode , draw , inner xsep=1.5 ] = [ circle , draw , fill , minimum size=1.0mm , inner sep=0pt , outer sep=0pt ] = [ circle , draw , fill , minimum size=1.2mm , inner sep=0pt , outer sep=0pt ] = [ draw = black!40 , inner sep=1.5 , text = black!40 ] = [ very thin , color = black!50 ]

sodium represents a neutron poison for the slow neutron capture ( @xmath9 ) process , particularly in massive stars with more than about eight solar masses ( @xmath108 m@xmath11 ) @xcite . the @xmath9 process in massive stars is particularly efficient in producing species in the mass range 60@xmath12a@xmath1290 , forming the weak @xmath9-process component in the inventory of the solar abundances . in addition to its importance for the neutron balance in massive stars , the ( @xmath1 ) cross section of @xmath0na is also needed to follow the production of sodium in low and intermediate mass asymptotic giant branch ( agb ) stars . in these stars , the @xmath13 @xmath9 process contributes most of the @xmath9 abundances in the solar system from zr to pb , and the @xmath14 @xmath9-process adds to the pb / bi abundances at the termination point of the @xmath9-process reaction path @xcite . the weak @xmath9 process in massive stars begins at the end of convective core he - burning ( @xmath15 ) , where @xmath16ne(@xmath17)@xmath18 mg operates as the principal neutron source . during that period , sodium is only marginally produced by neutron captures on @xmath16ne . during the subsequent convective c - shell burning , sodium is efficiently made via the @xmath19c(@xmath20)@xmath0na channel although most of the protons ( and sodium ) are consumed by @xmath0na(@xmath21)@xmath22ne reactions @xcite . nevertheless , the c - burning layers of massive stars ejected in the subsequent supernova ( sn ) are one of the major sources of sodium in the galaxy @xcite , together with stellar winds from agb stars ( e.g. @xcite ) . in the convective c - burning shell , neutrons are mainly released via @xmath16ne(@xmath17)@xmath18 mg reactions as @xmath16ne is present in the ashes of the convective he - burning core and @xmath23 particles are liberated in @xmath19c(@xmath24)@xmath22ne reactions ( e.g. @xcite ) . in the weak @xmath9 process most of the neutrons are captured by abundant light isotopes , which act as neutron poisons , and only a small fraction is available for captures on iron seed nuclei to feed heavy isotope nucleosynthesis . at solar metallicity , more than 70% of the available neutrons are captured by neutron poisons in the he - burning core , and more than 90% in the c - burning shell . for this reason , it is extremely important to quantify the neutron capture rates of light isotopes such as sodium to evaluate the impact of neutron poisons in the weak @xmath9 process . another relevant source for the production of sodium are thermally pulsing low - mass ( e.g. , @xcite ) and massive agb stars @xcite , where the @xmath9 process is related to the he shell burning stage of evolution . in a first episode , neutrons are produced by @xmath25c(@xmath17)@xmath26o reactions during the interpulse phase between he shell flashes at temperatures of @xmath27=0.9 ( @xmath28=8 kev ) @xcite . the mixing of protons with the top layer of the he shell , required to provide the necessary @xmath25c for neutron production , has the additional effect of activating the nena cycle in the partial mixing zone @xcite , which then continues in the h - burning shell throughout the interpulse phase @xcite . a second , weaker neutron exposure takes place during the he shell flash at higher temperatures of @xmath29=2.6 ( @xmath28=23 kev ) when the @xmath16ne(@xmath17)@xmath18 mg source is marginally activated . as the he flash engulfs the ashes of the h burning shell , further @xmath0na might be produced by neutron captures on the abundant @xmath16ne during this second phase of @xmath9-processing in thermally pulsing agb stars . recent studies by cristallo _ _ @xcite and bisterzo _ et al . _ @xcite confirm that neutron capture production of primary sodium is particularly efficient in low - mass agb stars of low metallicity . intermediate - mass agb models experience hot hydrogen burning ( hbb ) , which modifies the na abundance on the stellar surface depending on the attained temperature and on the interplay with the efficiency for third dredge up @xcite . despite of its relevance for nuclear astrophysics , the maxwellian averaged neutron capture cross section ( macs ) of @xmath0na is rather uncertain @xcite . in this work , we present new experimental data for @xmath0na measured in quasi - stellar neutron spectra at thermal energies of @xmath3 and 25 kev . appropriate spectra have been obtained via the @xmath4o(@xmath5)@xmath4f and @xmath6li(@xmath5)@xmath6be reactions to simulate stellar temperature conditions relevant to @xmath9-process nucleosynthesis . the experimental details and results of the activation measurements are given in secs . [ expsection ] and [ anasection ] . in sec . [ implications ] , macs values are derived for the full range of @xmath9-process temperatures on the basis of the present results . the implications of these data for the @xmath9-process abundances are discussed for massive stars as well as for agb stars . similar to many light nuclei , the @xmath0na(@xmath1)@xmath2na cross section is difficult to study experimentally given the high ratio of scattering to capture cross sections . in such cases , neutrons scattered on the sample and subsequently captured in or near the detector can induce a large background when measuring prompt capture gammas with the time of flight ( tof ) technique @xcite . these difficulties can be avoided with the activation method , because the induced activity of the product nucleus is counted only after the irradiation in a low background environment . therefore , the activation technique is well suited to measure ( @xmath30 ) cross sections of light nuclei with greater precision than reported previously from tof measurements . the experiment was carried out by a series of repeated irradiations with a set of different samples and by variation of the relevant activation parameters . in this way , corrections concerning the dimensions of the samples , self absorption , and the decay during irradiations could be constrained and the determination of systematic uncertainties improved . the samples for the individual runs were prepared from nacl ( 99.99% pure ) pressed into cylindrical pellets 6 , 8 , 10 , and 15 mm in diameter with varying thicknesses . as nacl is hygroscopic , care was taken to be sure that the water content of the material gave a negligible contribution to the mass . this was verified by heating a quantity of the nacl at 250 @xmath31c for two hours , showing that the sample mass before and after heating differed by less than 0.01% . the sodium cross section was measured relative to that of gold , which is commonly used as a reference in activation measurements . gold foils 0.03 mm in thickness were cut to the proper diameters and fixed to the front and back of the samples during irradiation . the sample masses , as well as those of the respective gold foils , are given in table [ tab1 ] . [ cols="<,^,^,^,^,^ " , ] the correction of these macs values for the effect of thermally excited nuclear states , the so - called stellar enhancement factor , is negligible over the entire range of @xmath9-process temperatures @xcite . the @xmath9 process in massive stars is known to produce most of the @xmath9 isotopes in the solar system between fe and sr ( see @xcite and references therein ) . in the convective he core , the neutron exposure starts to increase only in the last phase , close to he exhaustion , when the temperature is high enough to efficiently burn @xmath16ne via @xmath16ne(@xmath17)@xmath18 mg . the @xmath16ne available at the end of the he core phase is given by the initial abundance of the cno nuclei . as cno elements are converted to @xmath32n in the previous h - burning core , @xmath32n is converted to @xmath4o via the reaction channel @xmath32n(@xmath33)@xmath4f(@xmath34)@xmath4o at the beginning of the he - burning core and then to @xmath16ne by @xmath23-captures when the temperature exceeds @xmath27 = 2.5 . at the point of he exhaustion the most abundant isotopes are @xmath26o , @xmath19c , @xmath35ne and @xmath36 mg , where the final @xmath19c and @xmath26o abundances are defined by the @xmath19c(@xmath23,@xmath37)@xmath26o reaction . in he core conditions , @xmath0na is produced by the neutron capture channel @xmath16ne(@xmath1)@xmath0ne(@xmath38)@xmath0na , and it is depleted via @xmath0na(@xmath1)@xmath2na . in the convective c shell the neutron exposure starts to increase during c ignition at the bottom of the shell , where neutrons are mainly produced again by the @xmath16ne(@xmath39)@xmath18 mg reaction . typical temperatures at the bottom of the c shell are t@xmath40 1 gk , almost constant during the major part of the shell development ( e.g. , @xcite ) . in the last day(s ) before the sn , temperatures at the base of the c shell may increase due to thermal instabilities in the deeper o - burning layers , and if the c shell is still fully convective , c - shell nucleosynthesis will be revived @xcite . at the end of the convective c - burning shell the most abundant isotopes are @xmath26o , @xmath22ne , @xmath0na and @xmath2 mg . sodium is mainly produced via the c - burning reaction @xmath19c(@xmath19c , @xmath41)@xmath0na and marginally via @xmath16ne(@xmath42)@xmath0na and @xmath16ne(@xmath1)@xmath0ne(@xmath38)@xmath0na . the strongest sodium depletion reaction is @xmath0na(@xmath21)@xmath22ne , with smaller contributions from @xmath0na(@xmath42)@xmath2 mg and @xmath0na(@xmath1)@xmath2na . the impact of the new @xmath0na(@xmath1)@xmath2na cross section on the weak @xmath9-process distribution was studied with the nugrid post - processing code mppnp @xcite for a full 25 m@xmath43 stellar model of solar metallicity @xcite . the stellar structure was previously calculated using the genec stellar evolution code @xcite . by the end of the core he burning phase the @xmath9-process abundance distribution between @xmath44fe and @xmath45mo was found to be rather insensitive to the macs values for @xmath0na(@xmath1)@xmath2na . although the macs at 25 kev ( 25@xmath4630 kev is the temperature range of the @xmath9 process during core he@xmath46burning ) is about 10% lower compared to the previous rate @xcite , the final @xmath0na overabundance increases by only a few % and the effect on the @xmath9 abundances between fe and sr is limited to about 1% . this is explained by the fact that in he core conditions the @xmath0na production is marginal , and its abundance coupled with the low macs implies that the neutron poisoning effect of @xmath0na during core he burning is low . the final @xmath9-abundance distribution at the end of c shell burning between @xmath44fe and @xmath45mo obtained with the new @xmath0na(@xmath1)@xmath2na macs is compared in fig . [ fig3 ] with the distribution based on the previous rate @xcite . at this point , the entire isotopic distribution is affected with variations in the order of 5% . at 90 kev thermal energy ( typical for the c - burning phase ) the new macs of @xmath0na is lower by 13% compared to the previous rate of bao _ et al . while the effect on the final overabundance of @xmath0na increases only by about 1% , the effect of @xmath0na as an important neutron poison in the c shell becomes evident by the propagation effect beyond iron . -abundance distribution at the end of c shell burning for a 25 m@xmath11 star compared to the distribution obtained with the macs of @xmath0na from the kadonis compilation @xcite . bottom : isotopic ratios emphasizing the reduced neutron poison effect due to the smaller macs of @xmath0na from this work . ( isotopes of the same element are connected by solid lines.)[fig3],title="fig:",width=302 ] -abundance distribution at the end of c shell burning for a 25 m@xmath11 star compared to the distribution obtained with the macs of @xmath0na from the kadonis compilation @xcite . bottom : isotopic ratios emphasizing the reduced neutron poison effect due to the smaller macs of @xmath0na from this work . ( isotopes of the same element are connected by solid lines.)[fig3],title="fig:",width=302 ] interestingly , the neutron - rich isotopes @xmath47zn , @xmath48ge , and @xmath49se , which are traditionally considered to be of @xmath50-only origin , are affected by the new macs of @xmath0na as much as most of the @xmath9-only isotopes ( e.g. , @xmath47ge and @xmath48se ) . the reduced neutron poison effect of the lower sodium macs leads to an enhancement of the neutron density , thus increasing the neutron capture probability in the @xmath9-process branchings . by far the strongest change is obtained for the branching point at @xmath51se , where neutron capture on the unstable isotope @xmath51se becomes more probable . as a consequence , the @xmath52kr/@xmath49kr ratio is reduced by about 3% . the changes in the @xmath53kr branching affect mostly the final abundances of @xmath54kr and - by the later decay of @xmath53kr - of @xmath53rb , rather than those of the related @xmath9-only isotopes @xmath55sr . beyond the abundance peak around sr , the @xmath9-process production in massive stars becomes marginal , and the current macs of @xmath0na has a negligible effect . in the model used in this work the c shell is not convective during the last day before the sn . in models , where the c shell stays convective , the neutron density rises from a few 10@xmath56 up to a few 10@xmath19 because all the residual @xmath16ne is consumed in ( @xmath17 ) reactions at the final increase of the c - burning temperature @xcite . in this case , the higher neutron density will lead to a correspondingly larger modification of the abundance pattern in the @xmath9-process branchings . it is interesting to note that the higher @xmath9-process efficiency found with the reduced macs data for @xmath0na is partly compensated by the effect of revised ( @xmath1 ) data for the ne @xcite and mg @xcite isotopes . accordingly , we confirm the conclusion of heil _ et al . _ @xcite for the weak component , i.e. that `` the reproduction of the @xmath9 abundances in the solar system is far from being settled . '' accordingly , further improvements of the neutron capture cross sections for heavy species along the @xmath9-process path and for light neutron poisons are fundamental for constraining @xmath9-process nucleosynthesis predictions in massive stars . there are essentially two mechanisms for sodium production in agb stars . at solar metallicities , sodium is produced primarily during h shell burning where the mixing of protons with the he shell gives rise not only to the formation of a @xmath25c pocket ( where neutrons are produced via the @xmath25c(@xmath17)@xmath26o reaction ) , but also to related @xmath32n and @xmath0na pockets , thus activating the nena cycle in the latter mixing zone @xcite . under these conditions , neutron reactions on sodium are of minor importance . at low metallicities , however , large amounts of primary @xmath16ne are synthesized by conversion of primary @xmath19c into @xmath32n during h burning , which is then transformed during he burning by the sequence @xmath32n(@xmath33)@xmath4f(@xmath57)@xmath4o(@xmath58)@xmath16ne @xcite . this @xmath16ne contributes significantly to the primary production of light isotopes , as @xmath0na ( via @xmath16ne(@xmath1)@xmath0ne(@xmath38)@xmath0na ) and @xmath2 mg ( via @xmath0na(@xmath1)@xmath2na(@xmath38)@xmath2 mg ) . accordingly , @xmath16ne and @xmath0na are - together with @xmath19c , @xmath32n , and @xmath26o - major neutron poisons in the @xmath25c pocket . as shown in the nucleosynthesis studies of cristallo _ et al . _ @xcite neutron captures on @xmath16ne account for about 50% of the total sodium production at very low metallicity ( @xmath59 ) . for higher metallicities this effect decreases and becomes negligible at [ fe / h]@xmath60@xmath61 . with the larger neutron exposures in stars of low metallicity , which are characteristic of the strong @xmath9-process component , the highly abundant ne and na are either acting as seeds for the reaction flow ( enhancing the @xmath9-process production up to pb / bi @xcite ) or as neutron poisons , depending on the efficiency for neutron production in the @xmath25c pocket . -process yields of a 1.5 m@xmath11 star with z=0.0001 obtained with the present and old macs for @xmath0na . pure @xmath9-nuclei are highlighted by full circles , crosses outside the overall distribution are due to branchings in the reaction path . top : for neutron captures in the @xmath25c pocket @xmath0na acts as an additional seed . this contribution is reduced by the smaller macs of this work . bottom : in less efficient @xmath25c pockets the poisoning effect of @xmath0na dominates . consequently , more free neutrons are available due to the smaller macs , thus relaxing the poisoning effect . [ agb],title="fig:",width=302 ] -process yields of a 1.5 m@xmath11 star with z=0.0001 obtained with the present and old macs for @xmath0na . pure @xmath9-nuclei are highlighted by full circles , crosses outside the overall distribution are due to branchings in the reaction path . top : for neutron captures in the @xmath25c pocket @xmath0na acts as an additional seed . this contribution is reduced by the smaller macs of this work . bottom : in less efficient @xmath25c pockets the poisoning effect of @xmath0na dominates . consequently , more free neutrons are available due to the smaller macs , thus relaxing the poisoning effect . [ agb],title="fig:",width=302 ] the effect of the present macs is illustrated in fig . [ agb ] for the case of a 1.5 m@xmath11 star with a @xmath62200 times lower metallicity compared to solar ( z=0.0001 ) . for efficient @xmath25c - pockets ( e.g. for the standard @xmath25c pocket ( st ) adopted in ref . @xcite ) the ne - na abundances are acting as neutron seeds and are contributing to the @xmath9-process production up to pb / bi . with the smaller macs for @xmath0na these contributions are reduced , resulting in the relative reduction of the @xmath9-distribution indicated in the upper panel of fig . [ agb ] . in less efficient @xmath25c pockets ( st/12 ) the role of @xmath0na as a neutron poison becomes dominant as shown in the lower panel . due to the smaller macs , more free neutrons are now available for the @xmath9-process and are leading to an increase of the @xmath9-distribution . one of the major uncertainties of the @xmath9 process in low - mass agb stars is related to the mixing mechanisms that model the @xmath25c - pocket . a clear answer to the properties involved in such mixing , possibly resulting from the interplay between different physical processes in stellar interiors ( e.g. , overshooting , semi - convection , rotation , magnetic fields , see review by herwig @xcite and refs . @xcite ) has not been reached yet , thus leaving the structure of the @xmath25c - pocket a persisting problem . depending on the shape and extension of the @xmath25c pocket , the impact of light neutron poisons may affect the @xmath9 distribution in different ways . because the @xmath25c pocket is artificially introduced in our post - process agb models , the impact of the new @xmath0na macs could be explored by adopting different shapes and sizes of the @xmath25c pocket according to recent theoretical and observational indications . from the results obtained in these tests , the @xmath9-distribution was affected by less than @xmath625% , independent of the assumptions for the @xmath25c pocket . therefore , the improved accuracy of the present macs provides significant constraints for the neutron poisoning effect of @xmath0na in agb stars . the @xmath0na(@xmath1)@xmath2na cross section has been measured at the karlsruhe van de graaff accelerator in quasi - stellar thermal neutron spectra at @xmath3 and 25 kev . the resulting maxwellian averaged cross sections of @xmath63 mb and @xmath8 mb are significantly smaller compared to the recommended values of the kadonis - v0.3 compilation @xcite . after reducing the radiative width of the prominent s - wave resonance at 2.8 kev by 35% , the measured cross sections were found perfectly compatible with the set of resonance data in the endf / b - ii.1 library . with this modification , maxwellian averaged cross sections in the relevant range of thermal energies between @xmath64 kev were derived using the energy dependence obtained by an r - matrix calculation with the sammy code @xcite . the effect of the present cross section on the @xmath9 process abundances in massive stars ( weak @xmath9 process ) is quite small during the he core burning phase , but becomes significant during the carbon - shell burning phase where @xmath0na is synthesized in increasing quantities via the @xmath19c(@xmath20)@xmath0na reaction . the impact of the present macs measurement has been investigated within a massive star model . it was found that the new lower macs causes a propagation effect over the entire weak @xmath9-process distribution , with a general abundance increase of about 5% . the authors are thankful to d. roller , e .- knaetsch , w. seith , and the entire van de graaff group for their support during the measurements . eu would also like to acknowledge the support of jina ( joint institute for nuclear astrophysics ) , university of notre dame , notre dame , in , usa . sb acknowledges financial support from jina ( joint institute for nuclear astrophysics , university of notre dame , in ) and kit ( karlsruhe institute of technology , karlsruhe , germany ) . mp acknowledges support from nugrid by nsf grant phy 09 - 22648 ( joint institute for nuclear astrophysics , jina ) , nsf grant phy-1430152 ( jina center for the evolution of the elements ) , and eu mirg - ct-2006 - 046520 . he also appreciates support from the `` lendulet-2014 '' programme of the hungarian academy of sciences ( hungary ) and from snf ( switzerland ) . mp also acknowledges prace , through its distributed extreme computing initiative , for resource allocations on sisu ( csc , finland ) , archer ( epcc , uk ) , and beskow ( kth , sweden ) and the support of stfc s dirac high performance computing facilities ; dirac is part of the national e - infrastructure . ongoing resource allocations on the university of hull s high performance computing facility - viper - are gratefully acknowledged . cl acknowledges support from the science and technology facilities council uk ( st / m006085/1 ) . i. dillmann , r. plag , f. kppeler , and t. rauscher , in _ efnudat fast neutrons - scientific workshop on neutron measurements , theory & applications _ , edited by f .- hambsch ( jrc - irmm , geel , 2009 ) , pp . 55 58 , http://www.kadonis.org .

the cross section of the @xmath0na(@xmath1)@xmath2na reaction has been measured via the activation method at the karlsruhe 3.7 mv van de graaff accelerator . nacl samples were exposed to quasistellar neutron spectra at @xmath3 and 25 kev produced via the @xmath4o(@xmath5)@xmath4f and @xmath6li(@xmath5)@xmath6be reactions , respectively . the derived capture cross sections @xmath7 mb and @xmath8 mb are significantly lower than reported in literature . these results were used to substantially revise the radiative width of the first @xmath0na resonance and to establish an improved set of maxwellian average cross sections . the implications of the lower capture cross section for current models of @xmath9-process nucleosynthesis are discussed .

planets are formed in a circumstellar disk composed of gas and solid materials ( solids are of the order of 1% in mass ) . the solid material is initially sub - micron grains , which are controlled by an aerodynamical frictional force that is much stronger than the gravity of the central star ( * ? ? ? * hereafter ahn ) . as solid bodies grow , gas drag becomes relatively less important . once bodies get much larger than 1 m , they have keplerian orbits around the central star that are slightly altered by gas drag ; then , their orbits are characterized by orbital elements : semimajor axes @xmath1 , eccentricities @xmath2 , and inclinations @xmath3 . these bodies grow via collisions , and the collisional rates are given by relative velocities determined by @xmath2 and @xmath3 ( e.g. , * ? ? ? damping due to gas drag and stirring by the largest body in each annulus of the disk mainly control @xmath2 and @xmath3 , which evolve in the protoplanetary disk during planet formation . in addition , radial drift due to gas drag , which is expressed by @xmath4 , reduces small bodies , which stalls the growth of bodies ( e.g. , * ? ? ? * ; * ? ? ? * ) . therefore , the time derivative of @xmath1 , @xmath2 , and @xmath3 ( @xmath4 , @xmath5 , and @xmath6 ) caused by gas drag are very important for planet formation . protoplanets are formed out of collisions with kilometer - sized or larger bodies called planetesimals . while protoplanets grow , @xmath2 and @xmath3 of planetesimals are determined by the hill radius of the protoplanets , and their @xmath2 and @xmath3 are smaller than 0.3 unless the protoplanets are greater than ten earth masses ( see equation 15 of * ? ? ? * ) . therefore , ahn derived formulae of @xmath4 , @xmath5 , and @xmath6 due to gas drag for a body with low @xmath14 and @xmath15 . however , @xmath2 and @xmath3 may possibly increase when more massive planets are formed . indeed , in the solar system , some comets , asteroids , and kuiper belt objects have very high @xmath2 and @xmath3 ( e.g. , * ? ? ? in addition , if inclined and eccentric orbits of irregular satellites around jovian planets are originated from captures due to interaction with circumplanetary disks ( e.g. , * ? ? ? * ) , these captured bodies with high @xmath2 and @xmath3 evolve their orbits in the disks . therefore , analytic formulae for @xmath4 , @xmath5 , and @xmath6 for bodies with high @xmath2 and @xmath3 are helpful for the analysis of small bodies in the late stage of planet formation . in this paper , i first introduce a model for gaseous disks such as protoplanetary and circumplanetary disks , and then , i revisit the derivation of @xcite for the analytic formulae of @xmath4 , @xmath5 , and @xmath16 . next , i derive @xmath4 , @xmath5 , and @xmath6 for bodies with high @xmath2 and/or high @xmath3 . by combining these limited solutions , i construct approximate formulae for @xmath4 , @xmath5 , and @xmath6 , which are applicable for all @xmath2 and @xmath3 unless @xmath17 or @xmath18 . lastly , i discuss the orbital evolution of satellites captured by circumplanetary disks using the derived analytic formulae for @xmath4 , @xmath5 , and @xmath16 . in order to evaluate the drag force due to nebula gas , the disk model is set as follows . a gaseous disk rotates around a central object with mass @xmath19 , which is axisymmetric and in a steady state . in a cylindrical coordinate system ( @xmath20 ) , the gas density @xmath21 is defined from the force equilibrium in the @xmath22 direction in a vertical isothermal disk as @xmath23 where @xmath24 is the surface density of the nebula disk , @xmath25 is the disk scale height , @xmath26 is the keplerian angular velocity , and @xmath27 is the gravitational constant . for simplicity , the @xmath28-dependences of @xmath29 and @xmath30 are assumed as @xmath31 , @xmath32 , respectively . this relations give @xmath33 , where @xmath34 . in the minimum - mass solar nebula model @xcite , for example , @xmath35 and @xmath36 . the angular gas velocity @xmath37 is obtained from the force equilibrium in the @xmath28 direction as @xcite @xmath38 . \label{eq : anglar}\ ] ] in equation ( [ eq : anglar ] ) , the terms of @xmath39 and higher are ignored . this treatment is valid even for investigation of the gas drag effect on highly inclined orbits because the gas drag ( and the nebula gas ) is negligible at a high altitude ( @xmath40 ) . at the midplane of the disk , the relative velocity difference between the gas motion and the keplerian rotation is given by @xmath41 for a body with mass @xmath42 and radius @xmath43 , gas drag force per unit mass can be written as ahn @xmath44 where @xmath45 is the dimensionless gas drag coefficient , @xmath46 is the relative velocity vector between the body and the gas , @xmath47 , and @xmath48 . although @xmath49 depends on mach number @xmath50 and reynolds number @xmath51 , @xmath45 is a constant for high @xmath51 ( @xmath52 km in the minimum - mass solar nebula ) or for @xmath53 ( @xmath2 or @xmath3 is much larger than @xmath54 ) ( ahn ) . in this paper , i investigate the time variations of semimajor axis @xmath1 , eccentricity @xmath2 , and inclination @xmath3 of a body due to gas drag for constant @xmath45 ( and then constant @xmath58 ) . the time derivatives of @xmath1 , @xmath2 , and @xmath3 are given by ahn as @xmath59 , \label{eq : gauss_a } \\ \frac{de}{dt } & = & - a \rho u \left[2 \cos f + 2 e - \frac{2 \cos f + e + e \cos^2 f}{(1+e \cos f)^{1/2}}\kappa \cos i \right ] , \\ \frac{di}{dt } & = & - a \rho u \frac{\cos^2 ( f + \omega)}{(1+e \cos f)^{1/2}}\kappa \sin i , \label{eq : gauss_i}\end{aligned}\ ] ] where @xmath60 and @xmath61 are the true anomaly and the argument of pericenter , respectively , @xmath62 , @xmath63^{1/2}$ ] , @xmath64^{1/2 } , \label{eq : u}\end{aligned}\ ] ] @xmath65 is the midplane density at @xmath66 , and @xmath67 is the keplerian velocity . if the variation timescales of @xmath1 , @xmath2 , and @xmath3 are much longer than the orbital time , the evolution of @xmath1 , @xmath2 , and @xmath3 follows the averaged rate . the orbital averaging is taken as @xmath68 where @xmath69 is the keplerian period . the same averaging is taken for @xmath2 and @xmath3 . for @xmath72 , @xcite derived the averaged changes in @xmath1 , @xmath2 , and @xmath3 for three cases , ( i ) @xmath73 , ( ii ) @xmath74 , and ( iii ) @xmath75 , and summed up the leading terms for these cases . this method was used to treat @xmath76 in equations ( [ eq : gauss_a ] ) to ( [ eq : gauss_i ] ) analytically : the assumptions simplify as @xmath77 in case ( i ) , @xmath78 in case ( ii ) , and @xmath79 in case ( iii ) . other terms are also simplified , such as @xmath80 . then , the terms are easily averaged over the orbital period by equation ( [ eq : average ] ) . the derived formulae are in good agreement with the results of orbital integrations for @xmath7 and @xmath81 . while @xcite provided the term of @xmath82 in @xmath4 , they did not take into account the vertical dependence of @xmath21 , which includes other @xmath82 terms . since the sum of these @xmath82 terms is negligible , i thus exclude the @xmath82 term derived by ahn . @xcite found that the mean root squares of these limited solutions are in better agreement with the results of orbital evolution than the simple summation by @xcite . the averaged variation rates of @xmath1 , @xmath2 , and @xmath3 are therefore given by @xmath83 ^ 2 \right\}^{1/2 } , \label{eq : mod_adachi_a } \\ -\frac{\tau_0}{e } \left\langle \frac{de}{dt } \right\rangle_1 & = & \left [ \left ( \frac{3}{2}\eta\right)^2 + \left ( \frac{2}{\pi}i \right)^2 + \left ( \frac{2e}{\pi}e \right)^2 \right]^{1/2 } , \\ - \frac{\tau_0}{i } \left\langle \frac{di}{dt } \right\rangle_1 & = & \frac{1}{2 } \left\ { \eta^2 + \left ( \frac{8}{3\pi}i \right)^2 + \left [ \frac{2e}{\pi}e \left ( 1+\frac{2k-5e}{9e } \cos 2\omega \right ) \right]^2 \right\}^{1/2 } , \label{eq : mod_adachi_i}\end{aligned}\ ] ] where @xmath84 and @xmath85 are the first and second complete elliptic integrals of argument @xmath86 , respectively , and @xmath87 is the stopping time due to gas drag for @xmath88 . note that i corrected an error in the factor of the @xmath89 term for @xmath5 in @xcite , which was pointed out by @xcite . for @xmath90 , equations ( [ eq : mod_adachi_a ] ) to ( [ eq : mod_adachi_i ] ) are compared with the results of orbital integrations in figure [ fig1 ] . these formulae are valid unless @xmath91 . moreover , the @xmath3 dependence in these formulae are valid for @xmath92 ( see figure [ fig2 ] ) . -5pt , @xmath2 , and @xmath3 as a function of @xmath2 for @xmath93 and @xmath61 @xmath94 @xmath95 in the disk with @xmath96 , @xmath97 , and @xmath98 . analytic formulae for low @xmath2 ( gray dotted curves ) , given by equations ( [ eq : mod_adachi_a ] ) to ( [ eq : mod_adachi_i ] ) , and ones for high @xmath2 ( gray dashed curves ) , given by equations ( [ eq : da_he ] ) to ( [ eq : di_he ] ) , are in good agreement with the results of orbital integration ( open circles ) for low @xmath2 or high @xmath2 , respectively . the combined formulae ( solid curves ) , given by equations ( [ eq : da_high ] ) to ( [ eq : di_mid ] ) , are valid for the whole region.,width=377 ] -5pt -5pt , @xmath2 , and @xmath3 as a function of @xmath3 for @xmath99 , and @xmath100 in the same disk as figure [ fig1 ] . analytic formulae for low @xmath3 ( gray dotted curves ) , given by equations ( [ eq : mod_adachi_a ] ) to ( [ eq : mod_adachi_i ] ) , and those for high @xmath3 ( gray dashed curves ) , given by equations ( [ eq : da_high ] ) to ( [ eq : di_high ] ) , are in good agreement with the results of orbital integration ( open circles ) for @xmath3 @xmath101 and @xmath3 @xmath102 , respectively . the combined formulae ( solid curves ) , given by equations ( [ eq : da_high ] ) to ( [ eq : di_mid ] ) , represent within a factor of @xmath103.,width=377 ] -5pt here , let us consider the case where @xmath2 is almost equal to unity and @xmath3 is much smaller than @xmath104 . expanding equations ( [ eq : gauss_a ] ) to ( [ eq : gauss_i ] ) with respect to @xmath105 under the assumption of @xmath106 , keeping only the lowest - order terms of @xmath105 , and applying the orbital averaging such as equation ( [ eq : average ] ) to these equations , @xmath107 where @xmath108 the dependences of @xmath4 and @xmath5 on @xmath60 are seen in the integral @xmath109 , while a term proportional to @xmath110 in @xmath6 vanishes by the orbital averaging because of an odd function of @xmath60 . the integrals of @xmath109 , @xmath111 , and @xmath112 are functions of @xmath113 . in the minimum - mass solar nebular model , @xmath114 is 5/4 , and then , @xmath115 , @xmath116 , and @xmath117 . the @xmath2 dependences in these formulae are applicable for @xmath118 as shown in figure [ fig1 ] . although the effective range of these formulae is limited , the @xmath2 dependences improve the high @xmath2 parts in equations ( [ eq : mod_adachi_a ] ) to ( [ eq : mod_adachi_i ] ) as shown below . next , let us consider highly inclined orbits where @xmath119 is much larger than unity . bodies with such a high inclination penetrate the nebula disk twice around the ascending and descending nodes through an orbital period . gas drag is effective only around the nodes . since the body experiences significant gas drag around the ascending node ( @xmath120 ) , the leading terms of @xmath121 for equations ( [ eq : gauss_a ] ) to ( [ eq : gauss_i ] ) are @xmath122 where @xmath123 , \\ j(\omega ) & = & \tilde{r}^{-\alpha + 1 } \tilde{u } \left [ 2 ( e + \cos \omega ) - \left ( \cos \omega + \frac{\cos \omega + e}{1 + e\cos \omega } \right ) \cos i \sqrt{1+e \cos \omega } \right ] , \\ k(\omega ) & = & \tilde{r}^{-\alpha + 2 } \tilde{u } \sqrt{1 + e \cos \omega},\end{aligned}\ ] ] and @xmath124 for this derivation , @xmath125 , since the relative velocity is mainly determined by inclination . in order to apply averaging over half an orbit around the ascending node , @xmath4 , @xmath126 , and @xmath6 are integrated from @xmath127 to @xmath128 . since @xmath4 , @xmath5 , and @xmath6 are gaussian functions as shown in equations ( [ eq : high_i_a ] ) to ( [ eq : high_i_i ] ) , they are negligible for large @xmath129 and the integral is thus approximated to be that over interval [ @xmath130 as follows : @xmath131 where @xmath132 . using this , equations ( [ eq : high_i_a ] ) to ( [ eq : high_i_i ] ) are integrated around the ascending node , which results in the averaged variation rates of @xmath1 , @xmath2 , and @xmath3 in half an orbit . the variation rates due to the penetration near the descending node ( @xmath133 ) are obtained in the same way as above . summing up the changes at two penetrations , the averaged changes are given by @xmath134,\label{eq : da_high } \\ \left < \frac{de}{dt } \right>_{\rm high } & = & - \frac{1}{\tau_0 } \frac{h}{2 \sqrt{\pi } a ( 1-e^2 ) \sin i } [ j(\omega ) + j(\omega + \pi ) ] , \\ \left < \frac{di}{dt } \right>_{\rm high } & = & - \frac{1}{\tau_0 } \frac{h}{2\sqrt{\pi}a ( 1-e^2)^2 } [ k(\omega ) + k(\omega + \pi)].\label{eq : di_high}\end{aligned}\ ] ] the validity of equations ( [ eq : da_high ] ) to ( [ eq : di_high ] ) is shown in figures [ fig2 ] and [ fig3 ] . these formulae are applicable for @xmath135 . -5pt , but for @xmath136 and dotted lines given by equations ( [ eq : da_he ] ) to ( [ eq : di_he]).,width=377 ] -5pt the variation rates of @xmath1 , @xmath2 , and @xmath3 in two limited cases for @xmath137 are derived above . the formulae for low @xmath2 do not well reproduce the variation rate in @xmath138 , while high-@xmath2 formulae overestimate the values for low @xmath2 . combination of low - eccentricity formulae of equations ( [ eq : mod_adachi_a ] ) to ( [ eq : mod_adachi_i ] ) with the @xmath139 dependence derived in equations ( [ eq : da_he ] ) to ( [ eq : di_he ] ) gives @xmath140 these formulae are given in a very simple way , but they are in good agreement with the results of orbital integration if @xmath141 ( see figures [ fig1 ] to [ fig3 ] ) . if @xmath142 , the variation rates of @xmath1 , @xmath2 , and @xmath3 are obtained from combination of the low-@xmath3 formulae of equations ( [ eq : da_low ] ) to ( [ eq : di_low ] ) and the high-@xmath3 formulae of equations ( [ eq : da_high ] ) to ( [ eq : di_high ] ) . @xmath143 where @xmath144 is the smaller of @xmath145 and @xmath146 . in conclusion , the variation rates for @xmath1 , @xmath2 , and @xmath3 are approximately given by * equations ( [ eq : da_low ] ) to ( [ eq : di_low ] ) for @xmath147 , * equations ( [ eq : da_mid ] ) to ( [ eq : di_mid ] ) for the intermediate inclination ( @xmath148 ) , * equations ( [ eq : da_high ] ) to ( [ eq : di_high ] ) for @xmath135 . in the intermediate @xmath3 , the formulae tend to deviate from the right values but the accuracies are within a factor of 1.5 ( see figures [ fig1 ] to [ fig3 ] ) . it should be noted that these formulae are not applicable to the case of @xmath13 where a body experiences gas drag with relative velocity @xmath149 not only around the nodes but also for a whole orbit . jovian planets have many satellites , which may be formed in circumplanetary disks . satellites close to planets mainly have circular and coplanar orbits and may be formed in the disks . however , distant satellites tend to have inclined orbits . here , i discuss the possibility of the capture of satellites in the disks because the formulae for @xmath4 , @xmath5 , and @xmath6 that i derive in this paper are applicable to bodies with high @xmath2 and @xmath3 . orbital evolution of bodies with high @xmath2 is predicted from these analytic formulae . when a body is captured by gas drag in a circumplanetary disk , @xmath2 of the captured body is approximately @xmath0 . for @xmath150 , @xmath151 and @xmath152 are very large . variation rate of the pericenter distance @xmath153 is much smaller than those of @xmath1 and @xmath2 . indeed , @xmath154 is estimated to be zero in equations ( [ eq : da_he ] ) and ( [ eq : de_he ] ) . the result is caused by the neglect of the higher terms of @xmath105 , and these higher @xmath105 terms give @xmath155 a positive value but @xmath155 is much smaller than @xmath152 and @xmath151 . therefore , the orbital evolution occurs along with almost constant @xmath153 . with decreasing @xmath2 , the orbital evolution changes . since @xmath152 becomes smaller than @xmath151 for @xmath156 to 0.6 , @xmath2 decreases with almost constant @xmath1 . once @xmath157 , @xmath4 becomes dominant for the orbital evolution ; the body drifts to the host planet in the timescale of @xmath158 . the bodies that will be satellites are temporally captured by a planet at first @xcite , and the apocenter distances of the bodies decrease to less than the hill radius of the host planet during the temporal capture of bodies ( e.g. , * ? ? ? the change of orbital eccentricity in an orbit around the host planet is given by @xmath159 . the body is fully captured by gas drag if @xmath160 during the temporal capture , where @xmath161 is the number of close encounters with the planet during the temporal capture . using the combined formulae ( equations [ eq : da_high ] to [ eq : di_mid ] ) at @xmath162 , @xmath163 is given by @xmath164 , where @xmath165 and @xmath166 are @xmath167 and @xmath168 at the pericenter distance @xmath153 , respectively . therefore , the necessary condition for capture is given by @xmath169 where the interior density of bodies , @xmath170 , is assumed to be @xmath171 , the hill radius of jupiter is applied to @xmath153 , and @xmath161 is possibly approximately @xmath172 @xcite . as shown in figure [ fig4 ] , @xmath173 is mainly 0.1 to 10 . this density is comparable to or less than the ` minimum mass subnebula ' disk that contains a mass in solids equal to the mass of current jovian satellites and gas according to the solar composition @xcite . it should be noted that the temporally captured bodies are significantly affected by the central star . however , the temporally captured bodies rotate around the host planet , which means that the perturbation by the central star is roughly canceled out in a temporally captured orbit . therefore , the energy loss due to gas drag estimated above may lead to bound orbits . -5pt and @xmath174 derived from the combined equations ( equations [ eq : da_he ] to [ eq : di_he]).,width=377 ] -5pt inclination decreases during the full capture by gas drag , which is estimated as @xmath175_{e=1}$ ] in figure [ fig4 ] . the initial inclination is damped during capture for @xmath176 , while inclinations remain high after capture for other @xmath3 . however , inclinations keep decreasing due to gas drag after capture . a dissipation time of the disk , @xmath177 , that is shorter than the damping time of inclination is thus necessary for the formation of high - inclination satellites : @xmath178 where @xmath179 is the host planet mass . since the dissipation processes of circumplanetary disks are not clear yet @xcite , it is difficult to discuss the dissipation timescale . however , the dissipation timescale needed to form high - inclination satellites seems too short . therefore , the capture of high - inclination satellites might have occurred in the timescale estimated in equation ( [ eq : tdisk ] ) before the disk dissipation and the resulting satellites tend to have retrograde orbits ( see figure [ fig4 ] ) . i have investigated the time derivatives of orbital semimajor axis @xmath1 , eccentricity @xmath2 , and inclination @xmath3 of a body orbiting in a gaseous disk . * i have derived @xmath4 , @xmath5 , and @xmath6 for @xmath180 and @xmath141 ( equations [ eq : da_he ] to [ eq : di_he ] ) and for @xmath135 ( equations [ eq : da_high ] to [ eq : di_high ] ) . in addition , i have modified the formulae derived by ahn ; equations ( [ eq : mod_adachi_a ] ) to ( [ eq : mod_adachi_i ] ) are valid for @xmath181 and @xmath141 , where @xmath9 is the disk scale height . * i have combined the formulae in the limited cases and have constructed approximate formulae for @xmath4 , @xmath5 , and @xmath6 ( equations [ eq : da_high ] to [ eq : di_mid ] ) , which are applicable unless @xmath17 or @xmath13 . * using these formulae , i have discussed the orbital evolution of satellites captured by a circumplanetary disk . high - inclination satellites are formed if the bodies are captured in approximately @xmath182 years before the disk dissipation . the author declares that he has no competing interests . i acknowledge the useful discussion with k. nakazawa , s. ida , h. emori , and h. tanaka to derive the analytic solutions . i thank the reviewers for their comments that improved this manuscript . i gratefully acknowledge support from grant - in - aid for scientific research ( b ) ( 26287101 ) . tanaka h , takeuchi t , ward w ( 2002 ) three - dimensional interaction between a planet and an isothermal gaseous disk . i. corotation and lindblad torques and planet migration . j. 565:12571274 .

planets are formed from collisional growth of small bodies in a protoplanetary disk . bodies much larger than approximately @xmath0 m are mainly controlled by the gravity of the host star and experience weak gas drag ; their orbits are mainly expressed by orbital elements : semimajor axes @xmath1 , eccentricities @xmath2 , and inclinations @xmath3 , which are modulated by gas drag . in a previous study , @xmath4 , @xmath5 , and @xmath6 were analytically derived for @xmath7 and @xmath8 , where @xmath9 is the scale height of the disk . their formulae are valid in the early stage of planet formation . however , once massive planets are formed , @xmath2 and @xmath3 increase greatly . indeed , some small bodies in the solar system have very large @xmath2 and @xmath3 . therefore , in this paper , i analytically derive formulae for @xmath4 , @xmath5 , and @xmath6 for @xmath10 and @xmath8 and for @xmath11 . the formulae combined from these limited equations will represent the results of orbital integration unless @xmath12 or @xmath13 . since the derived formulae are applicable for bodies not only in a protoplanetary disk but also in a circumplanetary disk , i discuss the possibility of the capture of satellites in a circumplanetary disk using the formulae .

the possibility of observing large cp violating asymmetries in the decay of @xmath5 mesons motivates the construction of high luminosity @xmath5 factories at several of the world s high energy physics laboratories . the theoretical and the experimental signatures of these asymmetries have been extensively discussed elsewhere@xcite,@xcite,@xcite , @xcite,@xcite . at asymmetric @xmath5 factories , it is possible to measure the time dependence of @xmath5 decays and therefore time dependent rate asymmetries of neutral @xmath5 decays due to @xmath6 mixing . the measurement of time dependent asymmetries in the exclusive modes @xmath7 and @xmath8 will allow the determination of the angles in the cabbibo - kobayashi - maskawa unitarity triangle . this type of cp violation has been studied extensively in the literature . another type of cp violation also exists in @xmath5 decays , direct cp violation in the @xmath5 decay amplitudes . this type of cp violation in @xmath5 decays has also been discussed by several authors although not as extensively . for charged @xmath5 decays calculation of the magnitudes of the effects for some exclusive modes and inclusive modes have been carried out@xcite,@xcite , @xcite,@xcite , @xcite,@xcite,@xcite . in contrast to asymmetries induced by @xmath9 mixing , the magnitudes have large hadronic uncertainties , especially for the exclusive modes . observation of these asymmetries can be used to rule out the superweak class of models@xcite . in this paper we describe several quasi - inclusive experimental signatures which could provide useful information on direct cp violation at the high luminosity facilities of the future . one of the goals is to increase the number of events available at experiments for observing a cp asymmetry . in particular we examine the inclusive decay of the neutral and the charged @xmath5 to either a charged @xmath10 or a charged @xmath11 meson . by applying the appropriate cut on the kaon ( or @xmath11 ) energy one can isolate a signal with little background from @xmath12 transitions . furthermore , these quasi - inclusive modes are expected to have less hadronic uncertainty than the exclusive modes , would have larger branching ratios and , compared to the purely inclusive modes they may have larger cp asymmetries . in this paper we will consider modes of the type @xmath13 that have the strange quark only in the @xmath14-meson . in the sections which follow , we describe the experimental signature and method . we then calculate the rates and asymmetries for inclusive @xmath15 and @xmath16 decays . in the @xmath18 center of mass frame , the momentum of the @xmath19 from quasi - two body @xmath5 decays such as @xmath20 may have momenta above the kinematic limit for @xmath19 mesons from @xmath21 transitions . this provides an experimental signature for @xmath17 , @xmath22 or @xmath23 decays where @xmath24 denotes a gluon . this kinematic separation between @xmath21 and @xmath17 transitions is illustrated by a generator level monte carlo simulation in figure 1 for the case of @xmath25 . ( the @xmath26 spectrum will be similiar ) . this experimental signature can be applied to the asymmetric energy @xmath5 factories if one boosts backwards along the z axis into the @xmath18 center of mass frame . since there is a large background ( `` continuum '' ) from the non - resonant processes @xmath27 where @xmath28 , experimental cuts on the event shape are also imposed . to provide additional continuum suppression , the `` b reconstruction '' technique has been employed . the requirement that the kaon and @xmath29 other pions form a system consistent in beam constrained mass and energy with a @xmath5 meson dramatically reduces the background . after these requirements are imposed , one searches for an excess in the kaon momentum spectrum above the @xmath21 region . only one combination per event is chosen . no effort is made to unfold the feed - across between submodes with different values of n. 3.4 truein methods similar to these have been successfully used by the cleo ii experiment to isolate a signal in the inclusive single photon energy spectrum and measure the branching fraction for inclusive @xmath30 transitions and to set upper limits on @xmath31 transitions@xcite,@xcite . it is clear from these studies that the @xmath5 reconstruction method provides adequate continuum background suppression . the decay modes that will be used here are listed below : 1 . @xmath32 2 . @xmath33 3 . @xmath34 4 . @xmath35 5 . @xmath36 6 . @xmath37 7 . @xmath38 8 . @xmath39 in case of multiple entries for a decay mode , we choose the best entry on the basis of a @xmath40 formed from the beam constrained mass and energy difference ( i.e. @xmath41 ) . in case of multiple decay modes per event , the best decay mode candidate is picked on the basis of the same @xmath40 . cross - feed between different @xmath42 decay modes ( i.e. the misclassification of decay modes ) provided the @xmath19 is correctly identified , is not a concern as the goal is to extract an inclusive signal . the purpose of the @xmath5 reconstruction method is to reduce continuum background . as the multiplicity of the decay mode increases , however , the probability of misrecontruction will increase . the signal is isolated as excess @xmath19 production in the high momentum signal region ( @xmath43 gev ) above continuum background . to reduce contamination from high momentum @xmath44 production and residual @xmath21 background , we assume the presence of a high momentum particle identification system as will be employed in the babar , belle , and cleo iii experiments . we propose to measure the asymmetry @xmath45 where @xmath46 originates from a partially reconstructed @xmath5 decay such as @xmath47 where the additional pions have net charge @xmath48 and @xmath49 and one neutral pion is allowed and @xmath50 gev . we assume that the contribution from @xmath51 decays has been removed by cutting on the @xmath52 region in @xmath4 mass . it is possible that the anomalously large rate from this source@xcite could dilute the asymmetry . 5.0 truein in the standard model ( sm ) the amplitudes for hadronic @xmath5 decays of the type @xmath53 are generated by the following effective hamiltonian @xcite : @xmath54 + h.c.\;,\end{aligned}\ ] ] where the superscript @xmath55 indicates the internal quark , @xmath56 can be @xmath57 or @xmath58 quark . @xmath59 can be either a @xmath60 or a @xmath61 quark depending on whether the decay is a @xmath62 or @xmath63 process . the operators @xmath64 are defined as @xmath65 where @xmath66 , and @xmath67 is summed over u , d , and s. @xmath68 are the tree level and qcd corrected operators . @xmath69 are the strong gluon induced penguin operators , and operators @xmath70 are due to @xmath71 and z exchange ( electroweak penguins ) , and `` box '' diagrams at loop level . the wilson coefficients @xmath72 are defined at the scale @xmath73 and have been evaluated to next - to - leading order in qcd . the @xmath74 are the regularization scheme independent values obtained in ref . we give the non - zero @xmath72 below for @xmath75 gev , @xmath76 , and @xmath77 gev , @xmath78 where @xmath79 is the number of color . the leading contributions to @xmath80 are given by : @xmath81 and @xmath82 . the function @xmath83 is given by @xmath84 all the above coefficients are obtained up to one loop order in electroweak interactions . the momentum @xmath59 is the momentum carried by the virtual gluon in the penguin diagram . when @xmath85 , @xmath83 becomes imaginary . in our calculation , we use @xmath86 mev , @xmath87 mev , @xmath88 mev , @xmath89 gev @xcite . we assume that the final state phases calculated at the quark level will be a good approximation to the sizes and the signs of the fsi phases at the hadronic level for quasi - inclusive decays when the final state particles are quite energetic as is the case for the @xmath5 decays in the kinematic range of experimental interest@xcite . we proceed to calculate the matrix elements of the form @xmath92 which represents the process @xmath93 and where @xmath94 has been described above . the effective hamiltonian consists of operators with a current @xmath95 current structure . pairs of such operators can be expressed in terms of color singlet and color octet structures which lead to color singlet and color octet matrix elements . in the factorization approximation , one separates out the currents in the operators by inserting the vacuum state and neglecting any qcd interactions between the two currents . the basis for this approximation is that , if the quark pair created by one of the currents carries large energy then it will not have significant qcd interactions . in this approximation the color octet matrix element does not contribute because it can not be expressed in a factorizable color singlet form . in our case , since the energy of the quark pairs that either creates the @xmath10 or the @xmath4 state is rather large , factorization is likely to be a good first approximation . to accommodate some deviation from this approximation we treat @xmath79 , the number of colors that enter in the calculation of the matrix elements , as a free parameter . in our calculation we will see how our results vary with different choices of @xmath79 . the value of @xmath96 is suggested by experimental data on low multiplicity hadronic @xmath5 decays@xcite . the amplitude for @xmath97 can in general be split into a three body and a two body part . detailed expressions for the matrix elements , decay distributions and asymmetries can be found in @xcite in this section we discuss the results of our calculations . we find that there can be significant asymmetries in @xmath98 decays especially in the region @xmath99 gev which is also the region where an experimental signal for such decays can be isolated . the branching ratios are of order @xmath100 which are within reach for future b factories . the contribution of the amplitude with the top quark in the loop accounts for 60 - 75% of the inclusive branching fraction . however , since the top quark amplitude is large and has no absorptive part in contrast to the c quark amplitude , the top quark contribution reduces the net cp asymmetry from 30 - 50% to about 10% . this calculation includes the contribution from electroweak penguins . we find that the electroweak penguin contributions increase the decay rates by 10 - 20% but reduce the overall asymmetry by 20 - 30% . the main sources of uncertainties in our calculation are discussed extensively in @xcite . 2.8 truein 2.8 truein the asymmetries are sensitive to the values of the wolfenstein parameters @xmath101 and @xmath102 . the existing constraints on the values of @xmath101 and @xmath103 come from measurements of @xmath104 , @xmath105 in the k system and @xmath106 . ( see ref . @xcite for a recent review ) . in our calculation we will use @xmath107 mev and choose @xmath108 . in fig . 3 we show the asymmetries for @xmath10 and @xmath11 in the final state in charged @xmath5 decays for different values of @xmath79 . variation of the asymmetries with the different inputs in our calculation are presented in detail in @xcite . in table . [ tb_integrated ] we give the branching fractions and the integrated asymmetries for the inclusive decays for different @xmath79 , @xmath109 ( @xmath59 is the gluon momentum in the two body part of the amplitude ) , @xmath110 mev , @xmath111 . for the charged @xmath5 decays we also show the decay rates and asymmetries for @xmath112 ( @xmath113 ) gev as that is the region of the signal . .integrated decay rates and asymmetries for @xmath114 decay [ cols="^,^,^",options="header " , ] [ tb_integrated ] the above figures show that there can be significant asymmetries in @xmath97 decays , especially in the region @xmath99 gev which is the region of experimental sensitivity for such decays . as already mentioned , our calculation is not free of theoretical uncertainties . two strong assumptions used in our calculation are the use of quark level strong phases for the fsi phases at the hadronic level and the choice of the value of the gluon momentum @xmath115 in the two body decays . other uncertainties from the use of different heavy to light form factors , the use of factorization , the model of the b meson wavefunction , the value of the charm quark mass and the choice of the renormalization scale @xmath116 have smaller effects on the asymmetries @xcite . we find significant direct cp violation in the inclusive decay @xmath117 and @xmath118 for @xmath119 gev . the branching fractions are in the @xmath120 range and the cp asymmetries may be sizeable . these asymmetries should be observable at future @xmath5 factories and could be used to rule out the superweak class of models . this work was supported in part by national science and engineering research council of canada ( a. datta ) . a. datta thanks the organisers of m.r.s.t for hospitality and an interesting conference . for a review see a. ali , hep - ph/9610333 and nucl .instrum . meth . * a 384 * , 8 ( 1996 ) ; m. gronau , technion - ph-96 - 39 , hep - ph/9609430 and nucl . instrum . meth . * a 384 * , 1 ( 1996 ) t.e . browder and k. honscheid , progress in nuclear and particle physics , vol . k. faessler , p. 81 - 220 ( 1995 ) . a. buras , hep - ph/9509329 and nucl . . meth . * a 368 * , 1 ( 1995 ) ; j. l. rosner , hep - ph/9506364 and proceedings of the 1995 rio de janeiro school on particles and fields , 116 ; a. ali and d. london , desy 95 - 148 , udem - gpp - th-95 - 32 , hep - ph/9508272 and nuovo cim . * 109 a * , 957 ( 1996 ) . r. fleischer , z. phys . * c 58 * 438 ; z. phys . * c 62 * , 81 ; g. kramer , w. f. palmer and h. simma , nucl . * b 428 * , 77 ( 1994 ) ; z. phys . * c 66 * , 429 ( 1994 ) ; n.g . deshpande and x .- g . he , phys . lett . * b 336 * , 471 ( 1994 ) . m. lautenbacher and p. weisz , nucl . * b 400 * , 37 ( 1993 ) ; a. buras , m. jamin and m. lautenbacher , ibid , 75 ( 1993 ) ; m. ciuchini , e. franco , g. martinelli and l. reina , nucl . phys . * b 415 * , 403 ( 1994 ) .

we consider the possibility of observing cp violation in quasi - inclusive decays of the type @xmath0 , @xmath1 , @xmath2 and @xmath3 , where @xmath4 does not contain strange quarks . we present estimates of rates and asymmetries for these decays in the standard model and comment on the experimental feasibility of observing cp violation in these decays at future @xmath5 factories . we find the rate asymmetries can be quite sizeable .

high precision measurements at the @xmath0 pole at lep combined with polarized forward backward asymmetries at slc and other measurements of electroweak observables at lower energies have been used to place stringent limits on new physics beyond the standard model @xcite . under the assumption that the dominant effects of the new physics would show up as corrections to the gauge boson self - energies , the lep measurements have been used to parameterize the possible new physics in terms of three observables @xmath4 , @xmath5 , @xmath6 @xcite ; or equivalently @xmath7 , @xmath8 , @xmath9 @xcite . the difference between the two parameterizations is the reference point which corresponds to the standard model predictions . a fourth observable corresponding to the partial width @xmath10 has been analyzed in terms of the parameter @xmath11 @xcite or @xmath12 @xcite . in view of the extraordinary agreement between the standard model predictions and the observations , it seems reasonable to assume that the @xmath13 gauge theory of electroweak interactions is essentially correct , and that the only sector of the theory lacking experimental support is the symmetry breaking sector . there are many extensions of the minimal standard model that incorporate different symmetry breaking possibilities . one large class of models is that in which the interactions responsible for the symmetry breaking are strongly coupled . for this class of models one expects that there will be no new particles with masses below 1 tev or so , and that their effects would show up in experiments as deviations from the minimal standard model couplings . in this paper we use the latest measurements of partial decay widths of the @xmath0 boson to place bounds on anomalous gauge boson couplings . our paper is organized as follows . in section 2 we discuss our formalism and the assumptions that go into the relations between the partial widths of the @xmath0 boson and the anomalous couplings . in section 3 we present our results . finally , in section 4 we discuss the difference between our calculation and others that can be found in the literature , and assess the significance of our results by comparing them to other existing limits . detailed analytical formulae for our results are relegated to an appendix . we assume that the electroweak interactions are given by an @xmath13 gauge theory with spontaneous symmetry breaking to @xmath14 , and that we do not have any information on the symmetry breaking sector except that it is strongly interacting and that any new particles have masses higher than several hundred gev . it is well known that this scenario can be described with an effective lagrangian with operators organized according to the number of derivatives or gauge fields they have . if we call @xmath15 the scale at which the symmetry breaking physics comes in , this organization of operators corresponds to an expansion of amplitudes in powers of @xmath16 . for energies @xmath17 this is just an expansion in powers of the coupling constant @xmath18 or @xmath19 , and for energies @xmath20 it becomes an energy expansion . the lowest order effective lagrangian for the symmetry breaking sector of the theory is @xcite : ^(2)=v^2 4tr . [ lagt ] in our notation @xmath21 and @xmath22 are the @xmath23 and @xmath24 gauge fields with @xmath25 .. ] the matrix @xmath26 , contains the would - be goldstone bosons @xmath27 that give the @xmath28 and @xmath0 their masses via the higgs mechanism and the @xmath13 covariant derivative is given by : d_= _ + i 2 g w_^i ^i - i 2g^b__3 . [ covd ] eq . [ lagt ] is thus the @xmath13 gauge invariant mass term for the @xmath28 and @xmath0 . the physical masses are obtained with @xmath29 gev . this non - renormalizable lagrangian is interpreted as an effective field theory , valid below some scale @xmath30 tev . the lowest order interactions between the gauge bosons and fermions , as well as the kinetic energy terms for all fields , are the same as those in the minimal standard model . for lep observables , the operators that can appear at tree - level are those that modify the gauge boson self - energies . to order @xmath31 there are only three @xcite : ^(2gb)=_1v^24 ( tr ) ^2 + _ 8g^2 ( tr)^2 + g g^v^2 ^2 l_10 tr , [ oblique ] which contribute respectively to @xmath5 , @xmath6 and @xmath4 . notice that for the two operators that break the custodial @xmath32 symmetry we have used the notation of ref . @xcite . in this paper we will consider operators that affect the @xmath0 partial widths at the one - loop level . we will restrict our study to only those operators that appear at order @xmath31 in the gauge - boson sector and that respect the custodial symmetry in the limit @xmath33 . they are : ^(4 ) & = & v^2 ^2 \ { l_1 ( tr)^2 + l_2 ( tr)^2 + & & - i g l_9l tr - i g^ l_9r tr } , [ lfour ] where the field strength tensors are given by : w_&=&1 2(_w_- _ w_+ i 2g[w _ , w _ ] ) + b_&=&12(_b_-_b _ ) _ 3 . [ fsten ] although this is not a complete list of all the operators that can arise at this order , we will be able to present a consistent picture in the sense that our calculation will not require additional counterterms to render the one - loop results finite . our choice of this subset of operators is motivated by the theoretical prejudice that violation of custodial symmetry must be `` small '' in some sense in the full theory @xcite . we want to restrict our attention to a small subset of all the operators that appear at this order because there are only a few observables that have been measured . the operators in eq . [ oblique ] and eq . [ lfour ] would arise when considering the effects of those in eq . [ lagt ] at the one - loop level , or from the new physics responsible for symmetry breaking at a scale @xmath15 at order @xmath34 . we , therefore , explicitly introduce the factor @xmath35 in our definition of @xmath36 so that the coefficients @xmath37 are naturally of @xmath38 . violating couplings @xmath39 and @xmath40 since we do not concern ourselves with them in this paper . they are simply used as counterterms for our one - loop calculation . ] the anomalous couplings that we consider would have tree - level effects on some observables that can be studied in future colliders . they have been studied at length in the literature @xcite . in the present paper we will compute their contribution to the @xmath0 partial widths that are measured at lep . these operators contribute to the @xmath0 partial widths at the one - loop level . since we are dealing with a non - renormalizable effective lagrangian , we will interpret our one - loop results in the usual way of an effective field theory . we will first perform a complete calculation to order @xmath41 . that is , we will include the one - loop contributions from the operator in eq . [ lagt ] ( and gauge boson kinetic energies ) . the divergences generated in this calculation are absorbed by renormalization of the couplings in eq . [ oblique ] . this calculation will illustrate our method , and as an example , we use it to place bounds on @xmath42 . we will then place bounds on the couplings of eq . [ lfour ] by considering their one - loop effects . the divergences generated in this one - loop calculation would be removed in general by renormalization of the couplings in the @xmath43 lagrangian of those operators that modify the gauge boson self - energies at tree - level ; and perhaps by additional renormalization of the couplings in eq . [ oblique ] . this would occur in a manner analogous to our @xmath31 calculation . interestingly , we find that we can obtain a completely finite result for the @xmath44 partial widths using only the operators in eq . [ oblique ] as counterterms . however , our interest is to place bounds on the couplings of eq . so we proceed as follows . we first regularize the integrals in @xmath45 space - time dimensions and remove all the poles in @xmath46 as well as the finite analytic terms by a suitable definition of the renormalized couplings . we then base our analysis on the leading non - analytic terms proportional to @xmath47 . these terms determine the running of the @xmath48 couplings and can not be generated by tree - level terms at that order . it has been argued in the literature @xcite , that with a carefully chosen renormalization scale @xmath49 ( in such a way that the logarithm is of order one ) , these terms give us the correct order of magnitude for the size of the @xmath48 coefficients . we thus choose some value for the renormalization scale between the @xmath0 mass and @xmath15 and require that this logarithmic contribution to the renormalized couplings falls in the experimentally allowed range . clearly , the lep observables do not measure the couplings in eq . [ lfour ] , and it is only from naturalness arguments like the one above , that we can place bounds on the anomalous gauge - boson couplings . from this perspective , it is clear that these bounds are not a substitute for direct measurements in future high energy machines . they should , however , give us an indication for the level at which we can expect something new to show up in those future machines . we will perform our calculations in unitary gauge , so we set @xmath50 in eqs . [ lagt ] , [ oblique ] and [ lfour ] . this results in interactions involving three , and four gauge boson couplings , some of which we present in appendix a. those coming from eq . [ lagt ] are equivalent to those in the minimal standard model with an infinitely heavy higgs boson , and those coming from eq . [ lfour ] correspond to the `` anomalous '' couplings . for the lowest order operators we use the conventional input parameters : @xmath51 as measured in muon decay ; the physical @xmath0 mass : @xmath52 ; and @xmath53 . other lowest order parameters are derived quantities and we adopt one of the usual definitions for the mixing angle : s_z^2 c_z^2 . [ szdef ] we neglect the mass and momentum of the external fermions compared to the @xmath0 mass . in particular , we do not include the @xmath54-quark mass since it would simply introduce corrections of order @xmath55 and our results are only order of magnitude estimates . the only fermion mass that is kept in our calculation is the mass of the top - quark when it appears as an intermediate state . with this formalism we proceed to compute the @xmath56 partial width from the following ingredients . * the @xmath57 vertex , which we write as : i _ = -ie4 s_z c_z _ [ vertex ] where @xmath58 and @xmath59 . the terms @xmath60 and @xmath61 occur at one - loop both at order @xmath34 and at order @xmath48 and are given in appendix b. * the renormalization of the lowest order input parameters . at order @xmath34 it is induced by tree - level anomalous couplings and one - loop diagrams with lowest order vertices . at order @xmath48 it is induced by one - loop diagrams with an anomalous coupling in one vertex . we present analytic formulae for the self - energies , vertex corrections and boxes in appendix b. the changes induced in the lowest order input parameters are : & = & a_(q^2)q^2 _ q^2=0 + m_z^2m_z^2&=&a_zz(m_z^2)m_z^2 + g_fg_f & = & 2 _ w e _ 0 -a_ww(0)m_w^2+(z_f-1)+b_box [ shifts ] the self - energies @xmath62 receive tree - level contributions from the operators with @xmath42 , @xmath39 and @xmath40 . they also receive one - loop contributions from the lowest order lagrangian eq . [ lagt ] and from the operators with @xmath63 , @xmath64 , @xmath65 and @xmath66 . the effective @xmath67 vertex , @xmath68 , receives one - loop contributions from all the operators in eq . [ lfour ] . the fermion wave function renormalization factors @xmath69 and the box contribution to @xmath70 , @xmath71 , are due only to one - loop effects from the lowest order effective lagrangian and are thus independent of the anomalous couplings . notice that @xmath71 enters the renormalization of @xmath51 because we work in unitary gauge where box diagrams also contain divergences . * tree - level and one - loop contributions to @xmath72 mixing . instead of diagonalizing the neutral gauge boson sector , we include this mixing as an additional contribution to @xmath60 and @xmath61 in eq . [ vertex ] : l_f^ = r_f^ = - c_z s_z r_f a_z(m_z^2 ) m_z^2 [ mix ] * wave function renormalization . for the external fermions we include it as additional contributions to @xmath60 and @xmath61 as shown in appendix b. for the @xmath0 we include it explicitly . with all these ingredients we can collect the results from appendix b into our final expression for the physical partial width . we find : ( zf f ) & = & _ 0 z_z , [ full ] where @xmath73 is the lowest order tree level result , _ 0 ( zf f)= n_cf ( l_f^2 + r_f^2 ) g_f m_z^3 12 , [ widthl ] and @xmath74 is 3 for quarks and 1 for leptons . we write the contributions of the different anomalous couplings to the @xmath0 partial widths in the form : ( zf ) _ sm(zf ) ( 1 + _ f^l_i _ 0(zf ) ) . [ defw ] we use this form because we want to place bounds on the anomalous couplings by comparing the measured widths with the one - loop standard model prediction @xmath75 . using eq . [ defw ] we introduce additional terms proportional to products of standard model one - loop corrections and corrections due to anomalous couplings . these are small effects that do not affect our results . we will not attempt to obtain a global fit to the parameters in our formalism from all possible observables . instead we use the partial @xmath0 widths . we believe this approach to be adequate given the fact that the results rely on naturalness assumptions . specifically we consider the observables : _ e & = & 83.980.18 mev ref . @xcite + _ & = & 499.8 3.5 mev ref . @xcite + _ z & = & 2497.4 3.8 mev ref . @xcite + r_h & = & 20.7950.040 ref . @xcite + r_b & = & 0.22020.0020 ref . @xcite [ data ] the bounds on new physics are obtained by subtracting the standard model predictions at one - loop from the measured partial widths as in eq . we use the numbers of langacker @xcite which use the global best fit values for @xmath76 and @xmath77 with @xmath78 in the range @xmath79 gev . the first error is from the uncertainty in @xmath52 and @xmath80 , the second is from @xmath76 and @xmath78 , and the one in brackets is from the uncertainty in @xmath77 . _ e & = & 83.87 0.02 0.10 mev ref . @xcite + _ & = & 501.9 0.1 0.9 mev ref . @xcite + _ z & = & 2496 1 3 mev ref . @xcite + r_h & = & 20.782 0.006 0.004 ref . @xcite + _ bb^new & = & 0.022 0.011 ref . @xcite [ theory ] where @xmath81/\gamma(z \ra b\overline{b})^{(sm)}$ ] . we add all errors in quadrature . in this section we compute the corrections to the @xmath82 partial widths from the couplings of eq . [ lfour ] , and compare them to recent values measured at lep . we treat each coupling constant independently , and compute only its _ lowest _ order contribution to the decay widths . we first present the complete @xmath31 results . they illustrate our method and serve as a check of our calculation . we then look at the effect of the couplings @xmath83 which affect only the gauge - boson self - energies . we then study the more complicated case of the couplings @xmath84 . finally we isolate the non - universal effects proportional to @xmath85 . as explained in the previous section , we do not include in our analysis the operators that appear at @xmath31 that break the custodial symmetry . as long as one is interested in bounding the anomalous couplings one at a time , it is straightforward to include these operators . for example , we discussed the parity violating one in ref . @xcite . the operators in eq . [ oblique ] are the only ones that induce a tree - level correction to the gauge boson self - energies to order @xmath31 . this can be seen most easily by working in a physical basis in which the neutral gauge boson self - energies are diagonalized to order @xmath31 . this is accomplished with renormalizations described in the literature @xcite , and results in modifications to the @xmath86 and the @xmath87 couplings . this tree - level effect on the @xmath82 partial width is , of course , well known . it corresponds , _ at leading order _ , to the new physics contributions to @xmath88 or @xmath89 discussed in the literature @xcite . in this section we do not perform the diagonalization mentioned above , but rather work in the original basis for the fields . this will serve two purposes . it will allow us to present a complete @xmath31 calculation as an illustration of the method we use to bound the other couplings . also , because the gauge boson interactions that appear at this order have the same tensor structure as those induced by @xmath65 and @xmath66 , we will be able to carry out the calculation involving those two couplings simultaneously . in this way , even though the terms with @xmath84 are order @xmath48 , the calculation to order @xmath34 will serve as a check of our answer for @xmath84 . to recover the @xmath34 result we set @xmath90 ( and also @xmath91 but these terms are clearly different ) in the results of appendix b. as explained in section 2 , we have regularized our one - loop integrals in @xmath45 dimensions and isolated the ultraviolet poles @xmath92 . we find that we obtain a finite answer to order @xmath34 if we adopt the following renormalization scheme : , not for quantities like the self - energies . ] @xmath93pt theory for low energy strong interactions . ] l^r_10 ( ) & = & v^2 ^2 l_10 - 1 16 ^2 1 12 ( 1 + ) + _ 1^r ( ) & = & _ 1 - e^2 16 ^23 2 c^2_z ( 1 + ) . [ renorm ] we thus replace the bare parameters @xmath42 and @xmath39 with the scale dependent ones above . as a check of our answer , it is interesting to note that we would also obtain a finite answer by adding to the results of appendix b , the one - loop contributions to the self - energies obtained in unitary gauge in the minimal standard model with one higgs boson in the loop . equivalently , the expressions in eq . [ renorm ] correspond to the value of @xmath42 and @xmath39 at one - loop in the minimal standard model within our renormalization scheme . our result for @xmath42 at order @xmath34 is then : = e^2 c_z^2 s_z^2l^r_10 ( ) v^2 ^22r_f(l_f + r_f ) l_f^2+r_f^2c_z^2 s_z^2-c_z^2 [ shten ] once again we point out that , at this order , the contribution of @xmath42 to the lep observables occurs only through modifications to the self - energies that are proportional to @xmath94 . at this order it is therefore possible to identify the effect of @xmath42 with the oblique parameter @xmath4 or @xmath9 . if we were to compute the effects of @xmath42 at one - loop ( as we do for the @xmath84 ) , comparison with @xmath4 would not be appropriate . bounding @xmath42 from existing analyses of @xmath4 or @xmath9 is complicated by the fact that the same one - loop definitions must be used . for example , @xmath95 receives contributions from the standard model higgs boson that are usually included in the minimal standard model calculation . we will simply associate our definition of @xmath95 with _ new _ contributions to @xmath4 , beyond those coming from the minimal stardard model . at one loop with the precise definitions used to renormalize the standard model at one - loop . this does not matter for our present purpose . ] numerically we find the following @xmath96 confidence level bounds on @xmath42 when we take the scale @xmath97 tev : _ e & & -1.7 l^r_10(m_z)_new 3.3 + r_h & & -1.5 l^r_10(m_z)_new 2.0 + _ z & & -1.1 l^r_10(m_z)_new 1.5 [ tennum ] we can also bound the _ leading order _ effects of @xmath42 from altarelli s latest global fit @xmath98 @xcite . to do this , we subtract the standard model value obtained with @xmath99 gev and @xmath100 gev as read from fig . 8 in ref . we obtain the @xmath96 confidence level interval : -0.14 l^r_10(m_z)_new 0.86 [ sfalta ] we can also compare directly with the result of langacker @xmath101@xcite , to obtain @xmath96 confidence level limits : -0.46 l^r_10(m_z)_new 0.77 [ sflanga ] the results eqs.[sfalta ] , [ sflanga ] are better than our result eq . [ tennum ] because they correspond to global fits that include all observables . the couplings @xmath83 enter the one - loop calculation of the @xmath82 width through four gauge boson couplings as depicted schematically in figure 1 . our prescription calls for using only the leading non - analytic contribution to the process @xmath82 . this contribution can be extracted from the coefficient of the pole in @xmath46 . care must be taken to isolate the poles of ultraviolet origin ( which are the only ones that interest us ) from those of infrared origin that appear in intermediate steps of the calculation but that cancel as usual when one includes real emission processes as well . we thus use the results of appendix b with the replacement : = 2 4-n ( ^2 m_z^2 ) [ prescrip ] to compute the contributions to the partial widths using eq . [ full ] . since in unitary gauge @xmath83 modify only the four - gauge boson couplings at the one - loop level , they enter the calculation of the @xmath0 partial widths only through the self - energy corrections and eq . [ shifts ] . these operators induce a non - zero value for @xmath102 . for the observables we are discussing , this is the _ only _ effect of @xmath83 . we do not place bounds on them from global fits of the oblique parameter @xmath5 or @xmath7 , because we have not shown that this is the only effect of @xmath83 for the other observables that enter the global fits . it is curious to see that even though the operators with @xmath63 and @xmath64 violate the custodial @xmath32 symmetry _ only _ through the hypercharge coupling , their one - loop effect on the partial @xmath0 widths is equivalent to a @xmath103 contribution to @xmath104 , on the same footing as two - loop electroweak contributions to @xmath104 in the minimal standard model . the calculation to @xmath43 can be made finite with the following renormalization of @xmath39 : _ 1^r ( ) = _ 1 + 3 4 ^ 2 ( 1 + c_z^2)s_z^2 c_z^4 ( l_1 + 5 2 l_2)v^2 ^2 ( 1 + ) . [ shone ] using our prescription to bound the anomalous couplings , eq . [ prescrip ] , we obtain for the @xmath0 partial widths : = -3 2 ^ 2 ( 1 + c_z^2)s_z^2 c_z^4 ( l_1 + 5 2 l_2)v^2 ^2 ( ^2 m_z^2 ) ( 1 + 2 r_f ( l_f + r_f ) l_f^2 + r_f^2c_z^2 s_z^2 - c_z^2 ) . [ resonetwo ] using @xmath97 tev , and @xmath105 tev we find @xmath96 confidence level bounds : _ e & & -50 l_1 + 5 2l_2 26 + _ & & -28 l_1 + 5 2l_2 59 + r_h & & -190 l_1 + 5 2l_2 130 + _ z & & -36 l_1 + 5 2l_2 27 [ rhoana ] combined , they yield the result : -28 l_1 + 5 2 l_2 26 [ combineonetwo ] shown in figure 2 . as mentioned before , the effect of @xmath83 in other observables is very different from that of @xmath39 . it is only for the @xmath0 partial widths that we can make the @xmath106 calculation finite with eq . [ shone ] . the operators with @xmath65 and @xmath66 affect the @xmath0 partial widths through eqs . [ vertex ] , [ shifts ] , and [ mix ] . we find it convenient to carry out this calculation simultaneously with the one - loop effects of the lowest order effective lagrangian , eq . [ lagt ] , because the form of the three and four gauge boson vertices induced by these two couplings is the same as that arising from eq . [ lagt ] . this can be seen from eqs . [ conv ] , [ unot ] in appendix a. performing the calculation in this way , we obtain a result that contains terms of order @xmath34 ( those independent of @xmath84 ) , terms of order @xmath48 proportional to @xmath84 , and terms of order @xmath48 proportional to @xmath42 and @xmath39 . as mentioned before , we keep these terms together to check our answer by taking the limit @xmath107 . this also allows us to cast our answer in terms of @xmath108 , @xmath109 and @xmath110 which is convenient for comparison with other papers in the literature . it is amusing to note that the divergences generated by the operators @xmath84 in the one - loop ( order @xmath48 ) calculation of the @xmath111 widths can all be removed by the following renormalization of the couplings in eq . [ oblique ] ( in the @xmath112 limit ) : _ 1^r ( ) & = & _ 1 - e^296 s_z^4 c_z^4 v^2 ^2 ( 1 + ) + l^r_10()&= & l_10 - 196 s_z^2 c_z^2 ( 1 + ) [ amusing ] this proves our assertion that our calculation to order @xmath106 can be made finite by suitable renormalizations of the parameters in eq . [ oblique ] . however , we do not expect this result to be true in general . that is , we expect that a calculation of the one - loop contributions of the operators in eq . [ lfour ] to other observables will require counterterms of order @xmath48 . thus , eq . [ amusing ] does _ not _ mean that we can place bounds on @xmath84 from global fits to the parameters @xmath4 and @xmath5 . without performing a complete analysis of the effective lagrangian at order @xmath48 it is not possible to identify the renormalized parameters of eq . [ amusing ] with the ones corresponding to @xmath4 and @xmath5 that are used for global fits . combining all the results of appendix b into eq . [ full ] , and keeping only terms linear in @xmath84 , we find after using eq . [ renorm ] , and our prescription eq . [ prescrip ] : & = & ^2 24 1c_z^4 s_z^4v^2 ^2 ( ^2 m_z^2 ) + & & \ { + & & + 2(1 + 2r_f(l_f+r_f)l_f^2 + r_f^2 c_z^2 s_z^2-c_z^2 ) } + & & + ^2 12 ( ^2 m_z^2 ) 1 + 2c_z^2 c_z^4 s_z^4 ( l_b^2 + r_b^2 ) v^2 ^2 m_t^2 m_z^2_fb [ finalres ] the last term in eq . [ finalres ] corresponds to the non - universal corrections proportional to @xmath85 that are relevant only for the decay @xmath113 . using , as before , @xmath97 tev , @xmath105 tev we find @xmath96 confidence level bounds : _ e & & -92 l_9l+0.22l_9r 47 + _ & & -79 l_9l+1.02l_9r 170 + r_h & & -22 l_9l-0.17l_9r 16 + _ z & & -22 l_9l-0.04l_9r 17 [ rhonum ] we show these inequalities in figure 3 . if we bound one coupling at a time we can read from figure 3 that : -22 & l_9l & 16 + -77 & l_9r & 94 [ oneattime ] in a vector like model with @xmath114 we have the @xmath96 confidence level bound : -22<l_9l = l_9r < 18 . [ vectorbound ] we can relate our couplings of eq . [ lfour ] to the conventional @xmath108 , @xmath109 and @xmath110 by identifying our unitary gauge three gauge boson couplings with the conventional parameterization of ref . @xcite as we do in appendix a. however , we must emphasize that there is no unique correspondence between the two . our framework assumes , for example , @xmath115 gauge invariance and this results in specific relations between the three and four gauge boson couplings that are different from those of ref . @xcite which assumes only electromagnetic gauge invariance . furthermore , if one starts with the conventional parameterization of the three - gauge - boson coupling and imposes @xmath115 gauge invariance one does not generate any additional two - gauge - boson couplings . it is interesting to point out that within our formalism there are only two independent couplings that contribute to three - gauge - boson couplings ( @xmath84 ) but not to two - gauge - boson couplings ( as @xmath42 does ) . from this it follows that the equations for @xmath108 , @xmath109 and @xmath110 in terms of @xmath116 are not independent . in fact , within our framework we have : _ z = g_1^z + s_z^2 c_z^2 ( 1 - _ ) . [ depen ] the same result holds in the formalism of ref . @xcite . for the sake of comparison with the literature we translate the bounds on @xmath65 and @xmath66 into bounds on @xmath117 , @xmath118 and @xmath119 . for this exercise we set @xmath120 . we use @xmath121 to obtain the bound on @xmath117 . we then solve for @xmath65 and @xmath66 in terms of @xmath118 and @xmath119 , and bound each one of these assuming the other one is zero . we obtain the @xmath96 confidence level intervals : -0.08 < & g_1^z & < 0.1 + -0.3 < & _ z & < 0.3 + -0.3 < & _ & < 0.4 . [ ourdg ] similarly , if there is a non - zero @xmath42 , these couplings receive contributions from it . setting @xmath90 , we find from eq . [ tennum ] the bounds : @xmath122 , @xmath123 , and @xmath124 . these bounds are stronger by a factor of about 20 , just as the bounds on @xmath42 , eq . [ tennum ] are stronger by about a factor of 20 than the bounds on @xmath84 , eq . [ oneattime ] . however , these really are bounds on the oblique corrections introduced by @xmath42 ( which also contributes to three gauge boson couplings due to @xmath115 gauge invariance ) . it is perhaps more relevant to consider the couplings of operators without tree - level self - energy corrections . this results in eq . [ ourdg ] . as can be seen from eq . [ finalres ] , the @xmath125 partial width receives non - universal contributions proportional to @xmath85 . within our renormalization scheme , the effects that correspond to the minimal standard model do not occur . our result corresponds entirely to a new physics contribution of order @xmath48 proportional to @xmath84 . these effects have already been included to some extent in the previous section when we compared the hadronic and total widths of the @xmath0 boson with their experimental values . in this section we isolate the effect of the @xmath85 terms and concentrate on the @xmath125 width . keeping the leading non - analytic contribution , as usual , we find : -1 = ^2 121 + 2c_z^2 c_z^4 s_z^4 ( l_b^2 + r_b^2 ) v^2 ^2 m_t^2 m_z^2(^2 m_z^2 ) [ nonuni ] we use as before @xmath105 tev , and we neglect the contributions to the @xmath113 width that are not proportional to @xmath85 . we can then place bounds on the anomalous couplings by comparing with langacker s result @xmath126 for @xmath127 gev @xcite . bounding the couplings one at a time we find the @xmath96 confidence level intervals : -50 & l_9l & -4 + 90 & l_9r & 1200 . [ dbblim ] once again we find that there is much more sensitivity to @xmath65 than to @xmath66 . the fact that the @xmath128 vertex places asymmetric bounds on the couplings is , of course , due to the present inconsistency between the measured value and the minimal standard model result . clearly , the implication that the couplings @xmath84 have a definite sign can not be taken seriously . a better way to read eq . [ dbblim ] is thus : @xmath129 and @xmath130 . several studies that bound these `` anomalous couplings '' using the lep observables can be found in the literature . our present study differs from those in two ways : we have included bounds on some couplings that have not been previously considered , @xmath83 and we discuss the other couplings , @xmath84 within an @xmath131 gauge invariant formalism . we now discuss specific differences with some of the papers found in the literature . the authors of ref . @xcite obtain their bounds by regularizing the one - loop integrals in @xmath45 dimensions , isolating the poles in @xmath132 and identifying these with quadratic divergences . this differs from our approach where we keep only the ( finite ) terms proportional to the logarithm of the renormalization scale @xmath133 . to find bounds , the authors of ref . @xcite replace the poles in @xmath132 with factors of @xmath134 . we believe that this leads to the artificially tight constraints @xcite on the anomalous couplings quoted in ref . @xcite ( @xmath135 limits ) : @xmath136 and @xmath137 . we translate these into @xmath96 confidence level intervals : -1.0 & l_9r & 2.4 + -1.6 & l_9l & 4.2 [ vegas ] which are an order of magnitude tighter than our bounds . conceptually , we see the divergences as being absorbed by renormalization of other anomalous couplings . as shown in this paper , the calculation of the @xmath138 can be rendered finite at order @xmath139 by renormalization of @xmath39 and @xmath42 . thus , the bounds obtained by ref . @xcite , eq . [ vegas ] , are really bounds on @xmath39 and @xmath42 . they embody the naturalness assumption that all the coefficients that appear in the effective lagrangian at a given order are of the same size . our formalism effectively allows @xmath84 to be different from @xmath42 . the authors of ref . @xcite do not require that their effective lagrangian be @xmath115 gauge invariant , and instead they are satisfied with electromagnetic gauge invariance . at the technical level this means that we differ in the four gauge boson vertices associated with the anomalous couplings we study . it also means that we consider different operators . in terms of the conventional anomalous three gauge boson couplings , these authors quote @xmath140 results @xmath141 , @xmath142 , and @xmath143 . these constraints are tighter than what we obtain from the contribution of @xmath84 to @xmath117 , @xmath144 and @xmath145 , eq . [ ourdg ] ; they are weaker than what we obtain from the contribution of @xmath42 . the authors of ref . @xcite require their effective lagrangian to be @xmath13 gauge invariant , but they implement the symmetry breaking linearly , with a higgs boson field . the resulting power counting is thus different from ours , as are the anomalous coupling constants . their study would be appropriate for a scenario in which the symmetry breaking sector contains a relatively light higgs boson . their anomalous couplings would parameterize the effects of the new physics not directly attributable to the higgs particle . nevertheless , we can roughly compare our results to theirs by using their bounds for the heavy higgs case ( case ( d ) in figure 3 of ref . @xcite ) . to obtain their bounds they consider the case where their couplings @xmath146 and @xmath147 which corresponds to our @xmath120 , and @xmath114 . for @xmath148 gev they find the following @xmath96 confidence level interval @xmath149 , which we translate into : -7.8 l_9l = l_9r 18.8 [ zepp ] this compares well with our bound -22 l_9l = l_9r 18 [ uscomp ] finally , if we look _ only _ at those corrections that are proportional to @xmath150 and that would dominate in the @xmath151 limit , we find that they only occur in the @xmath113 vertex . this means that they can be studied in terms of the parameter @xmath12 of ref . @xcite or @xmath11 of ref . @xcite . converting our result of eq . [ dbblim ] to the usual anomalous couplings and recalling that only two of them are independent at this order , we find , for example : -0.28 & g_1^z & -0.03 + 0.6 & _ & 5.2 [ compdbb ] this result is very similar to that obtained in ref . @xcite . we now compare our results is different from that of ref . we have translated their results into our notation . ] with bounds that future colliders are expected to place on the anomalous couplings . in fig . 4 , we compare our @xmath152 confidence level bounds on @xmath65 and @xmath66 with those which can be obtained at lepii with @xmath153 gev and an integrated luminosity of @xmath154 @xcite . we find that lep and lepii are sensitive to slightly different regions of the @xmath65 and @xmath66 parameter space , with the bounds from the two machines being of the same order of magnitude . the authors of ref . @xcite find that the lhc would place bounds of order @xmath155 and a factor of two or three worse for @xmath66 . we find , eq . [ oneattime ] , that precision lep measurements already provide constraints at that level . we again emphasize our caveat that the bounds from lep rely on naturalness arguments and are no substitute for measurements in future colliders . the limits presented here on the four point couplings @xmath156 and @xmath157 are the first available for these couplings . they will be measured directly at the lhc . assuming a coupling is observable if it induces a @xmath158 change in the high momentum integrated cross section , ref . @xcite estimated that the lhc will be sensitive to @xmath159 , which is considerably stronger that the bound obtained from the @xmath0 partial widths . we have used an effective field theory formalism to place bounds on some non - standard model gauge boson couplings . we have assumed that the electroweak interactions are an @xmath13 gauge theory with an unknown , but strongly interacting , scalar sector responsible for spontaneous symmetry breaking . computing the leading contribution of each operator , and allowing only one non - zero coefficient at a time , our @xmath160 confidence level bounds are : -1.1 < & l_10^r(m_z)_new & < 1.5 + -28 < & l_1 & < 26 + -11 < & l_2 & < 11 + -22 < & l_9l & < 16 + -77 < & l_9r & < 94 . [ rescon ] two parameter bounds on ( @xmath161 ) and ( @xmath162 ) are given in the text . the bounds on @xmath161 are the first experimental bounds on these couplings . the bounds on @xmath65 and @xmath66 are of the same order of magnitude as those which will be obtained at lepii and the lhc . * acknowledgements * the work of g. v. was supported in part by a doe oji award . thanks the theory group at bnl for their hospitality while part of this work was performed . we are grateful to w. bardeen , j. f. donoghue , e. laenen , w. marciano , a. sopczak , and a. sirlin for useful discussions . we thank p. langacker for providing us with his latest numbers . we thank f. boudjema for providing us with the data file for the lepii bounds in figure 4 . it has become conventional in the literature to parameterize the three gauge boson vertex @xmath163 ( where @xmath164 ) in the following way @xcite : _ wwv&= & -ie c_zs_z g_1^z ( w_^ w^-w _ w^ ) z^-ie g_1^ ( w_^ w^-w _ w^ ) a^ + & & -ie c_zs_z _ z w_^ w_z^ -ie _ w_^ w_a^ + & & -e c_z s_z g_5^z ^ ( w_^-_w_^+-w_^+_w_^-)z_. [ conv ] terms of the form @xmath165 which are often included in the parameterization of the three gauge boson vertex do not appear in our formalism to the order we work . for calculations to order @xmath34 , it is most convenient to diagonalize the gauge - boson self - energies as done in ref . @xcite . this results in expressions for @xmath108 , @xmath110 and @xmath109 in terms of @xmath166 that we presented in ref . @xcite . for the present study , we do not keep the @xmath42 or @xmath39 terms as explained in the text . we thus use : g_1^z&=&1+e^22 c_z^2 s_z^2 l_9lv^2 ^ 2 + g_1^&= & 1 + _ z&=&1 + e^22 s_z^2c_z^2 ( l_9lc_z^2 -l_9rs_z^2)v^2 ^ 2 + _ & = & 1+e^2 2s_z^2 ( l_9l+l_9r)v^2 ^ 2 . [ unot ] the four gauge boson interactions derived from eqs . [ lagt ] and [ lfour ] after diagonalization of the gauge boson self - energies can be written as : _ wwv_i v_j&= & c_ij ( 2 w^+w^- v_iv_j -v_i w^+ v_jw^- -v_jw^+ v_iw^- ) + & & + e^4 s^4_z v^2 ^ 2[afbg ] where @xmath167 or @xmath168 and , c_&= & -e^2 + c_zz & = & -e^2 c_z^2s_z^2(g_1^z)^2 + c_z&=&-e^2c_zs_zg_1^z + c_ww&=&-e^2s_z^2(1 + 2 c_z^2(g_1^z-1 ) ) . [ fgbco ] as explained in the text , we will only consider the tree - level effects of @xmath42 . this means that for the one - loop calculation to order @xmath48 only @xmath84 appear in eq . [ unot ] . for the calculation to order @xmath34 presented in this paper , we do not use the diagonal basis , but rather obtain our results from the explicit factors of @xmath42 and @xmath39 that appear in the following expressions . the vector boson self energies can be written in the form : -i _ vv^(p^2)= a_vv(p^2)g^+ b(p^2)p^p^. [ defse ] we regularize in @xmath45 dimensions and keep only the poles of ultraviolet origin . for the case of fermion loops we treat all fermions as massless except the top - quark . we find : & = & -4 \ { p^23 m_w^2 - 1+_^212p^2m_w^4(p^2 - 2m_w^2 ) + _ m_w^2(p^2 - 6m_w^2 ) } + & & -12s_z^4_f n_cfr_f^2 + 8 l_10 + a_z(p^2)p^2&=&4 ( c_zs_z ) \{g_1^z(1-p^23 m_w^2)-_z_p^2 12 m_w^4(p^2 - 2m_w^2 ) + & -&(g_1^z_+_z)2 m_w^2 ( p^2 - 6 m_w^2 ) } + & & + 24 s_z^3 c_z _ f n_cf r_f ( r_f+l_f ) + 4l_10 + a_z z(p^2)p^2&=&4 ( c_zs_z)^2 \{g_1^z 2(1-p^23 m_w^2)-_z^2 p^2 12 m_w^4(p^2 - 2m_w^2)-g_1^z_zm_w^2 ( p^2 - 6 m_w^2 ) } + & & -8 s_z^2 c_z^2 -2m_z^2 p^2_1 - 8l_10 [ avv ] these results can be compared with the unitary gauge results of degrassi and sirlin in the standard model limit ( @xmath169 ) . when the contribution of the standard model higgs boson is included , eq . [ avv ] agrees with ref . @xcite for the renormalization of @xmath51 , we need @xmath170 , the @xmath67 vertex evaluated at @xmath171 , @xmath172 , the box contribution to @xmath173 , @xmath71 , and the charged lepton wavefunction renormalization , @xmath69 : a_ww(0)&= & 3 16 m_w^2 \ { _ ^2 + 2 + 2 _ -3s_z^2 ( 1 - 2 c_z^2 ) + & & + ( 1s_z^2 ) } + 38 s_z^2 m_t^2 + _ w e ( 0 ) & = & a_0 3 8 \ { ( 1 + 12 _ ) + c_z^22s_z^2 } + a_0&= & -g2 e^(1-_5 ) ^w _ + b_box&=&3 16 ( c_z^2s_z^2 ) ( 1+c_z^2)+4 + z_f-1 & = & -r_f^216 s_z^4 [ intermediate ] for massless fermions in dimensional regularization there is a cancellation between the ultraviolet and infrared divergent contributions , responsible for the familiar result that their wavefunction renormalization vanishes . we are isolating the ultraviolet divergences only , so we obtain a contribution to the fermion wavefunction renormalization . the corrections to the @xmath174 vertex from the diagrams shown in fig . 5 ( including in this term the wave function renormalization for the external fermion ) are : l_f & = & ( l_f - r_f)4 ( c_zs_z)^2 \{_zp^2m_w^2 ( 12 + p^212 m_w^2 ) + g_1^z ( 5p^26 m_w^2 ) } + & & + 16 s_z^2 m_t^2m_w^2_fb [ moreinter ] when the wavefunction renormalization is included in the definition of @xmath61 , we have @xmath175 from the diagrams of fig . the @xmath0 wavefunction renormalization is given by : z_z-1&=&-8 s_z^2 c_z^2_f n_cf 3 ( r_f ^2 + l_f^2 ) -8l_10 + & & + 4 ( c_zs_z)^2 \{g_1^z 2(1 - 23 c_z^2)-_z^2 12 ( 3c_z^4 - 4c_z^2 ) -g_1^z_z ( 2c_z^2 - 6 ) } [ wavefunct ]

we place bounds on anomalous gauge boson couplings from lep data with particular emphasis on those couplings which do not contribute to @xmath0 decays at tree level . we use an effective field theory formalism to compute the one - loop corrections to the @xmath1 decay widths resulting from non - standard model three and four gauge boson vertices . we find that the precise measurements at lep constrain the three gauge boson couplings at a level comparable to that obtainable at lepii and lhc . # 1#2#3_phys . rev . _ * d#1 * # 2 ( 19#3 ) # 1#2#3_phys . lett . _ * # 1b * # 2 ( 19#3 ) # 1#2#3_nucl . phys . _ * b#1 * # 2 ( 19#3 ) # 1#2#3_phys . rev . lett . _ * # 1 * # 2 ( 19#3 ) 0.0 in 0.0 in 6.0 in 8.75 in -1.0 in .5 in = -0.5 in bnl-60949 october , 1994 * bounds on anomalous gauge boson couplings from partial @xmath0 widths at lep * + * and g. valencia@xmath2 * + _ @xmath3 physics department , brookhaven national laboratory , upton , ny 11973 _ + _ @xmath2 department of physics , iowa state university , ames ia 50011 _ +

galaxy - galaxy interactions lead to the redistribution of stars and gas about each system , and an infusion of material into the local intergalactic medium , promoting star formation ( @xcite ) . tidal tails are signatures of galactic mergers ( @xcite ) , illuminated by the ignition of star formation ( @xcite ) . turbulent energy injected into the local hi through these mergers compresses the gas , forming new stars ( @xcite ) ; this is observationally shown in @xcite , who were able to link the presence of star clusters candidates ( sccs ) to turbulent regions of hi . while the young , in - situ formed sccs of tidal tails have been studied in the past through imaging and spectroscopy ( e.g. @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ) , the composition of the underlying stellar material remains a mystery . we know that gas is easily extracted from the parent galaxies during interactions ( e.g. @xcite ) , but whether or not stars follow suit has not yet been established . the extracted gas can collapse to form stars in a clustered manner , yet at the same time , simulations have shown that clusters in tails can be easily disrupted ( @xcite ) . this sparks an obvious question : what are tidal tails made of ? what is the relative fraction of gas and old stars within the interaction ejecta ? the answer can inform dynamical simulations of interactions , and therefore help refine our understanding of the enrichment of the intergalactic medium . in fact , current dynamical simulations ( e.g. @xcite ) use an older stellar component as a gravitational ` anchor ' for the gas . is this requirement justified by the observations ? existing studies have focused on the young stellar components of tidal tails , and therefore do not have the capability to probe deep into these regions . we have developed a new observing program designed to do just this , using deep , photometric _ ugriz _ imaging . to derive an accurate age estimate for a tidal tail , we plan our exposure times to view across the stellar sequence in our diffuse tidal tail light . these tails are imaged in each filter to an adequate signal to noise ratio , allowing us to derive an average colour and age of their diffuse light . this has the additional benefit of allowing us to age date star clusters within the tail . we choose the twin tidal tails of ngc 3256 as a case study for our method . the system is relatively nearby at a distance of 38 mpc ( @xcite ) , with an interaction age of 400 myr ( @xcite ) - not so young that our observations will be drowned out by ob stars , yet young enough so that the tidal structure is still visible . it has been well studied in the past through spectroscopic ( @xcite ; @xcite ; @xcite ) , photometric ( @xcite ; @xcite ; @xcite ) and hi ( @xcite ) observations , giving us benchmarks to compare to . we will begin in section [ sec:2 ] by describing our data and our reduction process . in section [ sec:3 ] we describe our analysis methods , and in section [ sec:4 ] we show our results . in section [ sec:5 ] we discuss our findings , and conclude our paper with the main points of our research in section [ sec:6 ] . images were obtained from the gemini - south observatory from march 2013 to june 2013 , using the gmos imager . the gmos field of view is superimposed on an optical image from the digital sky survey in figure [ fig : gmos_fov ] , measuring @xmath10 arcmin . exposure details are shown in table [ table : exposures ] . [ cols="^,^,^,^,^ " , ] a colour - colour diagram with only these sccs is shown in figures [ fig : w_scc_colour ] and [ fig : e_scc_colour ] . overlaid in gray are data points for the diffuse light of the respective tidal tail . the ssp models from @xcite do not include nebular emission . however , the nebular continuum , as well as emission lines from h@xmath11 , h@xmath12 , [ o iii ] and [ o ii ] , can have strong effects on our colours for young objects , with ages < 10 myr . we use ` starburst99 ` ( @xcite ) to include a nebular continuum , as well as emission from h@xmath11 and h@xmath12 . following @xcite and @xcite , we find the strengths of the [ o iii ] and [ o ii ] lines from the kiss sample of nearby low - mass star - forming galaxies ( e.g. , @xcite ) . we use the median ratios of [ o iii]/h@xmath12 and [ o ii]/h@xmath12 , listed at 0.08 and 0.56 , respectively . h@xmath11 emission falls in the _ r _ filter , causing the @xmath13 colour to appear bluer . both h@xmath12 and [ o iii ] fall in the _ g_-band , while [ o ii ] is in the _ u_-band . depending on the relative strengths of the oxygen lines , this can cause the @xmath14 colour to become bluer or redder . to show a possible range of nebular emission , we also include a track setting [ o iii]/h@xmath12 to its 90th percentile value ( 0.66 ) and [ o ii]/h@xmath12 to its 10th percentile value ( 0.22 ) . this provides a better fit for several data points in the eastern tail , although the position of these clusters on the diagram is likely due to a combination of varying emission line strengths and dust extinction . @xmath15 and @xmath16 . gray points correspond to diffuse light . for comparison , we show colours of objects outside the tail region on the right . nebular tracks with emission from h@xmath12 , h@xmath11 , [ o iii ] , [ o ii ] , and continuum emission are included . dotted lines indicate median values of [ o iii]/h@xmath12 and [ o ii]/h@xmath12 emission , while dashed lines indicate the 90th and 10th percentile of [ o iii]/h@xmath12 and [ o ii]/h@xmath12 emission , respectively , from the kiss galaxy sample . 31 sccs are detected in the western tail , with a median age of 8.15 log yrs . ] but for the eastern tail . several young objects show strong nebular emission . 19 sccs are detected in the eastern tail , with a median age of 7.96 log yrs . ] in examining figure [ fig : w_scc_colour ] , we see a clear overdensity of sccs in the western tail compared to objects outside of the tail . in the eastern tail ( [ fig : e_scc_colour ] ) , a number of sccs exist at blue values of @xmath14 and @xmath13 which are distinct from those outside the tail . sccs in the western tail overlap with the diffuse light ( though skewed to the blue end ) , while those in the eastern tail are distinct from the diffuse light in the host tail . to quantitatively test this , we apply the kolmogorov smirnov ( ks ) test to the colour distribution of the diffuse light and sccs . this test allows us to determine the probability that two samples were drawn from independent distributions . p_-value of less than 0.013 indicates the populations are distinct from one another at more than a 2.5@xmath17 confidence level . for the @xmath14 colours , we find _ p_-values of 0.028 and @xmath18 for the western and eastern tails , respectively . for the @xmath13 colours , we have _ p _ values of @xmath19 and @xmath20 , for the western and eastern tails . these results show the diffuse light and scc colour distributions are distinct from each other in both tails . ages and masses of our sources are found using the 3def method ( @xcite ) . this uses a maximum likelihood estimator to find the ages and masses of each cluster , by adjusting the colour excess @xmath21 to fit the observed colour in each filter to a given ssp evolutionary model . results for the total masses and median ages of sccs are shown in table [ table : mass_all ] , with masses and ages shown in figure [ fig : mass_age ] ; dashed lines represent our scc detection limits based on our colour and magnitude criteria from section [ sec : cluster_colours ] . error bars show the maximum and minimum age and mass for each data point . at low masses , there are several points which do not fall within the detection band . these objects are subject to internal extinction , which dims them to fall within our scc colour and magnitude cutoffs . there is a gap in age between the young objects in the eastern tail and the main distribution with ages of 8.0 - 8.8 log yrs . this indicates a recent small burst of isolated star formation in the eastern tail , small enough that only low mass clusters are present . other similar bursts could have occurred in either tail between 7.0 and 8.0 log yrs , but would have faded from view by the present . it is clear , however that there is not continuous star formation at the level of that seen during the main interaction period , 8.0 - 8.8 log yrs ago . spatial maps of tail colours and sccs are shown in figure [ fig : colour_tail_sccs ] for both tails . measurement boxes from figures [ fig : boxes ] are colour coded to indicate their @xmath14 colours . scc positions and @xmath14 values are added in a similar manner . the difference in age between the tails is evident in the abundance of younger regions , represented by cyan and yellow , in the western tail , compared to older regions , represented by orange and red , in the eastern tail . the spatial location of sccs does not appear to influence the colour of the diffuse light . this is not too surprising , as these objects were masked out in section [ sec : masking ] . both of the tails show colour gradients across their lengths , with bluer colours near the galactic bulges and redder colours at the far tips . likewise , the edges of the tails appear redder than their interiors . such gradients were also seen among several tails in @xcite . distributions of @xmath14 colours for diffuse light and sccs are shown in figure [ fig : colour_hist ] . colour distribution for sccs and diffuse light in ngc 3256w ( dark green ) and e ( purple ) , with bin size of 0.1 . hashed histograms indicate diffuse light measurements , while solid histograms represent the sccs colour distributions for the separate diffuse structure in the western tail are shown in the inset graph in orange , plotted alongside those of the western tail itself ( dark green ) . the diffuse light colours between the two tails are clearly distinct from one another . the distribution of scc colour in the western tail overlaps its diffuse light colour distribution , suggesting a common origin . sccs in the eastern tail are much younger than any of the sccs in the western tail . ] we can summarize figure [ fig : colour_hist ] with the following points : 1 . the difference in @xmath14 colour between the eastern and western tail shows the total diffuse stellar light of the two tails are separated in age by over 500 myr , with a significantly larger contribution from a young population in the western tail . 2 . assuming the two tails were formed at the same time , we find the stellar masses of the eastern and western tails to be dominated by old ( @xmath2210 gyr ) populations drawn from the parent galaxies . 3 . while the western tail contains a significant old ( @xmath2210 gyr ) population , the stellar light is largely comprised of a population which formed soon after the interaction , with an age of 8.29 log yrs . the similarity of the @xmath14 colour distribution of sccs in the western tail compared to the diffuse light gives credence to the idea that the western tail is comprised of disrupted star clusters formed shortly after the formation of the tail , with the ones we see today being the ones that survived . the stark contrast in @xmath14 colours for sccs in the eastern tail as compared to the diffuse light colours suggests some very recent star formation as compared to the age of the tail . stellar populations in the two tails of ngc 3256 have different compositions . we base our conclusions on the assumption that the tails were formed at the same time . it s possible to interpret figure [ fig : lsbphot ] as due to the eastern tail having formed first , with the stellar light we observe being formed from a burst of star formation caused by that earlier interaction . however , prograde collisions between galaxies are required to form tidal tails ( @xcite ; @xcite ) , as in the case of ngc 3256 . in such a case , these tails form simultaneously , which indicates a common age between ngc 3256 s tails . @xcite found three nuclei within the centre of ngc 3256 , leading them to construct a merger scenario involving two separate mergers and three galaxies . the eastern tail could be explained to have formed during the first merger , while the western tail formed later with a prograde interaction between the merger remnant and the third galaxy . however , @xcite suggests a more likely scenario with the two major galaxies initially merging , and a smaller satellite galaxy merging after the major interaction , supported in further work by @xcite . additionally , @xcite only observed two broad hi tails , leading them to conclude that ngc 3256 is likely created by a two galaxy prograde merger . given these works , it is very probable that the two tails were formed at the same time . we also find possible evidence for the small satellite merger in the separate diffuse structure , seen in the western tail . this structure is younger than either tail , supporting the theory of a late minor merger , occurring after the major merger . the diffuse light in the western tail is dominated by a younger population , formed soon after the formation of the tail . this population is also present in the eastern tail , although at a lower concentration . the source of the young population is uncertain , as stars may form in a variety of methods : in bound clusters , unbound stellar associations , down to near individual stellar formation . the fraction of stars forming in clusters can be up to 70% in regions of large gas density ( @xcite ) . however , these clusters are subject to frequent tidal shocks which will preferentially destroy low mass clusters ( @xcite ) , dispersing their material into the diffuse tidal light and making clusters an ideal candidate for the source of our young population . the destruction of star clusters is modeled in two parts : number loss ( removal of stars in a cluster ) and mass loss ( removal of mass in a cluster ) . number loss can be understood as effects from stellar feedback , such as stellar winds and supernovae , which expel gas in a cluster , causing it to become unbound . mass loss will remove stars from a cluster via two - body interactions . number loss , also known as infant mortality , will only be important for the first @xmath2210 myr , as the hot and massive stars evolve and explode . the remnants of these clusters are thus seen as the diffuse light of the tail . a similar effect is seen in ngc 7714 ages of hii regions within the tidal tails have been shown to be older than star clusters residing within them , suggesting previously formed clusters have been dispersed and surround the newly conceived clusters ( @xcite ) . the coincidence of peaks in histograms of @xmath14 colour between the diffuse light of the western tail and its sccs ( figure [ fig : colour_hist ] ) strongly suggests the two are intertwined . results of ks tests show the populations are distinct from each other , however , this is likely due to the presence of an old stellar population in the diffuse light of the tail . the timescale between tail formation and cluster formation is separated by @xmath22200 myr . we compare this to simulations of ngc 4038/9 , a similar merger , which found a peak in star formation @xmath2225 myr after the interaction ( @xcite ) . individual masses of sccs in the western tail are on average more massive than in the eastern tail , as shown in figure [ fig : mass_age ] , although the eastern tail has several very young , low mass objects absent in the western tail . the greater number of sccs for the western tail can be attributed to its larger hi mass . this can be related back to the diffuse light as well , as the western tail s higher abundance of gas led it to form more star clusters which could be disrupted and dispersed in the tail . we compare the ages of our clusters to those found in the nuclei from previous studies . @xcite spectroscopically studied 23 star clusters inside the centres of the galaxies , finding an average age of @xmath2210 myr . @xcite performed photometry on several hundred objects within the centre finding bright and blue objects . our sccs are substantially older than those found within the nuclei , suggesting the star formation in the tails was cut off earlier relative to the interior . this is consistent with spectroscopic observations of star clusters in the western tail by @xcite and photometric analysis of sccs in the eastern tail by @xcite , both of whom found ages of clusters to be older than those in the interior . a similar viewpoint is shared in the antennae simulations , which show a cessation of star formation in the tails , while star formation in the interiors is ongoing ( @xcite ) . star formation in the interiors can be stimulated as material previously thrown out during the initial encounters falls back into the centre of the potential well . of particular interest are the very young objects found in the eastern tails , with @xmath23 . these objects indicate relatively recent star formation in a tail whose diffuse light suggests a limited star formation history . it is possible these objects are now being formed as material falls back through the tail to the interiors , creating turbulence in the hi gas and sparking small bursts of star formation . we do not see these very young objects in the western tail . however , note that they are low mass , with masses @xmath24 . they will fade from detectability as they age ( see figure [ fig : mass_age ] ) ; additionally , they may subject to tidal shocks from regions of dense gas which can disrupt them . this suggests that there may be small star formation events periodically in both tails , but the evidence of these events will rapidly fade away . @xcite found that tidal tails with large hi line of sight velocity dispersion @xmath25 and high hi column densities were ideal locations for sccs . both the 3256 tails fit these criteron , and the majority of their sccs , as determined by @xcite using _ hst _ wfpc2 data , lie in hi pixels with these characteristics . however , the tails can be distinguished by measurements of shear ( @xmath26 ) , with the western tail s shear about one - third that of the eastern s . @xcite found that sccs were preferentially located in regions of low shear , which is suggestive that these areas can be ideal for star formation . we reproduce spatial maps of hi shear from figures 4.18 and 4.19 in @xcite , and compare them to our spatial @xmath14 maps in figure [ fig : e_shear ] for the eastern tail and figure [ fig : w_shear ] for the western tail . from figure [ fig : e_shear ] , two pockets of low shear are visible in yellow . the first of these , closest to the bulge , corresponds to the location of the youngest sccs in the eastern tail , visualized as beige circles . the second low shear region does not contain any sccs , although the diffuse light near this area is younger than its surroundings . the western tail has a ridge of low shear seen in figure [ fig : w_shear ] , running from the middle of the tail to the tip . we find the majority of sccs within the hi field of view reside in this region , with few ( @xmath223 out of 23 ) outside . the diffuse light appears to follow the shear as well . at the tip of the tail , the low hi shear seen in blue matches the yellow diffuse light boxes . as the shear increases to yellow and orange , the diffuse light reddens and the boxes shift to orange and red . the effect of dust on the diffuse light in these tails is similar to that of an older population ; in either case the colours are reddened . to investigate this effect , we redden the eastern tail colour to match the western s , by minimizing the distance between the two values . we find the amount of extinction needed for this is @xmath27 , which we find unrealistic , as tidal tails are regions of relatively low extinction ( @xcite ; @xcite ) . additionally , examination of archival _ galex _ nuv data ( @xcite ) show similar brightnesses for the two tails , suggesting similar levels of low extinction . extinction in the eastern tail would dim the _ galex _ data , with respect to the western tail , but this is not seen . @xcite examined the @xmath28 colour distribution within the eastern tail , finding negligible levels of reddening , supporting the lack of dust within the tails . we finally look at extinction in the three star clusters spectroscopically studied in the western tail from @xcite , which are listed at 0.0 , 0.3 , 0.5 @xmath29 mag , showing minimal extinction . additionally , as mentioned in section [ sec : cluster_mass ] , several of our young sccs are subject to internal extinction . extinction values of these objects range from 1.1 to 0.7 @xmath29 mag . dust can be expected to associate with star clusters , as dust is needed for star formation , so we expect to see the most extinction in these regions . however , such levels of extinction exist for a small handful of objects , and as these are masked and do not factor into our lsb measurements , we are lead to believe extinction does not significantly affect our analysis of the stellar age and mass distribution . our observational program has proven successful in allowing us to characterize the stellar populations of our tidal tails . we find ngc 3256w to be bluer than its twin tail , ngc 3256e . measured colours indicate that diffuse light in the western tail has a large contribution from a young population formed after the interaction , perhaps from dispersed star clusters , as compared to the eastern tail , which is primarily illuminated by an old population derived from the host galaxy . both tails exhibit colour gradients along their lengths , suggesting a gradient in the time scale of star formation . despite these colour differences , both tails appear to be dominated in mass by an old , underlying population , originating from the interacting galaxies . analysis of sccs shows a lack of old objects in either tail ( > @xmath30 yr ) , but a clustering of objects below 400 myr in the western tail and eastern tail . the eastern tail shows an interesting clustering of young objects , with ages < @xmath31 yr . these objects are low mass structures and are not likely to be detected as they age , disappearing as they fade beyond our detection limits , or are dispersed into the tail . the @xmath14 colour distribution of the western sccs is proven to be distinct from the diffuse stellar light in the western tail through ks probability tests , however the peaks of the colour distributions of the diffuse light and sccs match well , suggesting these objects and regions are intertwined . the fact that the ks test shows separate populations can be explained by the addition of an old , underlying stellar population to the diffuse light . ngc 3256 has been shown in past studies to contain a large number of sccs compared to other systems ( @xcite ; @xcite ) . we plan to apply our current observational program to additional tidal tail systems with varying amounts of sccs and hi properties , particularly those with low numbers of sccs . the presence of a tidal dwarf galaxy may also play a role in determining the composition of tidal tail diffuse light . _ galex _ observations of the antennae galaxy reveal a gradient in colour along the tail , with bluer colours near the tip of the southern tail , where two tidal dwarf galaxies reside ( @xcite ) . @xcite found that tails with tidal dwarfs did not contain as many sccs as those without them . have these structures already dispersed their clusters into their tails , or has there been a complete absence of star cluster formation ? this remains to be seen . we would like to thank the anonymous referee for helpful comments which have improved the quality and content of this paper . based on observations obtained at the gemini observatory ( program i d gs-2013a - q-57 , processed using the gemini iraf package ) , which is operated by the association of universities for research in astronomy , inc . , under a cooperative agreement with the nsf on behalf of the gemini partnership : the national science foundation ( united states ) , the national research council ( canada ) , conicyt ( chile ) , ministerio de ciencia , tecnologa e innovacin productiva ( argentina ) , and ministrio da cincia , tecnologia e inovao ( brazil ) . the digitized sky surveys 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 . the plates were processed into the present compressed digital form with the permission of these institutions . scg thanks the natural science and engineering research council of canada for support . kk is supported by an nsf astronomy and astrophysics postdoctoral fellowship under award ast-1501294 . the institute for gravitation and the cosmos is supported by the eberly college of science and the office of the senior vice president for research at the pennsylvania state university .

we have developed an observing program using deep , multiband imaging to probe the chaotic regions of tidal tails in search of an underlying stellar population , using ngc 3256 s 400 myr twin tidal tails as a case study . these tails have different colours of @xmath0 and @xmath1 for ngc 3256w , and @xmath2 and @xmath3 for ngc 3256e , indicating different stellar populations . these colours correspond to simple stellar population ages of @xmath4 myr and @xmath5 myr for ngc 3256w and ngc 3256e , respectively , suggesting ngc 3256w s diffuse light is dominated by stars formed after the interaction , while light in ngc 3256e is primarily from stars that originated in the host galaxy . using a mixed stellar population model , we break our diffuse light into two populations : one at 10 gyr , representing stars pulled from the host galaxies , and a younger component , whose age is determined by fitting the model to the data . we find similar ages for the young populations of both tails , ( @xmath6 and @xmath7 myr for ngc 3256w and ngc 3256e , respectively ) , but a larger percentage of mass in the 10 gyr population for ngc 3256e ( @xmath8 vs @xmath9 ) . additionally , we detect 31 star cluster candidates in ngc 3256w and 19 in ngc 2356e , with median ages of 141 myr and 91 myr , respectively . ngc 3256e contains several young ( < 10 myr ) , low mass objects with strong nebular emission , indicating a small , recent burst of star formation . = 1 [ firstpage ] galaxies : interactions galaxies : individual : ngc 3256 galaxies : star clusters : general

complex systems with interacting constituents are ubiquitous in nature and society . to understand the microscopic mechanisms of emerging statistical laws of complex systems , one records and analyzes time series of observable quantities . these time series are usually nonstationary and possess long - range power - law cross - correlations . examples include the velocity , temperature , and concentration fields of turbulent flows embedded in the same space as joint multifractal measures @xcite , topographic indices and crop yield in agronomy @xcite , temporal and spatial seismic data @xcite , nitrogen dioxide and ground - level ozone @xcite , heart rate variability and brain activity in healthy humans @xcite , sunspot numbers and river flow fluctuations @xcite , wind patterns and land surface air temperatures @xcite , traffic flows @xcite and traffic signals @xcite , self - affine time series of taxi accidents @xcite , and econophysical variables @xcite . a variety of methods have been used to investigate the long - range power - law cross - correlations between two nonstationary time series . the earliest was joint multifractal analysis to study the cross - multifractal nature of two joint multifractal measures through the scaling behaviors of the joint moments @xcite , which is a multifractal cross - correlation analysis based on the partition function approach ( mf - x - pf ) @xcite . over the past decade , detrended cross - correlation analysis ( dcca ) has become the most popular method of investigating the long - range power - law cross correlations between two nonstationary time series @xcite , and this method has numerous variants @xcite . statistical tests can be used to measure these cross correlations @xcite . there is also a group of multifractal detrended fluctuation analysis ( mf - dcca ) methods of analyzing multifractal time series , e.g. , mf - x - dfa @xcite , mf - x - dma @xcite , and mf - hxa @xcite . the observed long - range power - law cross - correlations between two time series may not be caused by their intrinsic relationship but by a common third driving force or by common external factors @xcite . if the influence of the common external factors on the two time series are additive , we can use partial correlation to measure their intrinsic relationship @xcite . to extract the intrinsic long - range power - law cross - correlations between two time series affected by common driving driving forces , we previously developed and used detrended partial cross - correlation analysis ( dpxa ) and studied the dpxa exponents of variable cases , combining the ideas of detrended cross - correlation analysis and partial correlation @xcite . in ref . @xcite , the dpxa method has been proposed independently , focussing on the dpxa coefficient . here we provide a general framework for the dpxa and mf - dpxa methods that is applicable to various extensions , including different detrending approaches and higher dimensions . we adopt two well - established mathematical models ( bivariate fractional brownian motions and multifractal binomial measures ) in our numerical experiments , which have known analytical expressions , and demonstrate how the ( mf-)dpxa methods is superior to the corresponding ( mf-)dcca methods . consider two stationary time series @xmath0 and @xmath1 that depend on a sequence of time series @xmath2 with @xmath3 . each time series is covered with @xmath4 $ ] non - overlapping windows of size @xmath5 . consider the @xmath6th box @xmath7 $ ] , where @xmath8 . we calibrate the two linear regression models for @xmath9 and @xmath10 respectively , @xmath11 where @xmath12^{\mathrm{t}}$ ] , @xmath13^{\mathrm{t}}$ ] , @xmath14 and @xmath15 are the vectors of the error term , and @xmath16 is the matrix of the @xmath17 external forces in the @xmath6th box , where @xmath18 is the transform of @xmath19 . equation ( [ eq : xy : z : rxy : betas ] ) gives the estimates @xmath20 and @xmath21 of the @xmath17-dimensional parameter vectors @xmath22 and @xmath23 and the sequence of error terms , @xmath24 we obtain the disturbance profiles , i.e. , @xmath25 where @xmath26 . we assume that the local trend functions of @xmath27 and @xmath27 are @xmath28 and @xmath29 , respectively . the detrended partial cross - correlation in each window is then calculated , @xmath30\left[r_{y , v}(k)-\widetilde{r}_{y , v}(k)\right],\ ] ] and the second - order detrended partial cross - correlation is calculated , @xmath31^{1/2}.\ ] ] if there are intrinsic long - range power - law cross - correlations between @xmath32 and @xmath33 , we expect the scaling relation , @xmath34 there are many ways of determining @xmath28 and @xmath29 . the local detrending functions could be polynomials @xcite , moving averages @xcite , or other possibilities @xcite . to distinguish the different detrending methods , we label the corresponding dpxa variants as , e.g. , px - dfa and px - dma . when the moving average is used as the local detrending function , the window size of the moving averages must be the same as the covering window size @xmath5 @xcite . to measure the validity of the dpxa method , we perform numerical experiments using an additive model for @xmath32 and @xmath33 , i.e. , @xmath35 where @xmath36 is a fractional gaussian noise with hurst index @xmath37 , and @xmath38 and @xmath39 are the incremental series of the two components of a bivariate fractional brownian motion ( bfbms ) with hurst indices @xmath40 and @xmath41 @xcite . the properties of multivariate fractional brownian motions have been extensively studied @xcite . in particular , it has been proven that the hurst index @xmath42 of the cross - correlation between the two components is @xcite @xmath43 this property allows us to assess how the proposed method perform . we can obtain the @xmath44 of @xmath32 and @xmath33 using the dcca method and the @xmath45 of @xmath38 and @xmath39 using the dpxa method . our numerical experiments show that @xmath46 . we use @xmath47 for theoretical or true values and @xmath48 for numerical estimates . in the simulations we set @xmath49 , @xmath50 , @xmath51 , and @xmath52 in the model based on eq . ( [ eq : dpxa : model ] ) . three hurst indices @xmath40 , @xmath41 , and @xmath37 are input arguments and vary from 0.1 to 0.95 at 0.05 intervals . because @xmath38 and @xmath39 are symmetric , we set @xmath53 , resulting in @xmath54 triplets of @xmath55 . the bfbms are simulated using the method described in ref . @xcite , and the fbms are generated using a rapid wavelet - based approach @xcite . the length of each time series is 65536 . for each @xmath56 triplet we conduct 100 simulations . we obtain the hurst indices for the simulated time series @xmath38 , @xmath39 , @xmath57 , @xmath32 , and @xmath33 using detrended fluctuation analysis @xcite . the average values @xmath58 , @xmath59 , @xmath60 , @xmath61 , and @xmath62 over 100 realizations are calculated for further analysis , which are shown in fig . [ fig : dpxa : dhxyz ] . a linear regression between the output and input hurst indices in fig . [ fig : dpxa : dhxyz](a c ) yields @xmath63 , @xmath64 , and @xmath65 , suggesting that the generated fbms have hurst indices equal to the input hurst indices . figure [ fig : dpxa : dhxyz](d ) shows that when @xmath66 , @xmath61 is close to @xmath60 . when it is not , @xmath67 . figure [ fig : dpxa : dhxyz](e ) shows that @xmath68 . because @xmath69 and @xmath70 [ see fig . [ fig : dpxa : dhxyz](a)(b ) ] , we verify numerically that @xmath71 note also that @xmath72 , and that @xmath73 is a function of @xmath58 , @xmath59 and @xmath60 . a simple linear regression gives @xmath74 which indicates that the dpxa method can be used to extract the intrinsic cross - correlations between the two time series @xmath32 and @xmath33 when they are influenced by a common factor @xmath57 . we calculate the average @xmath75 over different @xmath37 and then find the relative error @xmath76 figure [ fig : dpxa : dhxyz](f ) shows the results for different combinations of @xmath58 and @xmath59 . although in most cases we see that @xmath77 , when both @xmath58 and @xmath59 approach 0 , @xmath78 increases . when @xmath79 , @xmath80 , and when @xmath81 and @xmath82 , @xmath83 . for all other points of @xmath84 , the relative errors @xmath78 are less than 0.10 . in a way similar to detrended cross - correlation coefficients @xcite , we define the detrended partial cross - correlation coefficient ( or dpxa coefficient ) as @xmath85 as in the dcca coefficient @xcite , we also find @xmath86 for dpxa . the dpxa coefficient indicates the intrinsic cross - correlations between two non - stationary series . ( color online . ) detrended partial cross - correlation coefficients . ( a ) performance of different methods by comparing three cross - correlation coefficients @xmath87 , @xmath88 and @xmath89 of the mathematical model in eq . ( [ eq : dpxa : model ] ) . ( b ) estimation and comparison of the cross - correlation levels between the two return time series ( @xmath90 ) and two volatility time series ( @xmath91 ) of crude oil and gold when including and excluding the influence of the usd index.,title="fig : " ] ( color online . ) detrended partial cross - correlation coefficients . ( a ) performance of different methods by comparing three cross - correlation coefficients @xmath87 , @xmath88 and @xmath89 of the mathematical model in eq . ( [ eq : dpxa : model ] ) . ( b ) estimation and comparison of the cross - correlation levels between the two return time series ( @xmath90 ) and two volatility time series ( @xmath91 ) of crude oil and gold when including and excluding the influence of the usd index.,title="fig : " ] we use the mathematical model in eq . ( [ eq : dpxa : model ] ) with the coefficients @xmath92 and @xmath93 to demonstrate how the dpxa coefficient outperforms the dcca coefficient . the two components @xmath38 and @xmath39 of the bfbm have very small hurst indices @xmath94 and their correlation coefficient is @xmath95 , and the driving fbm force @xmath57 has a large hurst index @xmath96 . figure [ fig : dpxa : rho](a ) shows the resulting cross - correlation coefficients at different scales . the dcca coefficients @xmath88 between the generated @xmath38 and @xmath39 time series overestimate the true value @xmath95 . because the influence of @xmath57 on @xmath38 and @xmath39 is very strong , the behaviors of @xmath32 and @xmath33 are dominated by @xmath57 , and the cross - correlation coefficient @xmath97 is close to 1 when @xmath5 is small and approaches 1 when @xmath5 us large . in contrast , the dpxa coefficients @xmath89 are in good agreement with the true value @xmath95 . note that the dpxa method better estimates @xmath38 and @xmath39 than the dcca method , since the @xmath88 curve deviates more from the horizontal line @xmath95 than the @xmath89 curve , especially at large scales . to illustrate the method with an example from finance , we use it to estimate the intrinsic cross - correlation levels between the futures returns and the volatilities of crude oil and gold . it is well - documented that the returns of crude oil and gold futures are correlated @xcite , and that both commodities are influenced by the usd index @xcite . the data samples contain the daily closing prices of gold , crude oil , and the usd index from 4 october 1985 to 31 october 2012 . figure [ fig : dpxa : rho](b ) shows that both the dcca and dpxa coefficients of returns exhibit an increasing trend with respect to the scale @xmath5 , and that the two types of coefficient for the volatilities do not exhibit any evident trend . for both financial variables , fig . [ fig : dpxa : rho](b ) shows that @xmath98 for different scales . although this is similar to the result between ordinary partial correlations and cross - correlations @xcite , the dpxa coefficients contain more information than the ordinary partial correlations since the former indicate the partial correlations at multiple scales . an extension of the dpxa for multifractal time series , notated mf - dpxa , can be easily implemented . when mf - dpxa is implemented with dfa or dma , we notate it mf - px - dfa or mf - px - dma . the @xmath99th order detrended partial cross - correlation is calculated @xmath100^{1/q}\ ] ] when @xmath101 , and @xmath102~.\ ] ] we then expect the scaling relation @xmath103 according to the standard multifractal formalism , the multifractal mass exponent @xmath104 can be used to characterize the multifractal nature , i.e. , @xmath105 where @xmath106 is the fractal dimension of the geometric support of the multifractal measure @xcite . we use @xmath107 for our time series analysis . if the mass exponent @xmath104 is a nonlinear function of @xmath99 , the signal is multifractal . we use the legendre transform to obtain the singularity strength function @xmath108 and the multifractal spectrum @xmath109 @xcite @xmath110 to test the performance of mf - dpxa , we construct two binomial measures @xmath111 and @xmath112 from the @xmath17-model with known analytic multifractal properties @xcite , and contaminate them with gaussian noise . we generate the binomial measure iteratively @xcite by using the multiplicative factors @xmath113 for @xmath38 and @xmath114 for @xmath39 . the contaminated signals are @xmath115 and @xmath116 . figures [ fig : mfpx : pmodel](a)(c ) show that the signal - to - noise ratio is of order @xmath117 . figures [ fig : mfpx : pmodel](d)(f ) show a power - law dependence between the fluctuation functions and the scale , in which it is hard to distinguish the three curves of @xmath118 . figure [ fig : mfpx : pmodel](g ) shows that for @xmath119 and @xmath120 , the @xmath121 function an approximate straight line and that the corresponding @xmath122 spectrum is very narrow and concentrated around @xmath123 . these observations are trivial because @xmath119 and @xmath120 are gaussian noise with the hurst indices @xmath124 , and the multifractal detrended cross - correlation analysis @xcite fails to uncover any multifractality . on the contrary , we find that @xmath125 and @xmath126 . thus the mf - dpxa method successfully reveals the intrinsic multifractal nature between @xmath127 and @xmath128 hidden in @xmath119 and @xmath120 . in summary , we have studied the performances of dpxa exponents , dpxa coefficients , and mf - dpxa using bivariate fractional brownian motions contaminated by a fractional brownian motion and multifractal binomial measures contaminated by white noise . these mathematical models are appropriate here because their analytical expressions are known . we have demonstrated that the dpxa methods are capable of extracting the intrinsic cross - correlations between two time series when they are influenced by common factors , while the dcca methods fail . the methods discussed are intended for multivariate time series analysis , but they can also be generalized to higher dimensions @xcite . we can also use lagged cross - correlations in these methods @xcite . although comparing the performances of different methods is always important @xcite , different variants of a method can produce different outcomes when applied to different systems . for instance , one variant that outperforms other variants under the setting of certain stochastic processes is not necessary the best performing method for other systems @xcite . we argue that there are still a lot of open questions for the big family of dfa , dma , dcca and dpxa methods . this work was partially supported by the national natural science foundation of china under grant no . 11375064 , fundamental research funds for the central universities , and shanghai financial and securities professional committee . 69ifxundefined [ 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 10.1017/s0022112075000304 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1140/epjb / e2009 - 00402 - 2 [ * * , ( ) ] link:\doibase 10.1007/s10661 - 009 - 1083 - 6 [ * * , ( ) ] link:\doibase 10.1063/1.3427639 [ * * , ( ) ] link:\doibase 10.1016/j.physa.2010.06.025 [ * * , ( ) ] link:\doibase 10.1016/j.atmosres.2010.11.009 [ * * , ( ) ] link:\doibase 10.1007/s11071 - 009 - 9642 - 5 [ * * , ( ) ] link:\doibase 10.1016/j.physa.2011.06.018 [ * * , ( ) ] link:\doibase 10.1016/j.physa.2010.12.038 [ * * , ( ) ] link:\doibase 10.1016/j.physa.2008.01.119 [ * * , ( ) ] link:\doibase 10.1073/pnas.0911983106 [ * * , ( ) ] link:\doibase 10.1016/j.physa.2010.01.040 [ * * , ( ) ] link:\doibase 10.1016/j.physa.2010.08.029 [ * * , ( ) ] link:\doibase 10.1016/j.chaos.2010.11.005 [ * * , ( ) ] link:\doibase 10.1209/epl / i1996 - 00438 - 4 [ * * , ( ) ] link:\doibase 10.1209/epl / i2000 - 00170 - 7 [ * * , ( ) ] link:\doibase 10.1209/0295 - 5075/79/44001 [ * * , ( ) ] link:\doibase 10.1142/s0218348x12500259 [ * * , ( ) ] link:\doibase 10.1103/physreve.73.066128 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.100.084102 [ * * , ( ) ] link:\doibase 10.1209/0295 - 5075/94/18007 [ * * , ( ) ] link:\doibase 10.1103/physreve.77.036104 [ * * , ( ) ] link:\doibase 10.1088/1742 - 5468/2009/03/p03037 [ ( ) ] in link:\doibase 10.1109/icassp.2009.4960233 [ _ _ ] ( ) pp . link:\doibase 10.1088/1367 - 2630/12/4/043057 [ * * , ( ) ] link:\doibase 10.1016/j.physa.2010.11.011 [ * * , ( ) ] link:\doibase 10.1140/epjb / e2013 - 40705-y [ * * , ( ) ] link:\doibase 10.1016/j.physa.2014.03.015 [ * * , ( ) ] link:\doibase 10.1142/s0218348x14500078 [ * * , ( ) ] link:\doibase 10.1103/physreve.91.022802 [ * * , ( ) ] link:\doibase 10.1140/epjb / e2009 - 00310 - 5 [ * * , ( ) ] link:\doibase 10.1016/j.physa.2010.10.022 [ * * , ( ) ] link:\doibase 10.1103/physreve.84.066118 [ * * , ( ) ] link:\doibase 10.1103/physreve.77.066211 [ * * , ( ) ] link:\doibase 10.1103/physreve.84.016106 [ * * , ( ) ] link:\doibase 10.1209/0295 - 5075/95/68001 [ * * , ( ) ] link:\doibase 10.1155/2009/249370 [ * * , ( ) ] link:\doibase 10.1140/epjb / e2009 - 00384-y [ * * , ( ) ] link:\doibase 10.1371/journal.pone.0015032 [ * * , ( ) ] link:\doibase 10.1111/j.1467 - 842x.2004.00360.x [ * * , ( ) ] @noop master s thesis , ( ) , link:\doibase 10.1038/srep08143 [ * * , ( ) ] link:\doibase 10.1103/physreve.49.1685 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physreve.58.6832 [ * * , ( ) ] link:\doibase 10.1140/epjb / e20020150 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1016/j.physa.2007.02.074 [ * * , ( ) ] link:\doibase 10.1016/j.physa.2011.07.008 [ * * , ( ) ] link:\doibase 10.1103/physreve.82.011136 [ * * , ( ) ] link:\doibase 10.1016/j.spl.2009.08.015 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1006/acha.1996.0030 [ * * , ( ) ] link:\doibase 10.1016/s0378 - 4371(01)00144 - 3 [ * * , ( ) ] link:\doibase 10.1016/j.resourpol.2010.05.003 [ * * , ( ) ] link:\doibase 10.1016/j.econmod.2012.09.052 [ * * , ( ) ] link:\doibase 10.1080/14697688.2014.946660 [ * * , ( ) ] link:\doibase 10.1016/s0378 - 4371(02)01383 - 3 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physreve.74.061104 [ * * , ( ) ] link:\doibase 10.1103/physreve.76.056703 [ * * , ( ) ] link:\doibase 10.1209/0295 - 5075/90/68001 [ * * , ( ) ] link:\doibase 10.1016/j.physleta.2014.12.036 [ * * , ( ) ] link:\doibase 10.1038/srep00835 [ * * , ( ) ]

when common factors strongly influence two power - law cross - correlated time series recorded in complex natural or social systems , using classic detrended cross - correlation analysis ( dcca ) without considering these common factors will bias the results . we use detrended partial cross - correlation analysis ( dpxa ) to uncover the intrinsic power - law cross - correlations between two simultaneously recorded time series in the presence of nonstationarity after removing the effects of other time series acting as common forces . the dpxa method is a generalization of the detrended cross - correlation analysis that takes into account partial correlation analysis . we demonstrate the method by using bivariate fractional brownian motions contaminated with a fractional brownian motion . we find that the dpxa is able to recover the analytical cross hurst indices , and thus the multi - scale dpxa coefficients are a viable alternative to the conventional cross - correlation coefficient . we demonstrate the advantage of the dpxa coefficients over the dcca coefficients by analyzing contaminated bivariate fractional brownian motions . we calculate the dpxa coefficients and use them to extract the intrinsic cross - correlation between crude oil and gold futures by taking into consideration the impact of the us dollar index . we develop the multifractal dpxa ( mf - dpxa ) method in order to generalize the dpxa method and investigate multifractal time series . we analyze multifractal binomial measures masked with strong white noises and find that the mf - dpxa method quantifies the hidden multifractal nature while the mf - dcca method fails .

there have been many models for the evolution of the bias @xmath0 derived from empirical knowledge @xcite , theory @xcite , simulations and from observations which account for the growth and merging of collapsed structure @xcite . however , all of these bias fitting forms include the unknown free parameters which need to be fitted with the set of galaxy bias data and simulation . it is shown that an incorrect bias model causes a shift in measured values of cosmological parameters @xcite . thus , the accurate modeling to @xmath0 is prerequisite for the precision cosmology . we obtain the exact linear bias obtained from its definition and show its dependence both on cosmology and on gravity theory . we provide @xmath0 which can be obtained from both theory and observation . this analytic solution for the bias allows one to use it as a cosmological parameter instead of a nuisance one . the observed linear galaxy power spectrum using a fiducial model including the effects of bias and the redshift space distortions is given by @xcite p_gal^(k,z ) = b^2 p_m(k , z_0 ) ( 1 + ^2)^2 ( ) ^2 [ pgal ] , where @xmath2 ( ratio of the hubble parameter of the adopted fiducial model , @xmath3 to that of the true model , @xmath4 ) , @xmath5 ( ratio of the angular diameter distance of the true model , @xmath6 to that of the adopted fiducial model , @xmath7 ) , @xmath8 defining the linear bias factor , @xmath9 means the present matter power spectrum , the redshift space distortions ( rsd ) parameter , @xmath10 is defined as @xmath11 , and @xmath12 is the linear growth factor of the matter fluctuation , @xmath13 with @xmath14 meaning @xmath15 . if one adopts the definition of the linear bias as @xmath16 , then one obtains @xmath17 . both @xmath10 and @xmath18 are obtained from observations , and theories predict @xmath19 and @xmath20 . if one takes the derivative of @xmath21 with respect to @xmath22 , then one obtains ( _ we use @xmath23 for @xmath21 below _ ) b(k , z ) & = & ( z ) ^-1 + & = & ( z ) ^-1 [ bkz ] , where we use @xmath24 , @xmath25 , @xmath26 denoting the observed fractional rms in galaxy number density , @xmath27 , and @xmath28 ( under the assumption of the flat universe ) , respectively . one can refer the appendix for detail derivation . all quantities in the second equality of eq . ( [ bkz ] ) are measurable from galaxy surveys . both @xmath18 and @xmath29 are measured from galaxy surveys @xcite . also @xmath30 can be directly measured from @xmath18 and @xmath29 @xcite . thus , one can measure the time evolution of bias if there exists enough binned data to measure @xmath31 . future galaxy surveys will provide the sub - percent level accuracy in measuring @xmath29 @xcite and will make the accurate measurement of bias possible . ( [ bkz ] ) holds for any gravity theory because it is derived from its definition . from the above eq . ( [ bkz ] ) , one can understand the theoretical motivation for the formulae of @xmath32 @xcite . if one assumes @xmath18 is constant , then one obtains @xmath33 . thus , the magnitude of @xmath34 is determined by the measured value of @xmath18 which might depend on luminosity , color , and spectral type of galaxies . however , there is no reason to believe that @xmath18 is time independent . thus , we regard @xmath18 as a time dependent observable in eq . ( [ bkz ] ) . in addition , the time evolution of bias is completely determined from observations of @xmath18 and @xmath29 . we assume the form of @xmath35 to investigate its behavior where we assign the dependence of bias on galaxy properties into @xmath36 . in this case , the galaxy dependence on @xmath37 is absorbed in @xmath36 solely . the cosmological dependence on bias is represented by @xmath38 , @xmath39 , and @xmath23 . actually , @xmath40 depends on @xmath38 , @xmath30 , and the underlying gravity theory . we restrict our consideration for the linear regime and one can solve the sub - horizon solution for the @xmath13 to obtain the growth factor , @xmath12 for the given model . one can numerically solve this for given models . even though we just investigate the constant dark energy equation of state @xmath38cdm , @xmath41 , and dgp model in this _ letter _ , one can generalize the consideration for the any model by solving @xmath13 numerically . in this subsection , we investigate the evolution of bias for different cosmological parameters ( @xmath38 and @xmath30 ) under the general relativity ( gr ) . for the constant dark energy equation of state , @xmath38 , there exists the known exact analytic solution for the linear growth rate , @xmath12 @xcite . we adopt this solution to show both the cosmology and the astrophysics dependence on @xmath0 . one can generalize the time dependent @xmath38 by using the numerical solution for the @xmath12 . we depict the dependence of @xmath0 on @xmath38 and @xmath30 in fig . [ fig1 ] . in the left panle of fig . [ fig1 ] , we show the evolution of @xmath0 for different values of @xmath38 fixed @xmath44 , @xmath45 , and @xmath46 . the dashed , solid , and dotted lines correspond @xmath47 -1.2 , -1.0 , and -0.8 , respectively . as @xmath38 decreases , so does @xmath0 . this is due to the fact that if @xmath38 increases , then both @xmath12 and @xmath48 decrease . the difference of @xmath0 between models increases , as @xmath22 increases . the difference between @xmath49 and @xmath50 is about 4.4 ( 3.5 ) % at @xmath51 . we also show the @xmath0 dependence on @xmath30 for @xmath43cdm model in the right panel of fig . the dashed , solid , and dotted lines correspond @xmath52 0.35 , 0.3 , and 0.25 , respectively for @xmath43cdm model . as @xmath30 increases , so do @xmath12 and @xmath48 . thus , @xmath0 decreases as @xmath30 increases . the difference between @xmath53 and @xmath44 is about 3.8 ( 3.2 ) % at @xmath51 . even though we limit our consideration for the constant @xmath38 with the flat universe , one can generalize the investigation for the time varying @xmath38 and the non - flat universe by solving the sub - horizon equation numerically . also one can find the time varying @xmath38 model which produce the same cmb result for the constant @xmath38 models @xcite . [ cols="^,^ " , ] we obtain the exact analytic solution for the linear bias . this solution can investigate both cosmological and astrophysical dependence on bias without any ambiguity . from this solution , one can exactly estimate the time evolution of bias for different models . the different gravity theories provide the different bias . thus , this provides the consistent check for the cosmological dependence on the measured galaxy power spectrum for the given model . this solution can be generalized to many models including the modified gravity theories and the massive neutrino dark matter model by replacing the approximate solution used in this _ letter _ with the exact sub - horizon solutions for corresponding models . these cases are under investigation @xcite . this theoretical form of bias can be measured from measurements of @xmath18 and @xmath29 from galaxy surveys if we achieve enough binned data . also a known degeneracy between the equation of state @xmath38 and the growth index parameter @xmath54 due to the evolution of @xmath39 can be broken due to this exact form of bias and can be used to distinguish the dark energy from the modified gravity . we would like to thank xiao - dong li and hang bae kim for useful discussion . this work were carried out using computing resources of kias center for advanced computation . we also thank for the hospitality at apctp during the program trp . one takes the derivative of @xmath55 using their definitions , @xmath56 and @xmath57 to obtain = [ dfsig8dz ] , where we use the sub - horizon scale equation for the growth factor @xmath12 , @xmath58 where dot means the derivative with respect to the cosmic time @xmath59 . thus , one obtains an interesting relation between @xmath60 and @xmath40 , _ 8(z ) & = & + & = & [ sigma8 ] , where we explicitly express the @xmath61 using the observable quantity @xmath29 in the second equality . thus , if one achieves enough binned data for @xmath29 , then one can measure @xmath20 at each epoch . for example , the present value of @xmath20 is given by _ 8 ^ 0 & = & ( ) ^-1 + & = & [ sig80 ] . the value of @xmath62 derived from the cmb depends on the primordial amplitude , @xmath63 and the spectral index , @xmath64 . however , the right hand side of eq . ( [ sig80 ] ) depends only on the background evolution parameters , @xmath38 and @xmath30 . thus , one can the constraint @xmath63 and @xmath64 from the rsd measurement . @xmath60 and @xmath30 are degenerated in galaxy surveys , but one can break this from the above eq . ( [ sig80 ] ) . if one adopts the definition of linear bias @xmath65 , then one obtains @xmath0 from the above eq . ( [ dfsig8dz ] ) b^-1(z ) = [ binv ] . thus , one obtains the exact analytic solution for @xmath0 given by eq . ( [ bkz ] ) . one can generalize @xmath0 as @xmath37 if one substitute @xmath12 with @xmath66 even for sub - horizon scales . for example , if one considers @xmath41 model or the massive neutrino model , then one can obtain @xmath66 inside horizon scales at linear regime @xcite . o. lahav _ et al . _ , mon . not . . soc . * 333 * , 961 ( 2002 ) [ arxiv : astro - ph/0112162 ] . l. clerkin , d. kirk , o. lahav , f. b. abdalla , and e. gaztanaga , [ arxiv:1405.5521 ] . j. n. fry , astrophys . j. * 461 * , l65 ( 1996 ) s. matarrese , p. coles , f. lucchin , l. moscardini , mon . not . astron . soc . * 286 * , 115 ( 1997 ) [ arxiv : astro - ph/9608004 ] . m. teggmark and p. j. e. peebles , astrophys . j. * 500 * , l79 ( 1998 ) [ arxiv : astro - ph/9804067 ] . j. l. tinker _ et al . _ , astrophys . j. * 724 * , 878 ( 2010 ) [ arxiv:1001.3162 ] . s. m. croom _ et al . _ , mon . not . . soc . * 356 * , 415 ( 2005 ) [ arxiv : astro - ph/0409314 ] . s. basilakos , m. plionis , and a. pouri , phys . d * 83 * , 123525 ( 2011 ) [ arxiv:1106.1183 ] . seo and d. j. eisenstein , astrophys . j. * 598 * , 720 ( 2003 ) [ arxiv : astro - ph/0307460 ] . f. beutler _ et al . _ , mon . not . . soc . * 423 * , 3430 ( 2012 ) [ arxiv:1204.4725 ] s. lee , j. cosmol . astropart . phys.*02 * , 021 ( 2014 ) [ arxiv:1307.6619 ] . v. silveira and i. waga , phys . rev . d * 50 * , 4890 ( 1994 ) . s. lee and k .- w . ng , phys . rev . d * 82 * , 043004 ( 2010 ) [ arxiv:0907.2108 ] . s. lee , [ arxiv:1409.1355 ] . w. saunders , m. rowan - robinson , and a. lawrence , mon . not . . soc . * 258 * , 134 ( 1992 ) . planck collaboration ; p. a. r. ade _ et al . _ , astron . astrophys . * 571 * , 39 ( 2014 ) [ arxiv:1309.0382 ] . g. dvali , g. gabadadze , and m. porrati , phys . b * 485 * , 208 ( 2000 ) [ arxiv : hep - th/0005016 ] . s. lee and k .- w . ng , phys . lett . b * 688 * , 1 ( 2010 ) [ arxiv:0906.1643 ] . r. gannouji , b. moraes , and d. polarski j. cosmol . astropart . phys.*02 * , 034 ( 2009 ) [ arxiv:0809.3374 ] . f. simpson and j. a. peacock , phys . d * 81 * , 043512 ( 2010 ) [ arxiv:0910.3834 ] . s. lee , [ in preparation ] .

since kaiser introduced galaxies as a biased tracer of the underlying total mass field , the linear galaxies bias , @xmath0 appears ubiquitously both in theoretical calculations and in observational measurements related to galaxy surveys . however , the generic approaches to the galaxy density is a non - local and stochastic function of the underlying dark matter density and it becomes difficult to make the analytic form of @xmath0 . due to this fact , @xmath0 is known as a nuisance parameter and the effort has been made to measure bias free observable quantities . we provide the exact and analytic function of @xmath0 which also can be measured from galaxy surveys using the redshift space distortions parameters , more accurately unbiased observable @xmath1 . we also introduce approximate solutions for @xmath0 for different gravity theories . one can generalize these approximate solutions to be exact when one solves the exact evolutions for the dark matter density fluctuation of given gravity theories . these analytic solutions for @xmath0 make it advantage instead of nuisance .

a pattern @xmath2 is a non - empty word over an alphabet @xmath10 of capital letters called _ variables_. a word @xmath11 over @xmath1 is an instance of @xmath2 if there exists a non - erasing morphism @xmath6 such that @xmath12 . the avoidability index @xmath13 of a pattern @xmath2 is the size of the smallest alphabet @xmath1 such that there exists an infinite word @xmath0 over @xmath1 containing no instance of @xmath2 as a factor . bean , ehrenfeucht , and mcnulty @xcite and zimin @xcite characterized unavoidable patterns , i.e. , such that @xmath14 . we say that a pattern @xmath2 is @xmath15-avoidable if @xmath16 . for more informations on pattern avoidability , we refer to chapter 3 of lothaire s book @xcite . in this paper , we consider upper bounds on the avoidability index of long enough patterns with @xmath7 variables . bell and goh @xcite and rampersad @xcite used a method based on power series and obtained the following bounds : [ bgr ] let @xmath2 be a pattern with exactly @xmath7 variables . * if @xmath2 has length at least @xmath8 then @xmath17 . @xcite * if @xmath2 has length at least @xmath18 then @xmath19 . @xcite * if @xmath2 has length at least @xmath20 then @xmath21 . @xcite our main result improves these bounds : [ 2tok ] let @xmath2 be a pattern with exactly @xmath7 variables . * if @xmath2 has length at least @xmath8 then @xmath19 . * if @xmath2 has length at least @xmath9 then @xmath21 . theorem [ 2tok ] gives a positive answer to problem 3.3.2 of lothaire s book @xcite . the bound @xmath8 in theorem [ 2tok].(a ) is tight in the sense that for every @xmath22 , the pattern @xmath23 with @xmath7 variables in the family @xmath24 @xmath25 @xmath26 @xmath27 has length @xmath28 and is unavoidable . similarly , the bound @xmath9 in theorem [ 2tok].(b ) is tight in the sense that for every @xmath22 , the pattern with @xmath7 variables in the family @xmath29 aabaa,@xmath30 @xmath31 has length @xmath32 and is not 2-avoidable . hence , this shows that the upper bound 3 of theorem [ 2tok].(a ) is best possible . + the avoidability index of every pattern with at most 3 variables is known , thanks to various results in the literature . in particular , theorem [ 2tok ] is proved for @xmath33 : * for @xmath34 , the famous results of thue @xcite give @xmath35 and @xmath36 . * for @xmath37 , every binary pattern of length at least 4 contains a square , and is thus 3-avoidable . moreover , roth @xcite proved that every binary pattern of length at least 6 is 2-avoidable . * for @xmath38 , cassaigne @xcite began and the first author @xcite finished the determination of the avoidability index of every pattern with at most 3 variables . every ternary pattern of length at least 8 is 3-avoidable and every binary pattern of length at least 12 is 2-avoidable . so , there remains to prove the cases @xmath39 . section [ sec : preliminary ] is devoted to some preliminary results . we prove theorem [ 2tok].(a ) in section [ 3-a ] as a corollary of a result of bell and goh @xcite . in section [ 2-a ] , we prove theorem [ 2tok].(b ) using the so - called _ entropy compression method_. let @xmath2 be a pattern over @xmath40 . an _ occurrence _ @xmath41 of @xmath2 is an assignation of a non - empty words over @xmath1 to every variable of @xmath2 that form a factor . note that two distinct occurrences of @xmath2 may form the same factor . for example , if @xmath42 then the occurrence @xmath43 of @xmath2 forms the factor @xmath44 ; on the other hand , @xmath45 is a distinct occurrence of @xmath2 which forms the same factor @xmath44 . a pattern @xmath2 is _ doubled _ if every variable of @xmath2 appears at least twice in @xmath2 . a pattern @xmath2 is _ balanced _ if it is doubled and every variable of @xmath2 appears both in the prefix and the suffix of length @xmath46 of @xmath2 . note that if the pattern has odd length , then the variable @xmath47 that appears in the middle of @xmath2 ( i.e. in position @xmath48 ) must appear also in the prefix and in the suffix in order to make @xmath2 balanced . [ cl : balanced ] for every integer @xmath49 , every pattern with at most @xmath7 variables and length at least @xmath50 contains a balanced pattern @xmath51 with at most @xmath52 variables and length at least @xmath53 as a factor . we prove this claim by induction on @xmath7 . if @xmath34 , then @xmath2 has size at least @xmath49 and is clearly balanced . suppose this is true for some @xmath54 , i.e. @xmath2 with @xmath55 variables and length at least @xmath56 contains a balanced pattern @xmath51 as a factor with at most @xmath57 variables and length at least @xmath58 . let @xmath59 and let @xmath60 ( resp . @xmath61 ) be the prefix ( resp . the suffix ) of @xmath2 of size @xmath46 . if @xmath2 is not balanced , then there exists a variable @xmath47 in @xmath2 that does not occur in @xmath62 for some @xmath63 . thus , @xmath62 has at most @xmath64 variables and length at least @xmath56 . therefore , by induction hypothesis , @xmath2 contains a balanced pattern with at most @xmath57 variables and length at least @xmath53 as a factor . in the following , we will only use the fact that the pattern @xmath51 in claim [ cl : balanced ] is doubled instead of balanced . we prove theorem [ 2tok].(a ) as a corollary of the following result of bell and goh @xcite : [ k6doubled ] every doubled pattern with at least 6 variables is 3-avoidable . theorem [ 2tok].(a ) we want to prove that every pattern with exactly @xmath7 variables and length at least @xmath8 is 3-avoidable , or equivalently , that every pattern with at most @xmath7 variables and length at least @xmath8 is 3-avoidable . by claim [ cl : balanced ] , every such pattern contains a doubled pattern @xmath51 as a factor with at most @xmath52 variables and length at least @xmath65 . so there remains to show that every doubled pattern with at most @xmath7 variables and length at least @xmath8 is @xmath66-avoidable . as discussed in the introduction , the case of patterns with at most @xmath66 variables has been settled . now , it is sufficient to prove that doubled patterns of length at least @xmath67 are 3-avoidable . suppose that @xmath60 is a doubled pattern containing a variable @xmath47 that appears at least 4 times . replace @xmath68 occurrences of @xmath47 with a new variable to obtain a pattern @xmath61 . example : we replace the first and third occurrence of @xmath69 in @xmath70 by a new variable @xmath71 to obtain @xmath72 . then @xmath61 is a doubled pattern such that @xmath73 and @xmath74 , since every occurrence of @xmath60 is also an occurrence of @xmath61 . given a doubled pattern @xmath2 of length at least @xmath75 , we make such replacements as long as we can . we thus obtain a doubled pattern @xmath51 of length at least @xmath75 such that @xmath76 . moreover , every variable in @xmath51 appears either @xmath68 or @xmath66 times and therefore @xmath51 contains at least @xmath77 variables . so @xmath51 is @xmath66-avoidable by lemma [ k6doubled ] . thus @xmath2 is @xmath66-avoidable , which finishes the proof . we want to prove that every pattern with exactly @xmath7 variables and length at least @xmath78 is 2-avoidable , or equivalently , that every pattern with at most @xmath7 variables and length at least @xmath78 is 2-avoidable . by claim [ cl : balanced ] , every such pattern contains a doubled pattern @xmath51 as a factor with at most @xmath52 variables and length at least @xmath79 . so there remains to show that every doubled pattern with at most @xmath7 variables and length at least @xmath78 is @xmath68-avoidable . as discussed in the introduction , the case of patterns with at most @xmath66 variables has been settled . now , it is sufficient to prove theorem [ 2tok].(b ) for doubled patterns and @xmath39 . let @xmath80 be the alphabet . for the remaining of this section , let @xmath39 and @xmath81 . suppose by contradiction that there exists a doubled pattern @xmath2 on @xmath7 variables and length at least @xmath82 that is not @xmath68-avoidable . then there exists an integer @xmath55 such that any word @xmath83 contains @xmath2 . we put an arbitrary order on the @xmath7 variables of @xmath2 and call @xmath84 the @xmath85-th variable of @xmath2 . let @xmath86 be a vector of length @xmath15 . the following algorithm takes the vector @xmath87 as input and returns a word @xmath0 avoiding @xmath2 and a data structure @xmath88 that is called a _ record _ in the remaining of the paper . @xmath89 @xmath90 @xmath88 , @xmath0 the way we encode information in @xmath88 at lines 5 and 7 will be explained in subsection [ subsec : record ] . in the algorithm avoidpattern , let @xmath91 be the word @xmath0 after @xmath92 steps . clearly , @xmath91 avoids @xmath2 at each step . by contradiction hypothesis , the resulting word @xmath0 of the algorithm ( that is @xmath93 ) has length less than @xmath55 . we will prove that each output of the algorithm allows to determine the input . then we obtain a contradiction by showing that the number of possible outputs is strictly smaller than the number of possible inputs when @xmath15 is chosen large enough compared to @xmath55 . this implies that every pattern @xmath2 with at most @xmath7 variables and length at least @xmath82 is @xmath68-avoidable . to analyze the algorithm , we borrow ideas from graph coloring problems @xcite . these results are based on the moser - tardos @xcite entropy - compression method which is an algorithmic proof of the lovsz local lemma . an important part of the algorithm is to keep the record @xmath88 of each step of the algorithm . let @xmath94 be the record after @xmath92 steps of the algorithm avoidpattern . on one hand , given @xmath87 as input of the algorithm , this produces a pair @xmath95 . on the other hand , given a pair @xmath95 , we will show in lemma [ lem : inj ] that we can recover the entire input vector @xmath87 . so , each input vector @xmath87 produces a distinct pair @xmath95 . let @xmath96 be the set of input vectors @xmath87 of size @xmath15 , let @xmath97 be the set of records @xmath88 produced by the algorithm avoidpatternand let @xmath98 be the set of different outputs @xmath95 . after the execution of the algorithm ( @xmath15 steps ) , @xmath93 avoids @xmath2 by definition and therefore @xmath99 by contradiction hypothesis . hence , the number of possible final words @xmath93 is independent from @xmath15 ( it is at most @xmath100 ) . we then clearly have @xmath101 . we will prove that @xmath102 . the record @xmath88 is a triplet @xmath103 where @xmath104 is a binary word ( each element is @xmath105 or @xmath106 ) , @xmath107 is a vector of @xmath108-sets of non - zero integers and @xmath47 is a vector of binary words . at the beginning , @xmath104 , @xmath107 and @xmath47 are empty . at step @xmath92 of the algorithm , we append @xmath109 $ ] to @xmath110 to get @xmath111 . if @xmath111 contains no occurrence of @xmath2 , then we append @xmath105 to @xmath104 to get @xmath94 and we set @xmath112 . otherwise , suppose that @xmath111 contains an occurrence @xmath41 of @xmath2 that forms a factor @xmath4 of length @xmath113 ( @xmath4 is the @xmath113 last letters of @xmath111 ) . recall that @xmath84 is the @xmath85-th variable of @xmath2 . let @xmath114 in the factor @xmath4 , let @xmath115 , @xmath116 for @xmath117 . let @xmath118 be a @xmath108-set of non - zero integers . let @xmath119 be the binary word obtained from @xmath120 ( where `` @xmath121 '' is the concatenation operator ) followed by as many @xmath105 s as necessary to get length @xmath122 . note that we necessarily have @xmath123 since the pattern is doubled . eventually , to get @xmath94 , we append the factor @xmath124 to @xmath104 ; we add @xmath125 as the last element of @xmath107 ; finally we add @xmath119 as the last element of @xmath47 . * example : * let us give an example with @xmath38 , @xmath126 and @xmath127 $ ] . the variables of @xmath2 were initially ordered as @xmath128 . for the first @xmath129 steps , no occurrence of @xmath2 appeared , so at each step @xmath130 , we append @xmath109 $ ] to @xmath110 and we append one @xmath105 to @xmath104 . hence , at step @xmath129 , we have : * @xmath131 * @xmath132\\ x&=&[\ ] \end{array } \right.$ ] now , at step @xmath133 , we first append @xmath134 = 1 $ ] to @xmath135 to get @xmath136 . the word @xmath136 contains an occurrence @xmath137 of @xmath2 which forms a factor of length @xmath138 ( the @xmath138 last letters of @xmath136 ) . then we set @xmath139 and @xmath140 ( this is @xmath141 followed by four @xmath105 s in order to get a binary word of length @xmath142 ) . eventually , to get @xmath143 and @xmath144 , we erase the suffix of length @xmath138 of @xmath136 to get @xmath143 , we append the factor @xmath145 to @xmath104 , @xmath125 to @xmath107 , and @xmath119 to @xmath47 . this gives : * @xmath146 * @xmath147 \\ x&=&[0111000000 ] \end{array } \right.$ ] this is where our example ends . let @xmath148 be the vector @xmath87 restricted to its @xmath92 first elements . we will show that the pair @xmath149 at some step @xmath92 allows to recover @xmath148 . [ lem : inj ] after @xmath92 steps of the algorithm avoidpattern , the pair @xmath149 permits to recover @xmath148 . before step 1 , we have @xmath150 , @xmath151,[\ ] ) $ ] , and @xmath152 . let @xmath153 be the record after step @xmath92 , with @xmath154 . * suppose that @xmath105 is a suffix of @xmath104 . this means that at step @xmath92 , no occurrence of @xmath2 was found : the algorithm appended @xmath109 $ ] to @xmath110 to get @xmath91 . therefore @xmath109 $ ] is the last letter of @xmath91 , say @xmath11 . then the word @xmath110 is obtained from @xmath91 by erasing the last letter and the record @xmath155 is obtained from @xmath94 by removing the suffix @xmath105 of @xmath104 . we recover @xmath156 from @xmath157 by induction hypothesis and we obtain @xmath158 . * suppose now that @xmath124 is a suffix of @xmath104 . this means that an occurrence @xmath41 of @xmath2 which forms a factor @xmath4 of length @xmath113 has been created during step @xmath92 . the last element @xmath125 of @xmath107 is a @xmath108-set @xmath118 and the last element @xmath119 of @xmath47 is a binary word of length @xmath122 . let @xmath159 and for @xmath160 , let @xmath161 . so , for @xmath162 , @xmath163 is clearly the length of the variable @xmath164 of @xmath2 in the occurrence @xmath41 by construction of @xmath125 . we know the pattern @xmath2 , the total length of the factor @xmath4 ( that is @xmath113 ) and the lengths of the @xmath165 first variables of @xmath2 in @xmath4 , so we are able to compute the length @xmath166 of the last variable @xmath167 . so we are now able to recover the occurrence @xmath41 of @xmath2 : the first @xmath168 letters of @xmath119 correspond to @xmath169 , the next @xmath170 letters correspond to @xmath171 and so on ( @xmath119 may contain some @xmath105 s at the end which are not relevant ) . it follows that the factor @xmath4 is completely determined . so @xmath110 is obtained from @xmath172 by removing the last letter @xmath11 of @xmath4 , this letter @xmath11 being @xmath109 $ ] ( the appended letter to @xmath110 at step @xmath92 to get @xmath111 ) . the record @xmath155 is obtained from @xmath94 as follows : remove the suffix @xmath124 from @xmath104 ; remove the last element of @xmath107 and the last element of @xmath47 . we recover @xmath156 from @xmath157 by induction hypothesis and we obtain @xmath158 . now we compute @xmath174 . let @xmath175 be a given record produced by an execution of avoidpattern . let @xmath176 , @xmath177 and @xmath178 be the set of such binary words @xmath104 , of such @xmath108-sets of non - zero integers @xmath107 , and of such vectors of binary words @xmath47 , respectively . we thus have @xmath179 , @xmath180 , and @xmath181 . the algorithm runs in @xmath15 steps . at each step , one letter is appended to @xmath0 , so @xmath15 letters have been appended and therefore the number of erased letters during the execution of the algorithm is @xmath182 . at some steps , an occurrence of @xmath2 appears and forms a factor which is immediately erased . let @xmath183 be the number of erased factors during the execution of the algorithm . let @xmath184 , @xmath185 , be the @xmath183 erased factors . we have @xmath186 since each variable of @xmath2 is a non - empty word and @xmath2 has length at least @xmath82 . moreover , we have @xmath187 . each time a factor @xmath184 is erased , we add an element to @xmath107 and @xmath47 , so @xmath188 . in the binary word @xmath104 , each @xmath105 corresponds to an appended letter during the execution of the algorithm and each @xmath106 corresponds to an erased letter . therefore , @xmath104 has length @xmath189 . observe that every prefix in @xmath104 contains at least as many @xmath105 s as @xmath106 s . indeed , since a @xmath106 corresponds to an erased letter @xmath11 , this letter @xmath11 had to be added first and thus there is a @xmath105 before that corresponds to this @xmath106 . the word @xmath104 is therefore a partial dyck word . since any erased factor @xmath184 has length at least @xmath82 , any maximal sequence of @xmath106 s ( which is called a _ descent _ in the sequel ) in @xmath104 has length at least @xmath82 . so @xmath104 is a partial dyck words with @xmath15 @xmath105 s such that each descent has length at least @xmath82 . the following two lemmas due to esperet and parreau @xcite give an upper bound on @xmath190 . let @xmath191 ( resp . @xmath192 ) be the number of partial dyck words with @xmath15 @xmath105 s and @xmath193 @xmath106 s ( resp . dyck words of length @xmath194 ) such that all descents have length at least @xmath195 . hence , we have @xmath196 . @xcite[louis1 ] @xmath197 . hence , we have @xmath198 . let @xmath199 . @xcite[louis2 ] let @xmath195 be an integer such that the equation @xmath200 has a solution @xmath201 with @xmath202 , where @xmath203 is the radius of convergence of @xmath204 . then @xmath201 is the unique solution of the equation in the open interval @xmath205 . moreover , there exists a constant @xmath206 such that @xmath207 where @xmath208 . the solution of the equation @xmath209 is equivalent to @xmath210 . the radius of convergence @xmath203 of @xmath211 is @xmath106 and since @xmath212 and @xmath213 , @xmath214 has a solution @xmath201 in the open interval @xmath205 . by lemma [ louis2 ] , this solution is unique and , for some constant @xmath215 , we have @xmath216 with @xmath217 . we clearly have @xmath218 . so , we can compute @xmath219 for @xmath195 fixed . we will use the following bounds : @xmath220 , @xmath221 , and @xmath222 . note that when @xmath195 increases , @xmath219 decreases . so , by lemmas [ louis1 ] and [ louis2 ] , when @xmath15 is large enough , we have @xmath223 ( resp . @xmath224 , @xmath225 ) if the length of any descent is at least @xmath129 ( resp . @xmath226 , @xmath227 ) . each element @xmath119 of @xmath47 corresponds to an erased factor @xmath184 and by construction @xmath228 . so the sum of the lengths of the elements of @xmath47 is @xmath229 . thus , the vector @xmath47 corresponds to a binary word of length at most @xmath230 . therefore @xmath231 . each element @xmath118 of @xmath107 corresponds to an erased factor @xmath184 and by construction each @xmath232 corresponds to the sum of the lengths of the @xmath85 first variables of @xmath2 in @xmath184 . let @xmath233 be the number of such @xmath108-sets @xmath125 that correspond to factors of length @xmath113 . recall that @xmath186 , so @xmath233 is defined for @xmath39 and @xmath234 . each of the @xmath183 elements of @xmath107 corresponds to an erased factor , so @xmath235 . let @xmath236 defined for @xmath39 and @xmath234 . then @xmath237 . so , if we are able to upper - bound @xmath238 by some constant @xmath239 for all @xmath234 , then we get @xmath240 . now we bound @xmath238 using two different methods depending on the value of @xmath7 and the length @xmath82 of @xmath2 . [ [ par : l1 ] ] bound on @xmath238 for @xmath241 , @xmath242 or @xmath243 , @xmath244 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + for any factor @xmath184 , we have @xmath245 since @xmath2 is doubled ( each variable appears at least twice ) . for a given @xmath118 that corresponds to some factor @xmath184 , we have @xmath246 . therefore , @xmath125 is a @xmath108-set of distinct non - zero integers at most @xmath247 , i.e. @xmath165 integers chosen among integers between @xmath106 and @xmath247 ; thus @xmath248 and so @xmath249 . we can upper - bound @xmath250 by @xmath251 for @xmath234 . the function @xmath252 is decreasing for @xmath234 ; so @xmath253 for all @xmath234 . moreover , we can see that @xmath254 for all @xmath243 . computing this upper bound , we get @xmath255 for all @xmath243 and @xmath256 and @xmath257 for all @xmath242 . [ [ par : l2 ] ] bound on @xmath258 for @xmath259 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the second method to bound the size of @xmath258 is based on ordinary generating functions ( ogf ) . here , @xmath241 , so let @xmath260 be the four variables of @xmath2 and let @xmath261 be the number of apparitions of @xmath262 in @xmath2 . therefore , @xmath263 . recall that each variable appears at least twice in @xmath2 since @xmath2 is doubled , so @xmath264 . moreover , a factor of length @xmath113 , with @xmath265 , is necessarily an occurrence of a pattern of length between @xmath129 and @xmath266 . so we just have to consider patterns @xmath2 with @xmath267 . given @xmath268 an element of @xmath107 that corresponds to some factor @xmath184 , we can compute the lengths @xmath269 of each variable @xmath262 in @xmath184 ( @xmath270 , @xmath271 for @xmath272 and @xmath273 ) . recall that @xmath274 since each variable of @xmath2 is a non - empty word . let @xmath275 be the ogf of such sets @xmath125 , i.e. @xmath276 is the number of @xmath66-sets @xmath277 that corresponds to a factor of length @xmath92 formed by an occurrence of a pattern of length @xmath278 ( that is @xmath276 is the number of @xmath279-tuples @xmath280 with @xmath274 such that @xmath281 ) . so by definition of @xmath282 , we have @xmath283 . this kind of ogf has been studied and is similar to the well - known problem of counting the number of ways you can change a dollar @xcite : you have only five types of coins ( pennies , nickels , dimes , quarters , and half dollars ) and you want to count the number of ways you can change any amount of cents . so , let @xmath284 be the ogf of the problem and thus any @xmath285 is the number of ways you can change @xmath92 cents . then , for example , @xmath286 corresponds to the number of ways you can change a dollar . here , @xmath287 . in our case , we have four coins , each of them has value @xmath261 ( so we can have different types of coins with the same value ) and each type of coins appears at least once ( since @xmath274 ) . thus we get @xmath288 . we use maple for our computation . for each @xmath289 , for each @xmath279-tuple @xmath290 such that @xmath291 , we consider the associated ogf @xmath292 and we compute , using maple , the truncated series expansion up to the order @xmath227 , that gives @xmath293 with explicit values for the coefficients @xmath276 . so , for any @xmath259 , @xmath258 is upper - bounded by the maximum of @xmath294 taken oven all @xmath292 . maple gives that @xmath294 is maximal for @xmath295 , @xmath296 , and @xmath297 : in this case , @xmath298 ( i.e. there exist @xmath299 distinct @xmath66-sets @xmath125 that correspond to some factor of length @xmath300 formed by an occurrence of a pattern of length @xmath129 ) . so , @xmath301 for all @xmath259 , @xmath241 and @xmath289 . [ [ bound - gell - for - all - kge-4 ] ] bound @xmath302 for all @xmath39 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we can deduce from paragraphs [ par : l1 ] and [ par : l2 ] the following . if @xmath241 , then @xmath303 for @xmath265 and @xmath304 for @xmath242 . so for @xmath241 , we have @xmath305 . if @xmath243 , then @xmath306 for @xmath234 . so for @xmath243 , we have @xmath307 . aggregating the above analysis , we get the following . for @xmath243 , we have @xmath308 : then @xmath309 . for @xmath241 , we have @xmath310 : then @xmath311 . thus for all @xmath39 , @xmath312 and so we obtained the desired contradiction : @xmath313 in our results , we heavily use the fact that the patterns are doubled . the fact that the patterns are long is convenient for our proofs but does not seem so important . so we ask whether every doubled pattern is 3-avoidable . by the remarks in section [ sec : intro ] and by lemma [ k6doubled ] , the only remaining cases are doubled patterns with @xmath279 and @xmath314 variables . also , does there exist a finite @xmath7 such that every doubled pattern with at least @xmath7 variables is 2-avoidable ? we know that such a @xmath7 is at least 5 since abccbadd is not 2-avoidable . j. berstel . axel thue s work on repetitions in words . invited lecture at the 4th conference on formal power series and algebraic combinatorics , montreal , 1992 , june 1992 . available at http://www-igm.univ-mlv.fr/ berstel / index.html .

in combinatorics on words , a word @xmath0 over an alphabet @xmath1 is said to avoid a pattern @xmath2 over an alphabet @xmath3 if there is no factor @xmath4 of @xmath0 such that @xmath5 where @xmath6 is a non - erasing morphism . a pattern @xmath2 is said to be @xmath7-avoidable if there exists an infinite word over a @xmath7-letter alphabet that avoids @xmath2 . we give a positive answer to problem 3.3.2 in lothaire s book `` algebraic combinatorics on words '' , that is , every pattern with @xmath7 variables of length at least @xmath8 ( resp . @xmath9 ) is 3-avoidable ( resp . 2-avoidable ) . this improves previous bounds due to bell and goh , and rampersad . , word ; pattern avoidance .

the aim of the cdt approach is to evaluate the gravitational quantum amplitude @xmath0~{\rm e}^{is[{\bf g_{\mu\nu}};t ' ] } \label{e0}\ ] ] between initial and final geometries @xmath1 . in this version we do not include matter fields in the theory . we use the intuition based on methods of quantum field theory to view this amplitude as a path integral over space - time geometries . defining a path integral in this case requires solving a number of non - trivial conceptual problems : * definition of _ time evolution _ assigns a special role to be played by a proper time @xmath2 for each quantum space - time , giving a meaning to the idea of _ initial _ and _ final _ and introducing a time foliation of geometry . * at each time @xmath2 we should define a hilbert space of states - spatial geometries of the universe . * definition of the measure @xmath3 $ ] is related to the choice of _ the domain of integration _ ( space of admissible space - times we should include in the path integral ) and possibly also solving the problem of diffeomorphism invariance . at the same time one would like to obtain an approach which is * background independent - the background geometry may emerge dynamically , but should not be introduced a priori . * non - perturbative - meaning again that it is not obtained as a perturbation around some fixed background . * has a well - defined infrared limit , described by the general theory of relativity . cdt provides a construction satisfying these requirements @xcite . * we consider only space - time geometries which admit a global time foliation . causality means that the spatial topology of the universe is fixed during the time evolution . * we introduce a lattice regularization of geometries , assuming that both space and time are discretized . the time variable is indexed by an integer time . at a fixed time we construct the hilbert space of states defined as a set of states @xmath4 representing triangulations of a 3d topological sphere using regular simplices ( tetrahedra ) with a common edge length @xmath5 . this definition does not involve coordinates , being by construction diffeomorphism invariant . different triangulations @xmath4 correspond to different geometries , which can not be mapped onto each other . the states @xmath4 satisfy @xmath6 where @xmath7 is the order of the automorphism group of @xmath8 . in the construction of the hilbert space we consider only three - manifolds with the simplest topology of of a three - sphere @xmath9 . this space splits in a natural way into a simple sum of spaces labelled by the number of tetrahedra @xmath10 . the number of states for a fixed volume @xmath10 is finite , but large . it grows exponentially with @xmath10 for large @xmath10 . * tetrahedra at time @xmath2 are bases of four - simplices @xmath11 and @xmath12 with four vertices at time @xmath2 and one vertex at @xmath13 . to form a closed four - dimensional manifold we need also four - simplices @xmath14 and @xmath15 with three vertices ( triangle ) at @xmath2 and two ( spanning a link ) at @xmath13 . four - simplices have a common length of the _ time link _ equal to @xmath16 see fig . [ fig01 ] . + + the sum ( integral ) over space - times is regularized as a sum over simplicial manifolds with topology @xmath17 $ ] . wick rotation to imaginary time can be realized as analytic continuation in @xmath18 and can be performed for each space - time configuration . manifolds can be characterized by a set of global numbers , where @xmath19 and @xmath20 denote the numbers of four - simplices of a particular type and @xmath21 the number of vertices . other numbers of this type can be expressed by this triple using topological identities . after a wick rotation each space - time configuration appears in the sum with the _ real _ weight @xmath22 , where @xmath23 is the hilbert - einstein action calculated using the simplicial structure defined above . dimensionless coupling constants @xmath24 and @xmath25 are related respectively to the inverse gravitational constant and cosmological constant . the parameter @xmath26 is related to the ratio of the lattice spacings in time and spatial direction ( @xmath27 for @xmath28 ) . for the imaginary time the quantum amplitude has the form of a partition function of a statistical ensemble of discretized space - time geometries . one can observe that the amplitude ( partition function ) can be represented as a _ matrix _ product @xmath29 matrix elements @xmath30 depend on a number of distinct ways to connect geometric spatial states at times @xmath2 and @xmath31 . in practice the model can not be solved analytically , except in the simplest case of 1 + 1 dimensional space - time @xcite . for larger dimensionality we are forced to use numerical methods as a tool to obtain physical information about the system . the tools we use are monte carlo simulations . in the cdt model a very important ingredient of the theory becomes the entropy of configurations . for a fixed set of @xmath32 the number of space - time configurations grows exponentially like @xmath33 the critical parameter @xmath34 depends on @xmath24 and @xmath26 , which means that the parameter @xmath26 , which originally had a geometric origin , plays the role of an independent coupling constant . the critical parameter renormalizes the bare cosmological constant @xmath35 and the model is defined only for @xmath36 . in the limit @xmath37 the average total number of simplices goes to infinity . this is the limit relevant for the continuum , where we may discuss what happens when the discretization effects can be neglected . in practice , the numerical approach means that we must consider systems with a finite volume . we can recover the information about the continuum properties studying the scaling properties of observables for a sequence of large but finite @xmath38 . this reduces the parameter space of the model to a set of two bare coupling constants @xmath24 and @xmath26 . in our simulations we choose periodic boundary conditions , which on the one hand frees us from the necessity to define initial and final geometries , but on the other hand does not change the physical picture , as will become clear below . using the monte carlo program we perform a random walk in the space of configurations using 7 elementary local _ moves _ , which preserve the ( local and global ) topological restrictions on a manifold . the probability to perform a particular move is obtained by the detailed balance condition . this is a markov process with a stationary limiting distribution satisfying @xmath39 configurations separated by a large number of moves are _ statistically independent _ and the probability to obtain a particular configuration is given by the limiting distribution . expectation values of observables are measured as averages in the large but finite set of statistically independent space - time configurations obtained at a particular set of parameters @xmath24 and @xmath26 and ( approximately ) fixed @xmath19 . the measurements are repeated for an increasing sequence of @xmath19 to check the scaling . in the following we discuss a very useful observable characterizing each space - time configuration . it is the distribution of a three - volume @xmath40 as a function of discrete time @xmath2 . depending on the position in the @xmath41 plane our system appears to be in three physically distinct phases @xcite characterized by different behaviour of @xmath40 . on the fig . [ fig02 ] we show a sample distribution of the three - volume @xmath40 for one typical configuration in phases a , b and c. phase a is characterized be a sequence of slices @xmath40 with no correlation between the states at neighbouring times . in phase b the time dependence of the distribution is squeezed to one time value ( one may view it as a spontaneous compactification of the time variable ) . for other times the volume is close to minimal . it can not be completely zero , because we choose periodic boundary conditions and do not allow the volume to vanish at any fixed time @xmath2 . most interesting from a physical point of view is the c phase . the volume profile looks like a fluctuation superimposed over a regular classical background ( the red line on the plot ) . a typical configuration consists of a central _ blob _ and a _ stalk _ of cut - off size resulting again from our choice of boundary conditions ( periodicity in time ) . the red line is the average distribution over many configurations with the same volume . we can compare distributions for a sequence of volumes ( fig . [ fix ] ) and we find a universal scaling behaviour in the variable @xmath42 , with hausdorff dimension @xmath43 . the plot illustrates the universality of the volume distribution for the rescaled observable @xmath44 plotted vs. @xmath45 . we expect the averaged scaled distribution to be volume - independent . this distribution can be interpreted as a semi - classical limiting distribution of volume . note that in our numerical experiments it is obtained by integrating out all other degrees of freedom ( details of the geometry ) except the spatial volume @xmath40 . the analysis shows that the averaged geometry scales in a way consistent with dimension four . although this result may appear trivial , it is definitely not , since the distribution is obtained as the effect of a very delicate balance between the entropy of configurations and the physical action . in earlier studies , where causality was not imposed , typical geometries dominating the quantum amplitude had either @xmath46 ( branched polymer phase ) or @xmath47 ( collapsed phase ) . we can analyze further the properties of the distribution and try to fit the limiting curve by an analytic formula . the effect of the analysis is presented on the fig . [ fig03 ] , together with the fit . the analytic form of the fit suggests that the observed geometry can be interpreted as the volume dependence inside a four - dimensional ball , in this case the variable @xmath2 plays the role of the azimuthal angle . this would indicate that in phase c we see the appearance of a spherical four - dimensional de sitter geometry . the geometric properties can be analyzed using other observables . a useful example is that of the spectral dimension @xmath48 . to measure this quantity we analyze the return probability in the diffusion process on the geometry , as a function of the diffusion time @xmath49 @xcite . if the geometry was regular we would expect @xmath50 with a constant @xmath51 . the figure shows the observed behaviour of @xmath48 obtained by averaging over many starting points of the diffusion process and over many configurations . the plot on fig . [ fig04 ] shows that @xmath48 is not a constant , but depends on @xmath49 suggesting a scale dependence of the effective geometry , ranging between two at short scales and four at large scales . this illustrates the quantum character of geometry . a similar property was discovered in other approaches to quantum gravity ( c.f . e.g. @xcite ) . the geometry presented above was measured at a particular point on the @xmath41 plane . when the values of the bare coupling constants are changed inside phase c , the qualitative behaviour remains the same , up to a finite change in the scale . we are particularly interested in the critical behaviour near phase transitions . the qualitative behaviour of the phase structure of cdt was found to have strong similarity to the phase structure predicted by hoava - lifschitz gravity ( @xcite and @xcite ) . the simplest check of the analogy was to measure the order of the phase transitions , which we found to be first - order for the a - c transition and second order for the b - c transition @xcite . the regular semi - classical distribution of spatial volume observed in phase c ( de sitter phase ) suggests that it reproduces a saddle point of some effective action of the spatial volume @xmath52 or , equivalently , of the scale factor @xmath53 . a natural candidate for such an action is the mini - superspace action @xmath54 where @xmath2 takes the continuum value and @xmath55 plays the role of lagrange multiplier , necessary to fix the total volume to some target volume @xmath56 @xmath57 here @xmath58 sets the scale in the time direction . in a discrete setup we may expect this action to take the discretized form @xmath59 although it could have a more complicated form . here @xmath60 . measuring the covariance matrix of volume fluctuations around the semi - classical distribution and inverting this matrix we can determine the matrix of second derivatives of the effective action , assuming that higher - order terms ( higher than second order in fluctuations ) can be neglected . this method was successfully applied in @xcite . indeed the form ( 10 ) of the effective action was confirmed , at least in the range of large volumes , permitting us to determine the physical parameters @xmath61 and @xmath62 . the parameter @xmath62 was particularly difficult to measure , since the corresponding term in the action ( after differentiating it twice ) falls off very fast with the spatial volume . on the other hand , small volumes can be expected to be ( and are in fact ) very sensitive to finite - size effects and lattice artefacts . the numerical experiment described above produces values of the physical parameters @xmath63 as functions of the bare couplings @xmath41 . these physical parameters have a direct interpretation in terms of the gravitational constant @xmath64 ( up to the dimensionful parameters @xmath65 and @xmath5 ) . from a practical point of view the determination of these parameters becomes more difficult near the phase transitions , where we observe a critical slowing - down and large finite - size effects . as we observed , the neighbourhood of the phase transitions is particularly interesting from a physical point of view . approaching these lines we see that the parameter @xmath66 ( or @xmath67 ) @xcite , which can be interpreted as the limit where the lattice spacing approaches zero . in this limit we may hope to observe genuine quantum effects of gravity . the form of the effective action ( 10 ) suggests a formal decomposition of the quantum amplitude ( 4 ) in the simplified form @xmath68 where we have introduced the _ effective _ projection operators @xmath69 on the space of states with a fixed volume @xmath10 . the projection operators behave as genuine projection operators on a single state @xmath70 . they can be used to study the properties of the cdt geometry , assuming that the elements of the transfer matrix @xmath71 in ( 4 ) depend only on volume . we can check to what extent this is true . in the proposed approach @xcite we determine directly the elements @xmath72 using numerical simulations of periodic systems with very small time extent . the method is based on the observation that terms in the sum ( 11 ) have the interpretation ( up to a normalization ) of the probability to measure a particular sequence of volumes . for a system with periodicity 2 and periodic boundary conditions @xmath73 by measuring the number of times a particular set @xmath74 appears in the simulation we determine the matrix element @xmath75 . in practice the method is more complicated , because we also want to study a particular range of @xmath74 . details of the method are explained in @xcite . on the fig . [ fig06 ] we show the logarithm of the transfer matrix , obtained by gluing together the results of measurements at the neighbouring ranges of volume . on the plot we see the gaussian behaviour of the off - diagonal _ kinetic _ term and the diagonal _ potential _ terms . the kinetic term corresponds to the first line in ( 10 ) and the potential term to the second line . we can easily measure the parameters of the effective action and find consistency with the values determined by the indirect method described before . the advantage of the new approach is a much smaller numerical error and at the same time a much shorter computer time needed to perform the measurements . the presented plot corresponds to one particular point on the @xmath41 plane , well inside the de sitter phase . we are currently measuring the behaviour of physical parameters @xmath63 in the whole range of the c ( de sitter ) phase , in particular near the phase transitions . preliminary results confirm that @xmath66 at the a - c transition and inside the a phase and indicate that @xmath76 changes sign at the b - c transition lines . details of this behaviour are crucial to determine and understand the critical behaviour near the phase transition and the critical scaling properties of the model . particularly interesting is the perspective to study the neighbourhood of the triple point , where the three phases meet . the cdt model allows us to study properties of the lattice regularized quantum theory of geometry in a wick - rotated formulation ( imaginary time ) . obvious questions about the full properties of the model under analytic continuation to real time remain open . some features of the model are however common to a formalism with real and imaginary time . one important property is the crucial role played by the entropy of configurations , an aspect which is usually not appreciated in mini - superspace - type models . our approach is based on integrating out all degrees of freedom , apart from a finite set and permits us to study the true effective model of the scale factor . ja and ag thank the danish research council for financial support via the grant `` quantum gravity and the role of black holes '' and eu for support from the erc - advance grant 291092 , `` exploring the quantum universe '' ( equ ) . jj acknowledges partial support of the international phd projects programme of the foundation for polish science within the european regional development fund of the european union , agreement no . jg - s acknowledges the polish national science centre ( ncn ) support via the grant 2012/05/n / st2/02698 . m. reuter and f. saueressig : _ functional renormalization group equations , asymptotic safety , and quantum einstein gravity _ [ 0708.1317 , hep - th ] , + d.f . litim : _ fixed points of quantum gravity _ , phys . rev . lett . 92 ( 2004 ) 201301 [ hep - th/0312114 ] , j. ambjrn , s. jordan , j. jurkiewicz and r. loll : _ a second - order phase transition in cdt , _ phys . ( 2011 ) 211303 [ hep - th/1108.3932 ] , + j. ambjrn , s. jordan , j. jurkiewicz and r. loll : _ second- and first - order transitions in cdt , _ phys . * d85 * ( 2012 ) 124044 [ hep - th/1205.1229 ] . j. ambjrn , a. grlich , j. jurkiewicz , r. loll , j. gizbert - studnicki , t. trzesniewski : _ the semiclassical limit of causal dynamical triangulations , _ nucl . phys . b 849 ( 2011 ) 144 - 165 [ hep - th/1102.3929 ] .

the causal dynamical triangulation model of quantum gravity ( cdt ) is a proposition to evaluate the path integral over space - time geometries using a lattice regularization with a discrete proper time and geometries realized as simplicial manifolds . the model admits a wick rotation to imaginary time for each space - time configuration . using computer simulations we determined the phase structure of the model and discovered that it predicts a de sitter phase with a four - dimensional spherical semi - classical background geometry . the model has a transfer matrix , relating spatial geometries at adjacent ( discrete lattice ) times . the transfer matrix uniquely determines the theory . we show that the measurements of the scale factor of the ( cdt ) universe are well described by an effective transfer matrix where the matrix elements are labelled only by the scale factor . using computer simulations we determine the effective transfer matrix elements and show how they relate to an effective minisuperspace action at all scales . address = the niels bohr institute , copenhagen university , blegdamsvej 17 , dk-2100 copenhagen , denmark . , altaddress = radboud university , nijmegen , institute for mathematics , astrophysics and particle physics , heyendaalseweg 135 , 6525 aj nijmegen , the netherlands address = institute of physics , jagiellonian university , reymonta 4 , pl 30 - 059 krakow , poland address = institute of physics , jagiellonian university , reymonta 4 , pl 30 - 059 krakow , poland , altaddress = the niels bohr institute , copenhagen university , blegdamsvej 17 , dk-2100 copenhagen , denmark . address = institute of physics , jagiellonian university , reymonta 4 , pl 30 - 059 krakow , poland address = radboud university , nijmegen , institute for mathematics , astrophysics and particle physics , heyendaalseweg 135 , 6525 aj nijmegen , the netherlands

et al _ @xcite initiated the application of the `` partial wave - cutoff method '' , to be explained below , to the important class of @xmath0 symmetric fields first introduced by s. l. adler @xcite . we will work in euclidean metric with @xmath1 where @xmath2 is a t hooft symbol , @xmath3 and @xmath4 . in general , @xmath5 may be any arbitrary spherically symmetric function . however , the profile we have chosen for @xmath5 has the following important properties : * @xmath6 is finite . * @xmath7 , which is what we need to see the chiral anomaly term @xmath8 . according to m. fry @xcite , the following general remarks hold for the spinor qed effective action in the background ( [ defbackground ] ) with @xmath9 : let @xmath10 denote the ( scheme independent ) effective action obtained after subtraction of the two - point contribution . it behaves for small @xmath11 as @xmath12 the logarithmic term is determined entirely by the chiral anomaly , @xmath13 after decomposing the negative chirality part of the dirac operator into partial - wave radial operators with quantum numbers @xmath14 and @xmath15 , the corresponding effective action is : @xmath16 we concentrate on the negative chirality sector of the spinor effective action where @xmath17 is the degeneracy factor , and the @xmath18 sum comes from adding the contributions of each spinor component . the partial - wave cutoff method separates the sum over the quantum number @xmath14 into a low partial - wave contribution , each term of which is computed using the ( numerical ) gelfand - yaglom method , and a high partial - wave contribution , whose sum is computed analytically using wkb . then we apply a regularization and renormalization procedure and combine these two contributions to yield the finite and renormalized effective action . the gelfand - yaglom method @xcite , can be summarized as follows : let @xmath19 and @xmath20 denote two second - order radial differential operators on the interval @xmath21 and let @xmath22 and @xmath23 be solutions to the initial value problem @xmath24 then the ratio of the determinants is given by @xmath25 in our case @xmath26 \right ) \phi_{-}(r ) & = & 0 \ , . \nonumber\end{aligned}\ ] ] the high - mode contribution , which remains to be calculated calculated using wkb , is @xmath27 for the class of backgrounds considered here , the partial - wave - cutoff method works well for any value of the mass up to numerical accuracy . the effective action calculated as above is finite for any non - zero value of the mass . when we use on - shell ( ` os ' ) renormalization ( @xmath28 ) , its leading small - mass behavior contains the logarithmically divergent term @xcite @xmath29 thus for the study of this small @xmath11 regime we introduce a modified effective action , @xmath30 it turns out that @xmath31 is finite for @xmath32 , which supports fry s conjecture , mentioned above , for the case of the backgrounds with @xmath33 ( where the chiral anomaly term is absent ) . in fig . [ fig1 ] we contrast both variants of the effective action for the scalar qed case ( see @xcite for the fermionic case which is very similar ) . in this section we exhibit the leading and subleading terms in the inverse mass (= heat kernel ) expansion of the one - loop scalar qed effective action . the first two terms are ( we calculated them using the worldline formalism along the lines of @xcite ) @xmath34 where the coefficients in the limit @xmath35 are , up to cubic order in @xmath36 , @xmath37 the large - mass behavior of the effective action is shown in fig . [ fig2 ] for the scalar qed case ( see @xcite for the fermionic case ) . in this section we show that the four - point contribution to the effective action in the `` standard '' @xmath0 symmetric background , ( [ defbackground ] ) with @xmath9 and @xmath38 , is finite in the massless limit . this is a detail of some importance for fry s investigation that had been missing in the analysis of @xcite , although it has been anticipated in @xcite . in the worldline formalism , we can write this quartic contribution to the effective action as ( in either scalar or spinor qed ) @xmath39 = -\prod_{i=1}^4 \int \frac{d^4k_i}{(2\pi)^4}\bar a(k_i^2 ) ( 2\pi)^4\delta^4(\sum k_i ) \gamma[k_1,\varepsilon_1;\cdots;k_4,\varepsilon_4]\ ; , \label{gamma4fin}\end{aligned}\ ] ] where @xmath40 is the worldline path integral representation of the off - shell euclidean four - photon amplitude and @xmath41 where @xmath42 is the modified bessel function of the second kind . after performing the path integral , suitable integrations by parts , a rescaling @xmath43 and the elimination of the global @xmath44 integral , we obtain ( see @xcite for details ) @xmath45 = -\frac{e^4}{(4\pi)^2 } \int_0 ^ 1 du_1du_2 du_3 du_4\ \frac{q_4(\dot g_{b12},\ldots,\dot g_{b34 } ) } { \bigl(m^2 -\frac{1}{2 } \sum_{i , j=1}^4 g_{bij}k_i\cdot k_j\bigr)^2}\;. \label{4photfin}\end{aligned}\ ] ] here @xmath46 is the worldline green s function and @xmath47 its derivative . @xmath48 is a polynomial in the various @xmath49 s , as well as in the momenta and polarizations . now , the qed ward identity implies that the rhs of ( [ 4photfin ] ) is @xmath50 in each of the four momenta , which can also be easily verified using properties of the numerator polynomial @xmath48 . using this fact and ( [ gamma4fin ] ) we see that there is no singularity at @xmath51 , and convergence at large @xmath52 . we have continued and extended here the full mass range analysis of the scalar and spinor qed effective actions for the @xmath0 symmetric backgrounds , started in @xcite , by a more detailed numerical study of both the small and large mass behaviors . in @xcite only the unphysically renormalized versions @xmath53 of these effective actions were considered ( corresponding to @xmath54 ) , which are appropriate for the small mass limit , but have a logarithmic divergence in @xmath11 in the large @xmath11 limit . here we have instead used the physically renormalized effective actions @xmath55 for the study of the large mass expansions , which made it possible to achieve a numerical matching of both this leading and even the subleading term in the inverse mass expansions of the effective actions . in our study of the small mass limit , we have improved on @xcite by obtaining good numerical results for @xmath53 even at @xmath56 , and showing continuity for @xmath57 for various values of @xmath36 . moreover , we have presented numerical evidence that @xmath58 stays finite even in the limit @xmath59 . this fact is important in the spinor case , where it supports indirectly fry s conjecture @xcite that , for the case at hand , the only source of a divergence of @xmath53 for @xmath60 at @xmath57 should be the chiral anomaly term . as a side result , we have proved the finiteness of the massless limit four - point contribution to the effective action in scalar and spinor qed for the standard @xmath0 symmetric background ( @xmath60 , @xmath61 ) . + more details and results for the spinor qed case will be given in a forthcoming publication @xcite . 99 g. v. dunne , j. hur , c. lee and h. min , phys . lett . * 94 * , 072001 ( 2005 ) , arxiv : hep - th/0410190 . g. v. dunne , j. hur , c. lee and h. min , phys . d * 71 * , 085019 ( 2005 ) , arxiv : hep - th/0502087 . g. v. dunne , j. hur and c. lee , phys . d * 74 * , 085025 ( 2006 ) , arxiv : hep - th/0609118 . g. v. dunne , a. huet , j. hur and h. min , phys . d * 83 * , 105013 ( 2011 ) . s. l. adler , phys . d * 6 * , 3445 ( 1972 ) ; erratum - ibid . d * 7 * , 3821 ( 1973 ) . s. l. adler , phys . d * 10 * , 2399 ( 1974 ) ; erratum - ibid . d * 15 * , 1803 ( 1977 ) . m. p fry , phys . d * 75 * , 065002 ( 2007 ) , hep - th/0612218 ; erratum - ibid . d * 75 * 069902 ( 2007 ) . m. p fry , phys . d * 81 * , 107701 ( 2010 ) . n. ahmadiniaz , a. huet , a. raya and c. schubert , in preparation m.g . schmidt and c. schubert , phys . b * 318 * , 438 ( 1993 ) , hep - th/9309055 . c. schubert , phys . * 355 * , 73 ( 2001 ) , arxiv : hep - th/0101036 .

an interesting class of background field configurations in qed are the @xmath0 symmetric fields . those backgrounds have some instanton - like properties and yield a one - loop effective action that is highly nontrivial but amenable to numerical calculation , for both scalar and spinor qed . here we report on an application of the recently developed `` partial - wave - cutoff method '' to the numerical analysis of both effective actions in the full mass range . in particular , at large mass we are able to match the asymptotic behavior of the physically renormalized effective action against the leading two mass levels of the inverse mass ( or heat kernel ) expansion . at small mass we obtain good numerical results even in the massless case for the appropriately ( unphysically ) renormalized effective action after the removal of the chiral anomaly term through a small radial cutoff factor . in particular , we show that the effective action after this removal remains finite in the massless limit , which also provides indirect support for m. fry s hypothesis that the qed effective action in this limit is dominated by the chiral anomaly term . = 11.6pt

the rcb stars are a small group of hydrogen - deficient carbon - rich supergiants which undergo spectacular declines in brightness of up to 8 magnitudes at irregular intervals ( clayton 1996 ) . rcb star atmospheres are extremely deficient in hydrogen but very rich in carbon . dust is apparently forming within a couple of stellar radii of the stars , which have @xmath5 k. rcb stars are very rare . only about 35 are known in the galaxy ( clayton 1996 ) . their rarity may stem from the fact that they are in an extremely rapid phase of the evolution toward white dwarfs . understanding the rcb stars is a key test for any theory that aims to explain hydrogen deficiency in post - asymptotic giant branch stars . there are two major evolutionary models for the origin of rcb stars : the double degenerate and the final helium shell flash ( iben et al . 1996 ) . the former involves the merger of two white dwarfs , and in the latter a white dwarf / evolved planetary nebula ( pn ) central star is blown up to supergiant size by a final helium flash . in the final flash model , there is a close relationship between rcb stars and pn . this connection has recently become stronger , since the central stars of three old pn s ( sakurai s object , v605 aql and fg sge ) have been observed to undergo final - flash outbursts which transformed them from hot evolved central stars into cool giants with the spectral properties of rcb stars ( kerber et al . 1999 ; asplund et al . 1999 ; clayton & de marco 1997 ; gonzalez et al . two of these stars , fg sge and sakurai s object are in an rcb - like phase at present . during a decline , a cloud of carbon - rich dust forms along the line of sight , eclipsing the photosphere , and revealing a rich emission - line spectrum made up primarily of neutral and singly ionized species . the emission lines suggest at least two temperature regimes ; a cool ( @xmath25000 k ) inner region likely to be the site of neutral and singly - ionized species producing a narrow - line spectrum , and a much hotter outer region indicated by the presence of broad emission lines such as c iii @xmath01909 , c iv @xmath01550 and he i @xmath01083010830 line is a triplet . the vacuum wavelengths of the triplet are 1.083206 , 1.083322 and 1.083331 . vacuum wavelengths are plotted in the figures . however , we will continue with tradition and refer to this as the he i @xmath010830 line . ] ( wing et al . 1972 ; querci & querci 1978 ; zirin 1982 ; clayton et al . 1992 ; lawson et al . 1999 ) . other broad lines , such as na i d and ca ii h & k , imply a cooler region . the possible detection of c iv @xmath01550 implies the presence of a transition region with an electron temperature @xmath6 k ( jordan & linsky 1987 ) . but rcb stars do not exhibit a normal chromospheric spectrum so the presence of high excitation lines in the cool rcb stars has been perplexing . he i @xmath010830 was detected in r crb thirty years ago , when it was just below maximum light ( wing et al . 1972 ; querci & querci 1978 ; zirin 1982 ) . the line showed a p - cygni profile with a violet displacement of more than 200 km @xmath7 . this line is similar to that measured for sakurai s object in its rcb - phase ( eyres et al . 1999 ) . since 1978 , the he i @xmath010830 line in r crb has been observed only once . it was seen strongly in emission while r crb was recovering from a deep decline in 1996 ( rao et al . no further observations of he i @xmath010830 in rcb stars exist in the literature . observations of he i @xmath010830 in ten rcb stars were obtained on 15 june 2001 at ukirt using the grating spectrometer cgs4 with the echelle grating and a 09 slit . the two - pixel resolution , matching the slit width , was 0.5 ( 14 km s@xmath1 ) . the stars were ratioed with comparison stars to remove telluric features . the flux calibrations were done using standards with colors from koornneef ( 1984 ) . wavelength calibration was achieved using telluric absorption lines observed in the comparison stars . a quadratic fit was made to a selection of these lines covering the entire spectral range . the 1-@xmath8 wavelength uncertainty is 0.000005 0.000008 . the observed sample is listed in table 1 . the spectra , slightly smoothed to a resolution of 0.65 ( 18 km s@xmath1 ) , are shown in figure 1 . the rcb stars range in effective temperature from 5000 to 7000 k. their spectra have very different appearances depending on whether the rcb star is warm ( t@xmath9 = 60008000 k ) or cool ( t@xmath9 @xmath10 6000 k ) ( asplund et al . 2000 ) . in figure 1 , the warm star spectra are in the lefthand column and the cool star spectra in the righthand column . the warmer stars show mainly atomic absorptions of c i and singly ionized metals while the cooler stars , in addition , show strong c@xmath11 and cn absorption bands . see figure 2 . the line identifications are from hirai ( 1974 ) , and hinkle , wallace , & livingston ( 1995 ) . he i @xmath010830 is present in the spectra of all of our sample stars . it varies significantly in strength and shape , from a fully developed p - cygni profile ( es aql ) to blue - shifted asymmetric absorptions with small emission components , to even more complex structures . in all cases , the line indicates a mass outflow - with a range of intensity and velocity . given the temperatures of the rcb stars , it might be expected that the photospheric component of he i @xmath010830 would be small . however , absorption features of he i @xmath05876 are present in several rcb stars at velocities consistent with a photospheric origin ( rao & lambert 1996 ; asplund et al . 2000 ) . a photospheric component of the he i @xmath010830 line is not clearly present in any star in our sample . the p - cygni - type line profiles in our sample were modeled to obtain quantitative information on the mass loss and the outflow velocity . a detailed discussion of the individual objects is given below . the results are compiled in table [ tab_seifit ] and shown in figures [ fig_seifit_es ] and [ fig_seifit ] . model profiles were computed with the sei ( `` sobolev plus exact integration '' ) code , orginally developed by lamers , cerruti - sola , & perinotto ( 1987 ) for wind lines of hot stars in the uv range , and subsequently extended to the analysis of h@xmath12 lines ( bianchi et al . the code calculates the source function with the escape probability method and the exact solution of the transfer equation . we follow the notation used by bianchi , vassiliadas , & dopita ( 1997 ) and bianchi et al . the velocity of the outflow increases outwards - following a law characterized by the exponent @xmath13 - until a terminal velocity ( ) is reached ( eq . ( 1 ) of bianchi et al . 2000 ) . in all profiles analyzed , we found a rather steep acceleration , with @xmath13 @xmath14 2 . terminal velocities are a few hundred km s@xmath1 . in many cases , the profiles are complex and blend with other photospheric absorption lines particularly in the cooler rcb stars , making an accurate estimation of the parameters difficult . in figure 3 , we show an example of a profile that can be fit very successfully using the sei method . we also show , in figure 4 , other profiles for which sei is successful . in some cases , the he i absorption trough appears to be almost saturated , making the method insensitive to the measurement of the total optical depth . several stars could not be fit successfully bacuse of saturated or complex absorption profiles . the optical depth values obtained from the sei analysis indicate column densities of the he i outflow higher than 10@xmath15 @xmath4 for all objects . es aql is the only member of the observed sample presenting a classical p - cygni profile . its he i @xmath010830 spectral region is affected by photospheric absorptions but much less than any other star in the sample . therefore , the location of the continuum and the fit of the he i line profile were rather accurate . the best fit profile is shown in figure [ fig_seifit_es ] and the corresponding parameters are given in table [ tab_seifit ] . the uncertainties in the total optical depth and terminal velocity ( ) were estimated by computing several profiles , varying the parameters around the best fit solution , and narrowing the range of acceptable solutions considering both the s / n of the observed profile and the uncertainty in the continuum location . in ry sgr and r crb , the he i absorption is much broader than the emission , unlike in a classical p - cygni profile . the he i line profile fit is affected by two photospheric lines , si i @xmath010830.1 and another line which may be a blend of s i @xmath010824.2 and cr i @xmath010824.6 ( see figure [ fig_seifit ] ) . the emission is partially masked by photospheric absorption . the derived parameters are much more uncertain in this case than for the pure p - cygni profile of es aql . two acceptable fits are shown for ry sgr in figure [ fig_seifit ] - with widely varying exponents of the optical depth law . therefore , the optical depth remains very uncertain . wx cra and v517 oph have very similar he i line profiles . the flat bottoms of the absorptions indicate that the line is saturated and/or blended with other lines , both factors preventing an accurate measurement of its optical depth . the blue - shifted part of the absorptions and the extreme red wings of the emission are well fit by p - cygni profiles , provided a turbulence of about 20 - 30 % of the terminal velocity is included ( somewhat higher than the 10% value which is typical for radiation pressure winds of hot stars ) . for these cooler rcb stars , many additional absorptions due to cn are obviously present , partially masking the emission . we nonetheless attempted to fit the blue wings of the absorptions to obtain an estimate of the wind velocities . although the line shapes are similar for these two stars , the velocity structures are different . the model profile adopted for wx cra ( @xmath2225 km s@xmath1 ) also fits the blue wing of the v517 oph ( @xmath2300 km s@xmath1 ) profile . however , the analysis indicates optical depths of @xmath16 @xmath171 , and velocities for the he i shells in the range found for other sample objects . the line profiles of sv sge and u aqr are also similar to each other , but in the spectrum of u aqr the absorption is narrower , implying a lower outflow velocity . the line has a smaller displacement blueward than the previous cases . u aqr is located in the halo and has unusual abundances even for an rcb star ( bond , luck , & newman 1979 ) . the weakest he i features are seen in v854 cen , v cra and v482 cyg . see figure 1 . the p - cygni profile is present in v cra with the si i line cutting through the middle of it . also , v482 cyg seems to have its absorption hiding in a broadened si i line . only v854 cen shows little or no absorption , although there may be a weak broad absorption which is very blue - shifted . the lower state of the he i @xmath010830 transition is 20 ev above the ground state . this state is metastable as its transition probability is very small ( a@xmath18=1.27 x 10@xmath19 s@xmath1 ) ( sasselov & lester 1994 ) . it can be populated by two mechanisms , photoionization / recombination or collisional excitation . he ii @xmath01640 is not seen in rcb stars ( clayton 1996 ; lawson et al . 1999 ) . this indicates that the he i @xmath010830 line is not being formed by helium photoionization and recombination ( rossano et al . rao et al . ( 1999 ) observed the he i lines in emission at 3889 , 5876 , 7065 and 10830 during the 1995 - 96 decline of r crb . the 5876 line was much weaker than the 3889 and 7065 lines . this can be explained if the lines are optically thick and the electron density is high . we can not calculate density since we only have data on the 10830 line but rao et al . estimate t @xmath2 20000 k and @xmath20 = 10@xmath2110@xmath3 @xmath22 . in this regime , collisional excitation is important . it has been suggested that the he i @xmath010830 line seen in sakurai s object is the result of collisional excitation in shocked gas being dragged outward by the expanding dust cloud around the star ( eyres et al . 1999 ; tyne et al . the dust formation and expansion by radiation pressure is thought to be quite similar in the rcb stars . in these fast - moving clouds , excitation might take place through atomic collisions or shocks ( feast 2001 ) . blue - shifted high velocity absorption features ( 100 - 400 km s@xmath1 ) have been seen from time to time in the broad - line emission spectra of rcb stars both early in declines and just before return to maximum light ( alexander et al . 1972 ; cottrell , lawson , & buchhorn 1990 ; clayton et al . 1992 , 1993 , 1994 ; vanture & wallerstein 1995 ; rao & lambert 1997 ; goswami et al . the blue - shifted absorptions can be understood if the dust , once formed , is blown away from the star by radiation pressure , eventually dissipating and allowing the stellar photosphere to reappear . the gas is dragged along with the dust moving away from the star . the velocities seen in the p - cygni profiles agree well with those measured for the blue - shifted absorption features . there is a possible relationship between the lightcurve of an rcb star , which represents the mass - loss history of the star , and the he i @xmath010830 profile . the last column of table 1 lists the state of each rcb star when the he i spectra were obtained . none of the stars were in a deep decline but half of the stars ( v517 oph , wx cra , es aql , sv sge and u aqr ) were below maximum light and in the late stages of a decline . all of these stars show strong p - cygni or asymmetric blue - shifted profiles . in addition , both r crb and ry sgr , which show fairly strong profiles , had just recently returned to maximum light . two stars , v cra and v482 cyg , show weaker profiles . these stars had been continuously at maximum light for 800 and 1400 days , respectively , at the time the spectra were taken . v482 cyg , which had gone the longest without a decline , has the weakest he i feature in the sample . an exception to this trend is v854 cen . its spectrum shows no sign of he i except possibly a weak broad absorption . like r crb and ry sgr , v854 cen was just out of a decline , so one might have predicted a strong profile . v854 cen is an unusual rcb star . it is extremely active . along with v cra , which also has a weak he i line , v854 cen is a member of the minority abundance group of the rcb stars , and has a relatively high hydrogen abundance ( asplund et al . 2000 ) . however , helium is still the dominant element so it is not clear why the p - cygni profile is so weak in these stars . v482 cyg is not a member of this minority group . of the ten stars , only r crb has been measured in the he i @xmath010830 line at more than one epoch . in addition to the june 2001 observation reported here , it was also observed in march and may 1972 ( 6 mag below maximum light ) , january 1978 ( 1 mag below maximum light ) , july 1978 ( at maximum light for 100 days ) , and may 1996 ( 3.5 mag below maximum light ) ( wing et al . 1972 ; querci & querci 1978 , zirin 1982 , rao et al . 1999 ) . the he i @xmath010830 line is present at all four epochs . it is a p - cygni profile in january 1978 , a blue - shifted absorption in july 1978 , a strong emission line in may 1996 and a p - cygni profile in june 2001 . the value of seems to be fairly constant at @xmath2200 - 240 km s@xmath1 . the p - cygni or asymmetric blue - shifted profile in r crb , at least , seems to be present at all times , whether the star is at maximum light or in decline . the he i @xmath010830 line is a key to understanding the evolution of rcb stars and the nature of their of mass - loss . the observed line profiles can help distinguish between the final flash and double degenerate models for the evolution of rcb stars . in the double degenerate scenario , an rcb star is unlikely to be a binary ; thus the detection of a companion would favor the final flash scenario . rao et al . ( 1999 ) have made the suggestion that rcb stars are binaries and that the higher temperature lines are formed in an accretion disk wind around a white dwarf companion . they suggest that the he i lines may arise from the inner regions of the accretion disk while lower excitation lines such as the na i d lines would then arise in the outer parts of the disk . no evidence for binarity has ever been found in the rcb stars . if the rcb stars are single stars with mass - loss similar to sakurai s object , we expect to see p - cygni - type profiles , while if we are viewing an accretion disk directly , we expect to see pure emission lines . a third possibility exists where the accretion disk is seen through the material being lost by a cool companion . in this case , the profile would vary with orbital phase . in these he i profiles , we can see the mass - loss from the rcb stars for the first time . it has long been suggested that when dust forms around an rcb star , radiation pressure accelerates the dust away from the star dragging the gas along with it . but until now , we have only been able to measure the dust . the he i @xmath010830 profiles measured here allow us to study the velocity structure and optical depth of the gas escaping from the star . we plan to monitor these stars to see how the column densities and velocities vary with time and how they are related to the dust formation episodes . the united kingdom infrared telescope is operated by the joint astronomy centre on behalf of the u.k . particle physics and astronomy research council . trg s research is supported by the gemini observatory , which is operated by the association of universities for research in astronomy , inc . , on behalf of the international gemini partnership . gcc appreciates the hospitality of the australian defence force academy and the mount stromlo and siding springs observatories . lb acknowledges support from nasa grant nra-99 - 01-ltsa-029 . we thank albert jones for providing photometry of the rcb stars . we also thank the referee for useful suggestions . alexander , j. b. , andrews , p. j. , catchpole , r. m. , feast , m. w. , lloyd evans , t. , menzies , j. w. , wisse , p. n. j. , & wisse , m. 1972 , mnras 158 , 305 asplund , m. , gustafsson , b. , rao , n.k . , & lambert , d.l . 1998 , a&a , 332 , 651 asplund , m. , gustafsson , b. , rao , n.k . , lambert , d.l . , & rao , n.k . 2000 , a&a , 353 , 287 asplund , m. , lambert , d.l . , kipper , t. , pollacco , d. , & shetrone , m.d . 1999 , a&a , 343 , 507 bianchi , l. et al . 2000 , , 538 , l57 bianchi , l. , lamers , h.j.g.l.m . , hutchings , j.b . , massey , p. , kudritzki , r. , herrero , a. , & lennon , d.j . 1994 , a&a , 292 , 213 bianchi , l. , vassiliadis , e. , & dopita , m. 1997 , , 480 , 290 bond , h.e . , luck , r.e . , & newman , m.j . 1979 , apj , 233 , 205 clayton , g.c . 1996 , pasp , 108 , 225 clayton , g. c. , & de marco , o. 1997 , aj , 114 , 2679 clayton , g. c. , lawson , w.a . , cottrell , p.l . , whitney , b. a. , stanford , s. a. , & de ruyter , f. 1994 , apj , 432 , 785 clayton , g. c. , lawson , w.a . , whitney , b. a. , & pollacco , d.l . 1993 , mnras , 264 , p13 clayton , g. c. , whitney , b. a. , stanford , s. a. , drilling , j. s. , & judge , p. g. 1992 , apj , 384 , l19 cottrell , p. l. , lawson , w. a. , & buchhorn , m. 1990 , mnras , 244 , 149 eyres , s. p. s. , smalley , b. , geballe , t. r. , evans , a. , asplund , m. , & tyne , v. h. 1999 , mnras , 307 , l11 feast , m. 2001 , in eta carina and other mysterious stars : the hidden opportunities of emission spectroscopy , , asp conf . 242 , p. 381 gonzalez , g. , lambert , d.l . , wallerstein , g. , rao , n.k . , smith , v.v . , & mccarthy , j.k . 1998 , apjs , 114 , 133 goswami , a. , rao , n.k , lambert , d.l , & smith , v.v . 1997 , pasp , 109 , 270 hinkle , k. , wallace , l , & livingston , w. 1995 , pasp , 107 , 1042 hirai , m. 1974 , pasj , 26 , 163 iben , i. , tutukov , a. v. , & yungelson , l. r. 1996 , apj , 456 , 750 jordan , c. , & linsky , j. l. 1987 , exploring the universe with the _ iue _ satellite , y. kondo , dordrecht : kluwer , 259 kerber , f. , koppen , j. , roth , m. , & trager , s.c .. 1999 , a&a , 344 , l79 koornneef , j. 1984 , a&a , 128 , 84 lamers , h. , cerruti - sola , m. & perinotto , m. 1987 , , 314 , 726 lawson , w.a . , et al . 1999 , aj , 117 , 3007 rao , n.k . , & lambert , d.l . 1996 , in hydrogen deficient stars , asp conf . 96 , 43 rao , n.k . , & lambert , d.l . 1997 , mnras , 284 , 489 rao , n.k . et al . 1999 , mnras , 310 , 717 rossano , g.s . , rudy , r.j . , puetter , r.c . , & lynch , d.k . 1994 , aj , 107 , 1128 querci , m. , & querci , f. 1978 , a&a , 70 , l45 sasselov , d.d . , & lester , j.b . 1994 , apj , 423 , 785 tyne , v.h . , eyres , s.p.s . , geballe , t.r . , evans , a. , smalley , b. , duerbeck , h.w . , & asplund , m. 2000 , mnras , 315 , 595 vanture , a.d . , & wallerstein , g. 1995 , pasp , 107 , 244 wing , r. f. , baumert , j. h. , strom , s. e. , & strom , k. m. 1972 , pasp , 84 , 646 zirin , h. 1982 , apj , 260 , 655 lllllll es aql&11.7&cool&355@xmath2315 & 1.8@xmath230.2 & 0.4@xmath230.05&3 mag below max + ry sgr&6.2&warm&275@xmath2350 & 5 - 20 : & 2@xmath231.5&just at end of last decline + u aqr&11.2&cool&175@xmath2340 & 2 - 5 : & 0.5@xmath230.3&2 mag below max + v517 oph&11.5&cool&300 : & & & 0.5 mag below max + wx cra&11.5&cool&225@xmath2330 & @xmath1750 & @xmath241:&0.5 mag below max + sv sge&11.0&cool&230@xmath2330 & 5@xmath233 & 0.7@xmath230.5&almost at end of last decline + r crb&5.8&warm&200 : & & & 100 d since end of last decline + vcra&10.0&warm&295:&&&800 d since end of last decline + v482 cyg&11.1&warm&260:&&&1400 d since end of last decline + v854 cen&7.0&warm&&&&just at end of last decline [ tab_seifit ]

we present new spectroscopic observations of the he i @xmath010830 line in r coronae borealis ( rcb ) stars which provide the first strong evidence that most , if not all , rcb stars have winds . it has long been suggested that when dust forms around an rcb star , radiation pressure accelerates the dust away from the star , dragging the gas along with it . the new spectra show that nine of the ten stars observed have p - cygni or asymmetric blue - shifted profiles in the he i @xmath010830 line . in all cases , the he i line indicates a mass outflow - with a range of intensity and velocity . around the rcb stars , it is likely that this state is populated by collisional excitation rather than photoionization / recombination . the line profiles have been modeled with an sei code to derive the optical depth and the velocity field of the helium gas . the results show that the typical rcb wind has a steep acceleration with a terminal velocity of = 200 - 350 km s@xmath1 and a column density of n @xmath210@xmath3 @xmath4 in the he i @xmath010830 line . there is a possible relationship between the lightcurve of an rcb star and its he i @xmath010830 profile . stars which have gone hundreds of days with no dust - formation episodes tend to have weaker he i features . the unusual rcb star , v854 cen , does not follow this trend , showing little or no he i absorption despite high mass - loss activity . the he i @xmath010830 line in r crb itself , which has been observed at four epochs between 1978 and 2001 , seems to show a p - cygni or asymmetric blue - shifted profile at all times whether it is in decline or at maximum light .

@xmath0 nuclei ( nuclear systems with a bound anti - kaon ) have recently become a hot topic in hadron and nuclear physics . with a phenomenological @xmath2 potential , it was suggested that the @xmath0 nuclei could exist as deeply bound states with small width @xcite . experiments performed in search for such states have so far been inconclusive @xcite . an important prototype is the @xmath1 system , the simplest @xmath0-nuclear cluster . recently this system has been studied using faddeev @xcite and variational @xcite approaches with @xmath2 interactions constrained by scattering data and properties of the @xmath7 . an essential ingredient to study @xmath0 nuclei is the @xmath2 interaction below threshold , which is only accessible through the subthreshold extrapolation of the amplitude adjusted to @xmath2 scattering data . theoretical guidance is required for this extrapolation . here we report on the study of a variational calculation of @xmath1 system @xcite using the effective @xmath2 interaction based on chiral su(3 ) dynamics @xcite . the present variational investigation focuses on the @xmath1 system with spin and parity @xmath9 and isospin @xmath10 , where the parity assignment includes the intrinsic parity of the antikaon . our model wave function for this @xmath1 state , @xmath11 , has two components : @xmath12_{t_n=1 } \ , \bar{k } \ , \right]_{t=1/2 , t_z=1/2 } \right\rangle , \label{phi+ } \\ & & |\phi_- \rangle \equiv \phi_- ( \bm{r}_1 , \bm{r}_2 , \bm{r}_k ) \ ; \left| s_n = 0 \right\rangle \times \ ; \left| \ , \left [ \ , [ nn]_{t_n=0 } \ , \bar{k } \ , \right]_{t=1/2 , t_z=1/2 } \right\rangle , \label{phi-}\end{aligned}\ ] ] where @xmath13 is a normalization factor . the first , second and third terms in eqs . ( [ phi+ ] ) and ( [ phi- ] ) correspond to the spatial wave function , the spin wave function of the two nucleons ( assuming @xmath14 ) , and the isospin wave function of the total system , respectively . we consider two different isospin states of the two nucleons ( @xmath15 in @xmath16 and @xmath17 in @xmath18 ) , while the @xmath19 system in both cases has total isospin and third component @xmath10 . the dominant contribution is the @xmath15 component corresponding to the leading @xmath1 configuration . the mixing with the @xmath17 component is caused by the difference between the @xmath2 interactions in @xmath20 and @xmath21 . the spatial part of the wave functions are products of single particle wave packets and two - particle correlation functions . the @xmath22 correlation function permits an adequate treatment of a realistic @xmath22 potential with its strong short - range repulsion . the parameters in the model wave function are determined by minimization of the energy . the hamiltonian used in the present study is of the form @xmath23 where @xmath24(@xmath25 ) stands for the @xmath22(@xmath2 ) interaction . here @xmath26 is the total kinetic energy . the energy of the center - of - mass motion , @xmath27 , is subtracted . as a realistic nucleon - nucleon interaction @xmath24 we choose the argonne v18 potential ( av18 ) @xcite . we employ the central , @xmath28 and spin - spin parts of the av18 potential for the singlet - even ( @xmath29 ) and singlet - odd ( @xmath30 ) channel , since the total spin of the two nucleons is restricted to zero in our model . we use the @xmath2 interaction @xmath25 derived from chiral su(3 ) dynamics @xcite . this complex and energy - dependent interaction is parametrized by a gaussian spatial distribution : @xmath31 , \nonumber\end{aligned}\ ] ] where @xmath32 is the isospin projection operator for the @xmath2 pair . the interaction strength @xmath33 is a function of the center - of - mass energy variable @xmath34 of the @xmath2 subsystem . the strength @xmath35 and the range parameter @xmath36 are systematically determined within the chiral coupled - channel approach . the energy dependence of the @xmath2 interaction requires the self - consistency in the variational procedure @xcite . we introduce an auxiliary ( non - observable ) antikaon binding energy " @xmath37 to control the energy @xmath34 of the @xmath2 subsystem within the @xmath1 cluster . this @xmath37 is defined as @xmath38 where @xmath39 is the nucleonic part of the hamiltonian . the relation between the @xmath2 two - body energy @xmath34 and @xmath37 within the three - body system is not @xmath40 @xmath41 fixed . in general , @xmath34 can take values @xmath42 , where @xmath43 is a parameter describing the balance of the antikaon energy between the two nucleons of the @xmath19 three - body system . one expects @xmath44 . the upper limit ( @xmath45 ) corresponds to the case in which the antikaon field collectively surrounds the two nucleons , a situation encountered in the limit of static ( infinitely heavy ) nucleon sources . in the lower limit ( @xmath46 ) the antikaon energy is split symmetrically half - and - half between the two nucleons . we investigate both cases and label them type i " and type ii " , respectively : @xmath47 our calculation is then carried out such that self - consistency for @xmath34 is achieved , namely , the @xmath34 used in the effective @xmath2 potential is made to coincide with the @xmath34 evaluated with the finally obtained wave function . due to the elimination of the @xmath48 and @xmath49 channels , the effective @xmath2 potential is complex . we perform the variational calculation with the real part of the potential to obtain the wave function . the decay width is then calculated perturbatively by taking the expectation value of the imaginary part of the potential : @xmath50 , which represents the mesonic decay channels ( @xmath51 ) . the dispersive effect induced by the imaginary part of the potential and the non - mesonic absorption width for @xmath52 are treated separately . relative density in @xmath1 for a chiral model @xcite with the type i ansatz . solid ( dashed ) line shows @xmath20 ( @xmath21 ) @xmath2 density . solid line with diamond shows the @xmath2 density of @xmath7 in the same model . all densities are displayed with @xmath53-multiplied . , title="fig:",scaledwidth=50.0% ] relative density in @xmath1 for a chiral model @xcite with the type i ansatz . solid ( dashed ) line shows @xmath20 ( @xmath21 ) @xmath2 density . solid line with diamond shows the @xmath2 density of @xmath7 in the same model . all densities are displayed with @xmath53-multiplied . , title="fig:",scaledwidth=45.0% ] here we present the results of the variational calculation . for an estimate of theoretical uncertainties , we have used four effective @xmath2 potentials derived from different versions of chiral models , and employed both the type i and the type ii ansatz , as described in ref . @xcite . in all cases the @xmath1 system turns out to be rather weakly bound as compared to previous calculations . as shown in the left panel of fig . [ fig:1 ] , the total binding energies range from 17 mev to 23 mev , and the mesonic decay width @xmath54 ( @xmath55 ) lies between 40 and 70 mev . the different versions of the chiral models give similar results within a relatively small window of uncertainties , while type ii ansatz gives slightly deeper binding than type i by a few mev . the reason for the shallow binding is found in the relatively weak @xmath2 potentials based on chiral dynamics . the chiral low energy theorem in the su(3 ) meson - baryon sector dictates strong @xmath48 attraction , and the coupled - channel dynamics locates the resonance structure in the @xmath2 amplitude at 1420 mev , displaced from the 1405 mev measured in the @xmath48 spectrum . the binding energy of the isolated @xmath6 system is about 12 mev measured from @xmath2 threshold . table [ tab:1 ] shows a typical result of @xmath1 calculated with a chiral model @xcite and the type i ansatz . the results of the other cases under study are essentially the same . the mean distance between two nucleons , @xmath56 , is about 2.2 fm which is smaller than that of the deuteron ( about 4 fm ) and close to the @xmath22 distance in normal nuclei , but the system is obviously not much compressed . it is interesting to compare the @xmath6 component in @xmath1 with the @xmath7 as the @xmath6 two - body quasibound state . the mean distance of the @xmath2 pair in @xmath1 is found to be close to that for @xmath7 , namely @xmath57 fm and @xmath58 fm . calculating the expectation value of the relative @xmath2 orbital angular momentum , it turns out that the @xmath6 pair is dominated by @xmath59-wave , just as the @xmath2 pair forming the @xmath7 . the structure of the @xmath6 pair in the @xmath1 system is thus similar to that of the @xmath7 . the right panel in fig . [ fig:1 ] shows the @xmath2 relative density distribution of @xmath60 and 1 ) components extracted from @xmath1 which are normalized to compare with that of @xmath7 . apparently , the distribution of @xmath6 pair in @xmath1 is very similar to that of the @xmath2 two - body quasibound state . cc|cc|cc||cc b. e. ( @xmath1 ) & @xmath61 & @xmath56 & @xmath62 & @xmath63 & @xmath64 & b. e. ( @xmath65 ) & @xmath66 + 16.9 & 47.0 & 2.21 & 1.97 & 1.82 & 2.33 & 11.5 & 1.86 + based on the wavefunction obtained above , we estimate the following contributions to the results which have not been taken into account so far : 1 ) dispersive corrections by the imaginary part of the potential , 2 ) effect of the @xmath8-wave @xmath2 interaction , and 3 ) decay width from the two - nucleon absorption process @xcite . first , we consider the dispersive correction induced by the imaginary part of the @xmath2 potential . this effect can be calculated explicitly for the two - body @xmath2 system , by comparing the bound state solution of the real part of the potential with the resonance structure observed in the scattering amplitude with original complex potential . examining four chiral models , we find an attractive shift of the binding energy @xmath67 mev in the two - body @xmath2 system . we therefore estimate that the dispersive correction would add another @xmath68 mev to the binding energy of the @xmath1 system . secondly , the contribution of the @xmath8-wave @xmath2 interaction is estimated perturbatively with the @xmath8-wave @xmath2 interaction : @xmath69\ , \nabla~.\end{aligned}\ ] ] the coefficient @xmath70 is complex and a detailed expression is given in ref . a prominent feature in the @xmath8-wave interaction is the @xmath71 resonance below the threshold . since the @xmath1 system is weakly bound and the energy variable @xmath34 lies slightly above the @xmath71 resonance , the @xmath8-wave contribution to the binding energy is repulsive , about @xmath72 mev . the decay width is increased by 10 @xmath73 35 mev , because of the large imaginary part around the @xmath71 resonance structure . next , we estimate the contribution of the two - nucleon absorption process ( non - mesonic decay width , @xmath74 ) . the width is calculated with the correlated three - body density @xmath75 as @xmath76 this is a generalization of the formula for @xmath77 absorption on proton pairs in a heavy nucleus , where the coupling constant is constrained by a global fit to the kaonic atom data @xcite . for the application to the few - body system , we modify the delta function type interaction to the finite range gaussian form . this procedure is necessary to account for short range correlations of the nucleons in the few - body system , and physically motivated by the underlying mechanism of the meson - exchange picture . using the correlation density obtained from the wave function of the @xmath1 , the two nucleon absorption width is estimated to be 4 - 12 mev . we have investigated the @xmath1 system with a variational method , employing a realistic @xmath22 potential ( av18 potential ) and an effective @xmath2 potential based on chiral su(3 ) dynamics . with theoretical uncertainties in the model , the binding energy and decay width of the @xmath1 turns out to be @xmath78 as a consequence of the strong @xmath79 interaction in chiral scheme , the strength of the @xmath2 interaction is reduced and therefore we find a weakly bound state . the @xmath6 pair in the obtained wave function of the @xmath1 system exhibits a similar structure as the @xmath2 two - body quasibound state in vacuum . we have estimated corrections , such as dispersive correction , the @xmath8-wave @xmath2 potential , and the two - nucleon absorption process . taking these effects into account , the total binding energy increases slightly and the total decay width becomes as large as 60 - 120 mev . our result should be compared with another three - body faddeev calculation with chiral interaction @xcite , where the @xmath1 state was found with 80 mev binding energy . while faddeev approach treats the coupled channels explicitly , our variational calculation works by eliminating the @xmath80 channel . although the two - body @xmath48 dynamics is fully incorporated in the effective @xmath2 interaction , the dynamics of @xmath81 three - body system may generate additional attraction ( see also ref . @xcite ) . in the coupled - channel framework , the obtained state is the mixture of the @xmath19 and @xmath80 components , as the @xmath7 resonance in @xmath2-@xmath48 system . in this sense , our strategy is to focus on the @xmath19 component , and the present framework may not be sensitive to the @xmath80 component . based on a recent experimental analysis , a broad structure at about 100 mev below the @xmath19 threshold is reported @xcite , the maximum of which coincides with the @xmath80 threshold . in the chiral framework , such a broad state in the deep subthreshold region would be interpreted in terms of @xmath80 dynamics , driven by the strong @xmath48 attraction . the present investigation is however not capable to deal with the @xmath80 component , since we have eliminated this channel . while the new report @xcite is an interesting observation , more careful analysis is needed to answer the question about its detailed structure . this project is partially supported by bmbf , gsi , by the dfg excellence cluster origin and structure of the universe . " , by the japan society for the promotion of science ( jsps ) , and by the grant for scientific research ( no . 19853500 , 19740163 ) from the ministry of education , culture , sports , science and technology ( mext ) of japan . this research is part of the yukawa international program for quark - hadron science . y. akaishi and t. yamazaki , phys . c * 65 * , 044005 ( 2002 ) . m. agnello _ et al . _ , ( finuda collaboration ) , phys . rev . lett * 94 * , 212303 ( 2005 ) ; v. k. magas , e. oset , a. ramos , and h. toki , phys . c * 74 * , 025206 ( 2006 ) ; t. kishimoto _ et al . . phys . * 118 * , 181 ( 2007 ) . n. v. shevchenko , a. gal , j. mares , and j. rvai , phys . c * 76 * , 044004 ( 2007 ) . y. ikeda and t. sato , phys . c * 76 * , 035203 ( 2007 ) . t. yamazaki and y. akaishi , phys . c * 76 * , 045201 ( 2007 ) . a. dot and w. weise , prog . suppl . * 168 * , 593 ( 2007 ) ; nucl - th/0701050 : proceedings of hyp06 . a. dot , t. hyodo and w. weise , nucl . phys . * a804 * , 197 ( 2008 ) ; arxiv:0806.4917 . t. hyodo and w. weise , phys . c * 77 * , 035204 ( 2008 ) . r. b. wiringa , v. g. j. stoks and r. schiavilla , phys . c * 51 * , 38 ( 1995 ) . t. hyodo , s. i. nam , d. jido , and a. hosaka , phys . c * 68 * , 018201 ( 2003 ) ; prog . theor . phys . * 112 * , 73 ( 2004 ) . j. mares , e. friedman and a. gal , nucl . a * 770 * , 84 ( 2006 ) . y. ikeda and t. sato , arxiv:0809.1285 . t. yamazaki _ _ , arxiv:0810.5182 [ nucl - ex ] , in these proceedings .

the prototype of a @xmath0 nuclear cluster , @xmath1 , has been investigated using effective @xmath2 potentials based on chiral su(3 ) dynamics . variational calculation shows a bound state solution with shallow binding energy @xmath3 mev and broad mesonic decay width @xmath4 - @xmath5 mev . the @xmath6 pair in the @xmath1 system exhibits a similar structure as the @xmath7 . we have also estimated the dispersive correction , @xmath8-wave @xmath2 interaction , and two - nucleon absorption width . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

several decay modes of @xmath6 mesons with a @xmath7 in the final state have been measured at the @xmath6-factories . the amplitudes governing these decays are interesting because none of the constituent flavors of the @xmath7 are present in the initial state . for example , the decays @xmath8 @xcite and @xmath9 @xcite , observed with branching fractions in the range @xmath10 , can proceed via a @xmath11 @xmath12-exchange diagram . here we study the related decays @xmath13 and @xmath14 . the former proceeds via cabibbo - suppressed @xmath12-exchange and has not yet been observed ; theoretical calculations predict a branching fraction ranging from @xmath15@xmath16 @xcite up to @xmath15@xmath17 @xcite . the latter of the two above decays proceeds via a cabibbo - favored tree diagram ; the ratio of its branching fraction to that for @xmath18 can be used to test the factorization hypothesis for exclusive non - leptonic decays of @xmath6 mesons @xcite . however , previous measurements of @xmath19 @xcite have large uncertainties , which limit the usefulness of this method at present . in this paper we report an improved measurement of @xmath0 decays and a search for @xmath20 decays with the belle detector @xcite at the kekb asymmetric - energy @xmath21 collider @xcite . charge conjugate modes are implied throughout this paper . the results are based on a @xmath22 fb@xmath23 data sample collected at the center - of - mass ( cm ) energy of the @xmath24 resonance , corresponding to @xmath25 pairs . we assume equal production of @xmath26 and @xmath27 pairs . to study backgrounds , we use a monte carlo ( mc ) simulated sample @xcite of @xmath28 events and continuum events , @xmath29 ( @xmath30 , @xmath31 , @xmath32 and @xmath33 quarks ) . the belle detector is a large - solid - angle magnetic spectrometer that consists of a multi - layer silicon vertex detector ( svd ) , a 50-layer central drift chamber ( cdc ) , an array of aerogel threshold cherenkov counters ( acc ) , a barrel - like arrangement of time - of - flight scintillation counters ( tof ) , and an electromagnetic calorimeter ( ecl ) comprised of csi(tl ) crystals located inside a superconducting solenoid coil that provides a @xmath34 t magnetic field . an iron flux - return located outside of the coil is instrumented to detect @xmath35 mesons and to identify muons ( klm ) . the detector is described in detail in ref . two different inner detector configurations were used . for the first 152 million @xmath2 pairs , a @xmath36 cm radius beampipe and a 3-layer silicon vertex detector were used ; for the latter 297 million @xmath2 pairs , a @xmath34 cm radius beampipe , a 4-layer silicon detector and a small - cell inner drift chamber were used @xcite . charged tracks are selected with loose requirements on their impact parameters relative to the interaction point ( ip ) and the transverse momentum of the tracks . for charged particle identification ( pid ) we combine information from the cdc , tof and acc counters into a likelihood ratio @xmath37 @xcite . a selection imposed on this ratio results in a typical kaon ( pion ) identification efficiency ranging from 92% to 97% ( 94% to 98% ) for various decay modes , while 2% to 15% ( 4% to 8% ) of kaon ( pion ) candidates are misidentified pions ( kaons ) . we use the @xmath38 , @xmath39 and @xmath40 modes to reconstruct @xmath41 mesons and @xmath42 , @xmath43 , and @xmath44 for the @xmath45 mesons , where the @xmath46 , @xmath47 and @xmath48 decay to @xmath49 , @xmath50 and @xmath51 , respectively . combinations of oppositely - charged kaons with @xmath52 mev/@xmath53 and of oppositely - charged kaons and pions with @xmath54 mev/@xmath53 , originating from a common vertex , are retained as @xmath46 and @xmath55 candidates , where @xmath56 and @xmath57 are the nominal masses of the two mesons @xcite . neutral kaons ( @xmath48 ) are reconstructed using pairs of oppositely - charged tracks that have an invariant mass within 30 mev@xmath58 of the nominal @xmath59 mass , and originate from a common vertex , displaced from the ip . all @xmath60 candidates with invariant masses within a @xmath61 ( @xmath62 ) interval around the nominal @xmath63 ( @xmath64 ) mass are considered for further analysis , where @xmath63 ( @xmath64 ) signal resolutions ( @xmath65 ) range from @xmath66 mev/@xmath53 to @xmath67 mev/@xmath53 ( @xmath68 mev/@xmath53 to @xmath69 mev/@xmath53 ) . a decay vertex fit with a mass constraint is applied to the selected @xmath60 candidates to improve their momentum resolution . for the decay @xmath4 we also add an additional constraint on the value of the cosine of a helicity angle , @xmath70 for the @xmath71 decay mode , where @xmath72 is defined as the angle between the direction of the @xmath73 and the @xmath74 originating from the vector - meson ( @xmath46 or @xmath55 ) in the vector - meson rest frame . the distribution in @xmath75 is expected to be proportional to @xmath76 for the signal and uniform for the combinatorial background . pairs of @xmath41 and @xmath77 meson candidates are combined to form @xmath78 meson candidates . these are identified by their cm energy difference , @xmath79 , and the beam - energy constrained mass , @xmath80 , where @xmath81 is the cm beam energy and @xmath82 and @xmath83 are the reconstructed energy and momentum of the @xmath6 meson candidate in the cm frame . the signal region is @xmath84 gev@xmath58 @xmath85 @xmath86 @xmath85 @xmath87 gev@xmath58 for the @xmath88 , and @xmath89 gev@xmath58 @xmath85 @xmath86 @xmath85 @xmath90 gev@xmath58 and @xmath91 @xmath92 gev for the @xmath4 decays . to suppress the large combinatorial background dominated by the two - jet - like @xmath29 continuum process , variables characterizing the event topology are used . we require the ratio of the second to zeroth fox - wolfram moments @xcite , @xmath93 and the thrust value of the event , @xmath94 . simulation shows that this selection retains more than 95% of @xmath2 events and rejects about 55% of @xmath95 events and 65% of @xmath96 , @xmath97 and @xmath98 events . the above selection criteria and signal regions are determined by maximizing the figure of merit ( fom ) , @xmath99 , where @xmath100 and @xmath6 are the numbers of signal and background events determined from mc . for optimization of the fom we assume @xmath101 . the fraction of events with more than one @xmath88 ( @xmath4 ) candidate is 4.9% ( 2.8% ) . as the best candidate we select the one with the minimal @xmath102 value , where @xmath103 and @xmath104 are @xmath105 s of the mass - constrained vertex fit . distribution for reconstructed @xmath0 events in the @xmath86 signal region . the curve shows the result of the fit . the normalized distribution for the events in the sidebands of both @xmath63 and @xmath64 invariant masses is shown as the hatched histogram.,scaledwidth=45.0% ] the @xmath106 distribution of events in the @xmath86 signal region , obtained after applying all selection criteria described above is shown in fig . [ fig_8 ] . apart from the signal peak at @xmath107 , contributions from two other specific decay modes were identified using the mc : @xmath108 and @xmath109 . these events cluster around @xmath110 and @xmath111 due to the unreconstructed @xmath112 or @xmath113 from the @xmath114 meson . the @xmath106 distribution is described by two gaussians with the same mean for the signal , two gaussians for the @xmath108 , @xmath115 background events , and a linear function for the rest of the background . the normalizations , positions and widths of the gaussians are free parameters of the binned likelihood fit . the solid line in fig . [ fig_8 ] shows the result of the fit . the positions and widths of the @xmath108 , @xmath115 background components agree with the values expected from the mc . in addition , we perform separate fits to the @xmath106 distributions for each @xmath63 decay mode using the same function with the widths and means of all four gaussian functions fixed to the values obtained by the overall @xmath106 fit . we use events in the @xmath63 and @xmath64 meson invariant mass sidebands in order to check for peaking backgrounds . for this check the masses of @xmath63 and @xmath64 candidates are not constrained to their nominal masses . the @xmath64(@xmath116 ) invariant mass sidebands are @xmath117 mev/@xmath53 intervals around @xmath64(@xmath116 ) nominal mass , excluding the @xmath64(@xmath116 ) signal region . due to common final states used to reconstruct @xmath64 and @xmath63 candidates we exclude the @xmath116(@xmath64 ) signal regions and a @xmath118 mev/@xmath53 @xmath119 mass region from @xmath64(@xmath116 ) sidebands . the @xmath106 and @xmath86 distributions obtained by simultaneously using events in the sidebands of both the @xmath64 and @xmath63 mesons are in agreement with the observed combinatorial background under the @xmath0 signal . a significant signal is present only in the @xmath63 sideband , for @xmath63 s reconstructed in the @xmath120 decay mode . this is due to the three - body @xmath121 decay , reported in ref . the fraction of these events in the signal peak was evaluated by fitting the @xmath106 distribution in the @xmath63 sideband . we observe no peaking background when using the @xmath64 mass sideband . the signal in fig . [ fig_8 ] also includes contributions from @xmath122 , @xmath123 and @xmath124 , which all have a common final state , as well as a small contribution ( @xmath125 ) from @xmath126 decays , where one of the @xmath127 decays in - flight to a @xmath128 and @xmath129 and the @xmath128 is misidentified as the @xmath127 . we evaluate these fractions using simulated events . the contribution of these decays is around five times larger than the contribution of @xmath121 decays . we take into account the relative contributions of individual @xmath63 and @xmath64 decay modes and determine the overall fraction of peaking background events ( @xmath130 ) to be ( @xmath131)% . the uncertainty includes the statistical uncertainty in @xmath63 sideband fits , non - uniformity of @xmath132 in @xmath121 decays , limited mc statistics and uncertainties in the corresponding branching fractions @xcite . the signal yield for @xmath0 is thus @xmath133 , where @xmath134 is the number of events in the signal peak obtained from the fit to the @xmath106 distribution ( fig . [ fig_8 ] ) . the @xmath106 distribution for @xmath4 decays obtained after applying all selection criteria described above is shown in fig . [ fig_12](a ) . distribution for the @xmath20 decay mode . two vertical dashed lines show the interval excluded from the fit , as described in the text , and two dotted lines show the @xmath106 signal region . ( b ) @xmath106 distribution for reconstructed events obtained by inverting the kaon identification requirements in data and in the mc sample.,scaledwidth=45.0% ] the expected width of the narrower signal gaussian , which describes @xmath135 of the events , is @xmath136 mev . this value is obtained from the mc sample and rescaled by a factor obtained after a comparison of parameters from @xmath88 data and mc samples . the @xmath106 signal region includes around @xmath137 of the signal . while the @xmath106 distribution of the combinatorial background is well described by a first order polynomial , there is a significant cross - feed contribution from @xmath88 , @xmath138 , and @xmath139 decays , where the @xmath45 decays into a @xmath140 or @xmath141 final state and one of the pions is misidentified as a kaon . figure [ fig_12](b ) shows the @xmath106 distribution of these cross - feed events , as obtained in both data and mc samples by selecting one of the kaon tracks in the @xmath63 decay chain with a pion pid requirement . events peaking around @xmath142 gev are due to @xmath143 decays , while the events clustering around @xmath144 gev are due to @xmath145 and @xmath146 decays without a reconstructed @xmath112 or a photon . the @xmath106 distribution of cross - feed events is described by the sum of two gaussian functions and a constant . the solid line in fig . [ fig_12](b ) shows the result of the fit . the widths and means of the two gaussian functions are statistically consistent with the values obtained from mc . the expected number of background events populating the @xmath106 signal region is determined by a binned likelihood fit to the @xmath106 distribution sidebands ( @xmath147 mev region indicated by the two vertical dashed lines in fig . [ fig_12](a ) ) . while normalizations are free parameters of the fit , the widths and means of the two gaussian functions are fixed to the values obtained from a fit to the @xmath106 distribution of the misidentified data ( fig . [ fig_12](b ) ) . the fit result is then integrated across the @xmath106 signal region ( indicated by the two dotted lines in fig . [ fig_12](a ) ) to obtain the number of background events , @xmath148 , where the systematic error is evaluated by varying values of the fixed fit parameters by one standard deviation . since only three events are observed in the @xmath106 signal region , the result for @xmath149 indicates that there is no statistically significant signal present in this @xmath106 interval . thus the expected @xmath150 tail of the signal , which might populate the fitted region ( parameterized as background only ) , can be safely neglected . the average efficiency of the selection criteria @xmath151 is evaluated from mc , where the intermediate branching fractions @xmath152 and @xmath153 are taken from ref . @xcite , and @xmath154 is taken from ref . @xcite . to check for a possible peaking background we use events in the @xmath63 mass sidebands . no peaking structures are observed in any of the @xmath86-@xmath106 distributions . .sources of systematic uncertainty in @xmath155 and @xmath156 measurements . [ cols="^,^,^",options="header " , ] the number of signal @xmath0 events , @xmath157 , is converted into a branching fraction using the mc efficiency @xmath158 and the number of @xmath2 events . the measured branching fraction is given in table [ tab_dsd ] . we use the world average of @xmath159 @xcite and calculate the ratio @xmath160 before comparing this result to the numerical prediction of @xmath161 given in ref . @xcite in which the calculation is performed in the generalized factorization scheme and includes penguin effects we rescale it by a factor @xmath162 , where @xmath163 is the average value of @xmath63 meson decay constant given in refs . @xcite and @xmath164 is the value used in the original calculation . the expected value is @xmath165 , where the uncertainty originates from the dependence on the decay constant @xmath166 and form - factors , the former being the main source . the ratio @xmath167 is consistent with unity . if one does not include the penguin contributions @xcite to the amplitude for @xmath88 decay , the above ratio would be @xmath168 . we observe no statistically significant signal in the @xmath4 decay mode . the central value for the measured branching fraction is @xmath169\times 10^{-5}$ ] . we infer an upper limit on the @xmath156 from the total measured number of reconstructed events and the number of background events in the @xmath106 signal region ( @xmath170 and @xmath171 , respectively ) , and the measured sensitivity , @xmath172 . the latter error includes all systematic uncertainties given in table [ tab_4 ] . to estimate the upper limit we use bayes s theorem with a flat - prior for the signal following the prescription in ( section 32.3.1 in ref . @xcite ) : @xmath173 the number of observed events @xmath174 is poisson distributed around the sum of @xmath175 and @xmath176 : @xmath177 , where @xmath175 and @xmath176 are the expected number of signal and background events , respectively . in particular @xmath175 can be written as @xmath178 , where @xmath179 and @xmath100 are true values of @xmath156 and the sensitivity @xmath180 , respectively . the true value of @xmath100 can only take non - negative values and is gaussian distributed around @xmath181 with variance @xmath182 . hence @xmath183 is a gaussian function with a cut - off for @xmath184 . the prior probability density @xmath185 is assumed to be factorizable , @xmath186 . for @xmath187 we use a flat - prior , and @xmath188 is again a gaussian function centered at @xmath149 , with a width of @xmath189 and with a cut - off for @xmath190 . integrating out the nuisance parameters @xmath100 and @xmath176 we obtain the posterior @xmath191 , which already takes into account the statistical error on @xmath149 , the systematic error due to the parameterization of @xmath106 distribution in the fit , and systematic uncertainties on the efficiency and on the number of @xmath2 pairs . the 90% c. l. upper limit on @xmath156 following from this posterior is found to be @xmath192 in conclusion , we have measured the branching fraction for @xmath0 decays . the measured value is @xmath193\times 10^{-3}$ ] , which represents a large improvement in accuracy as compared to previous measurements @xcite . combining this result with the world average for @xmath194 @xcite we obtain the ratio @xmath167 . with present experimental and theoretical uncertainties , the results are consistent with the factorization hypothesis for non - leptonic exclusive decays of @xmath6 mesons . if one does not include the penguin contributions @xcite to the amplitude for @xmath88 decay , the above ratio is not consistent with unity . for @xmath4 decays we found no statistically significant signal . we set an upper limit of @xmath195 at 90% c.l . this result puts even more stringent limits on @xmath156 than the recent measurement by the babar collaboration @xcite , severely challenges recent theoretical estimates in refs . @xcite and implies that the weak annihilation contributions in decay modes with two charmed mesons are small , as suggested in ref . @xcite . we thank the kekb group for excellent operation of the accelerator , the kek cryogenics group for efficient solenoid operations , and the kek computer group and the nii for valuable computing and super - sinet network support . we acknowledge support from mext and jsps ( japan ) ; arc and dest ( australia ) ; nsfc and kip of cas ( china ) ; dst ( india ) ; moehrd , kosef and krf ( korea ) ; kbn ( poland ) ; mist ( russia ) ; arrs ( slovenia ) ; snsf ( switzerland ) ; nsc and moe ( taiwan ) ; and doe ( usa ) . 99 p. krokovny _ et al . _ [ belle collaboration ] , phys . lett . * 89 * , 231804 ( 2002 ) . b. aubert _ et al . _ [ babar collaboration ] , phys . lett . * 90 * , 181803 ( 2003 ) . a. drutskoy _ et al . _ [ belle collaboration ] , phys . rev . lett . * 94 * , 061802 ( 2005 ) . y. li , c. d. lu and z. j. xiao , j. phys . g * 31 * , 273 ( 2005 ) . j. o. eeg , s. fajfer and a. prapotnik , eur . phys . j. c * 42 * , 29 ( 2005 ) . c. s. kim , y. kwon , j. lee and w. namgung , phys . d * 65 * , 097503 ( 2002 ) . d. bortoletto _ et al . _ [ cleo collaboration ] , phys . d * 45 * , 21 ( 1992 ) . h. albrecht _ et al . _ [ argus collaboration ] , z. phys . c * 54 * , 1 ( 1992 ) . d. gibaut _ et al . _ [ cleo collaboration ] , phys . rev . d * 53 * , 4734 ( 1996 ) . b. aubert _ et al . _ [ babar collaboration ] , phys . rev . d * 74 * , 031103 ( 2006 ) . a. abashian _ et al . _ [ belle collaboration ] , nucl . instrum . meth . a * 479 * , 117 ( 2002 ) . s. kurokawa and e. kikutani , nucl . instrum . a * 499 * , 1 ( 2003 ) , and other papers included in this volume . we use the evtgen @xmath6-meson decay generator developed by the cleo and babar collaborations , see : http://www.slac.stanford.edu/@xmath15lange / evtgen/. the detector response is simulated by a program based on geant-3 , cern program library writeup w5013 , cern , ( 1993 ) . a small fraction of events are generated with the cleo qq generator , see : http://www.lns.cornell.edu/public/cleo/soft/qq ) . for these events the detector response is also simulated with geant , r. brun _ et al . _ , geant 3.21 , cern report dd / ee/84 - 1 , 1984 . z. natkaniec _ et al . _ , [ belle svd2 group ] nucl . instrum . meth . a * 560 * , 1 ( 2006 ) . e. nakano , nucl . instrum . a * 494 * , 402 ( 2002 ) . yao _ et al . _ [ particle data group ] , j. phys . g * 33 * , 1 ( 2006 ) . g. c. fox and s. wolfram , phys . lett . * 41 * , 1581 ( 1978 ) . a. drutskoy _ et al . _ [ belle collaboration ] , phys . b * 542 * , 171 ( 2002 ) . n. adam _ et al . _ [ cleo collaboration ] , arxiv : hep - ex/0607079 . m. artuso _ et al . _ [ cleo collaboration ] , arxiv : hep - ex/0607074 . b. aubert _ et al . _ [ babar collaboration ] , phys . rev . d * 72 * , 111101 ( 2005 ) . c. h. chen , c. q. geng and z. t. wei , eur . j. c * 46 * , 367 ( 2006 ) .

we reconstruct @xmath0 decays using a sample of @xmath1 @xmath2 pairs recorded by the belle experiment , and measure the branching fraction to be @xmath3\times 10^{-3}$ ] . a search for the related decay @xmath4 is also performed . since we observe no statistically significant signal an upper limit on the branching fraction is set at @xmath5 ( 90% c.l . ) .

consider an anosov diffeomorphism @xmath3 of a compact smooth manifold . structural stability asserts that if a diffeomorphism @xmath4 is @xmath1 close to @xmath3 then @xmath3 and @xmath4 are topologically conjugate . the conjugacy @xmath5 is unique in the homotopy class of identity . @xmath6 it is known that @xmath5 is hlder continuous . there are simple obstructions for @xmath5 being smooth . namely , let @xmath7 be a periodic point of @xmath3 , @xmath8 . then @xmath9 and if @xmath5 were differentiable then @xmath10 i.e. @xmath11 and @xmath12 are conjugate . we see that every periodic point carries a modulus of @xmath1-differentiable conjugacy . suppose that for every periodic point @xmath7 , @xmath8 , differentials of return maps @xmath11 and @xmath12 are conjugate then we say that periodic data ( p. d. ) of @xmath3 and @xmath4 coincide . suppose that p. d. coincide , is @xmath5 differentiable ? a positive answer for anosov diffeomorphisms of @xmath13 was given in @xcite , @xcite . de la llave @xcite observed that the answer is negative for anosov diffeomorphisms of @xmath14 , @xmath15 . he constructed two diffeomorphisms with the same p. d. which are only hlder conjugate . we provide positive answer to the previous question in dimension three under an extra assumption . the authors would like to thank a.katok for suggesting us the problem , numerous discussions and constant encouragement . let @xmath3 be an anosov diffeomorphism of @xmath14 . it is known @xcite that @xmath3 is topologically conjugate to a linear torus automorphism @xmath16 . it is also known that anosov diffeomorphisms of @xmath17 are the only anosov diffeomorphisms on three dimensional manifolds @xcite , @xcite . let @xmath16 be a hyperbolic automorphism of @xmath17 with real eigenvalues . it is easy to show that absolute values of these eigenvalues are distinct . for the sake of notation we also assume that the eigenvalues are positive . this is not restrictive . we will always assume that the anosov diffeomorphisms that we are dealing with are at least @xmath0 . given @xmath16 as above there exists a @xmath1-neighborhood @xmath18 of @xmath16 such that any @xmath3 and @xmath4 in @xmath18 having the same p. d. are @xmath2 conjugate , @xmath19 . the constant @xmath20 depends on the size of @xmath18 and provided sufficient smoothness of @xmath3 and @xmath4 can be made as close as desired to @xmath21 ( see the definition in the next section ) by shrinking the size of @xmath18 . we do nt know how to bootstrap regularity of @xmath5 to the regularity @xmath3 and @xmath4 like it was done in dimension two . a result about integrability of central distribution @xcite allows to show a stronger statement . let @xmath3 and @xmath4 be anosov diffeomorphisms of @xmath17 and @xmath22 where @xmath5 is a homeomorphism homotopic to identity . suppose that p. d. coincide . also assume that @xmath3 and @xmath4 can be viewed as partially hyperbolic diffeomorphisms : there is an @xmath3-invariant splitting @xmath23 and constants @xmath24 , @xmath25 such that for @xmath26 @xmath27 analogous conditions with possibly different set of constants hold for a @xmath4-invariant splitting @xmath28 . then the conjugacy @xmath5 is @xmath2 , @xmath19 . here and further in the paper we assume that the unstable distribution has dimension two . obviously one can formulate the counterpart of theorem 2 in the case when stable distribution has dimension two . here we outline the proof of theorem 1 . let @xmath29 , @xmath30 and @xmath31 be the eigenvalues of the linear automorphism @xmath16 , @xmath32 . we choose @xmath18 in such a way that every @xmath33 is partially hyperbolic , satifying ( [ phd ] ) with constants @xmath34 independent on the choice of @xmath3 , @xmath35 and @xmath36 first we concentrate on a single diffeomorphism @xmath3 in @xmath18 . it is well known that distributions @xmath37 , @xmath38 and @xmath39 integrate uniquely to stable , unstable and strong unstable foliations @xmath40 , @xmath41 and @xmath42 respectively . we denote by @xmath43 the leaf of @xmath44 passing through @xmath7 , @xmath45 and later @xmath46 . by @xmath47 we denote the local leaf of size @xmath48 , i. e. , a ball of radius @xmath48 inside of @xmath43 centered at @xmath7 , @xmath49 . let @xmath50 be conjugacy between @xmath3 and @xmath16 , @xmath51 . stable and unstable foliations can be characterized topologically , e.g. @xmath52 as a consequence we have that @xmath53 and @xmath54 . in other words @xmath50 maps leaves of foliations for @xmath3 into leaves of corresponding foliations for @xmath16 . we prove two simple lemmas . let @xmath3 be in @xmath18 . then the distribution @xmath55 integrates uniquely to the foliation @xmath56 . define @xmath50 as above . then @xmath57 . now let @xmath3 and @xmath4 be as in theorem 1 . for each of them we have the system of one dimensional invariant foliations . we know that @xmath58 . also from lemma 2 we have @xmath59 since @xmath60 . consider restrictions of @xmath5 to the leaves of @xmath40 and @xmath56 . these restrictions are one dimensional maps . we show that they are smooth . the conjugacy @xmath5 is @xmath2 along @xmath40 . which means that @xmath5 is differentiable along the stable foliation and the derivative is a hlder continuous function on @xmath17 with exponent @xmath20 the general strategy of the proof of theorem 1 is similar to de la llave s strategy for anosov diffeomorphisms of @xmath13 @xcite . one proves smoothness of @xmath5 along one dimensional stable and unstable foliations . in particular proof of lemma 3 can be carried out in the same way as in dimension two . the hard part is showing smoothness of @xmath5 along two dimensional unstable foliation . we would like to show the same for the foliation @xmath56 but we split the proof into two steps . the conjugacy @xmath5 is uniformly lipschitz along @xmath56 . the conjugacy @xmath5 is @xmath2 along @xmath56 . after that we deal with the remaining foliation . @xmath61 . we would like to remark that lemma 6 requires only the coincidence of p. d. in the weak unstable direction . it is not true in general that strong unstable foliations match . the conjugacy @xmath5 is @xmath2 along @xmath42 . proofs of smoothness along the foliations @xmath62 and @xmath42 are similar and use the coincidence of periodic data in corresponding directions . showing smoothness along the weak unstable foliation is more subtle . now smoothness of @xmath5 is a simple consequence of a regularity result . @xcite let @xmath63 be a manifold and @xmath64 , @xmath65 be continuous transverse foliations with uniformly smooth leaves , @xmath66 . suppose that @xmath67 is a homeomorphism that maps @xmath68 into @xmath69 and @xmath70 into @xmath71 . moreover assume that the restrictions of @xmath5 to the leaves of these foliations are uniformly @xmath72 , then @xmath5 is @xmath73 . first we apply the lemma on every unstable leaf of @xmath41 for the pair of foliations @xmath74 . after we know that @xmath5 is @xmath2 along @xmath41 we finish by applying the lemma to stable and unstable foliations . the structure of the next chapter is the following . we prove lemmas 1 and 2 in section 4.1 . section 4.2 is devoted to the proof of lemma 4 . sections 4.3 and 4.4 are the heart of our argument and contain proofs of lemmas 5 and 6 correspondingly . first we prove theorem 1 . for every @xmath90 consider a cone @xmath91 . the assumption ( [ angles ] ) tells us that @xmath92 . hence a leaf of @xmath42 can be considered as a graph of a lipschitz function over @xmath93 . the lipschitz constant depends only on @xmath94 . it follows that @xmath42 is quasi - isometric : and since @xmath101 and @xmath102 lie in the same unstable leaf but not in the same weak unstable leaf we get @xmath103 finally since @xmath104 we have that @xmath105 for any @xmath7 and @xmath106 such that @xmath107 . hence * @xmath114 is well defined and hlder continuous . * @xmath115 . * @xmath116 . * the function @xmath117 is the only continuous function satisfying @xmath118 and property 3 . * @xmath119 such that @xmath120 whenever @xmath121 . fix an arbitrary point @xmath123 . let @xmath124 be the restriction of @xmath5 to @xmath125 . we would like to show that @xmath126 is lipschitz with a constant that does not depend on @xmath123 . let @xmath127 be the induced volume on @xmath125 . consider the function @xmath128 @xmath129 we integrate along the leaf with respect to the measure @xmath127 . * @xmath131 , * @xmath132 , * @xmath133 such that @xmath134 whenever @xmath135 . * the function @xmath128 is continuous . to state this property precisely we consider lift of @xmath128 . we speak about lifts of points and leaves . @xmath136 the lift of the conjugacy @xmath5 satisfies the equation ( [ periodicity ] ) which implies the following @xmath139 also we know that weak unstable foliation is quasi - isometric which gives us the same for the distance in weak unstable foliations @xmath140 this tells us that @xmath126 is lipschitz for points that are far enough . so we need to estimate @xmath141 for @xmath7 and @xmath106 close . note that ( d3 ) allows us to use @xmath137 and @xmath128 in these estimates instead of @xmath142 and @xmath143 . if @xmath144 is a transitive anosov diffeomorphism and @xmath145 are hlder continuous functions such that @xmath146 then there is a function @xmath147 , unique up to a multiplicative constant , such that @xmath148 moreover @xmath149 is hlder continuous . choose points @xmath7 and @xmath106 close on the leaf @xmath125 . choose the smallest @xmath153 such that @xmath154 . then @xmath155 here we used ( [ farlipschitz ] ) and ( d3 ) for @xmath128 and @xmath137 . function @xmath149 is bounded away from zero and infinity so we get that @xmath5 is uniformly lipschitz along the weak unstable foliation . [ [ transitive - point - argument - and - construction - of - a - measure - absolutely - continuous - with - respect - to - weak - unstable - foliation ] ] transitive point argument and construction of a measure absolutely continuous with respect to weak unstable foliation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ we divide the proof of lemma 5 into several steps . the conjugacy @xmath5 is lipschitz along @xmath56 and hence @xmath46-differentiable at almost every point with respect to lebesgue measure on the leaves of @xmath56 . it is obvious that @xmath46-differentiability of @xmath5 at @xmath7 implies @xmath46-differentiability of @xmath5 at any point from the orbit @xmath156 . moreover : the problem now is to show existence of such a transitive point @xmath7 . we know that almost every point is transitive with respect to a given ergodic measure with full support . on the other hand @xmath5 is @xmath46-differentiable at almost every point with respect to lebesgue measure on the leaves . unfortunately it can happen that for natural ergodic `` physical measures '' these two `` full measure '' sets do not intersect . in other words weak unstable foliation is not absolutely continuous with respect to a `` physical measure '' . let us explain this phenomenon in more detail . consider a volume preserving @xmath1 small perturbation @xmath158 of @xmath16 , @xmath159 . the lyapunov exponents of @xmath158 are defined on a full volume set of regular points @xmath160 and are given by the formula @xmath161 the perturbation @xmath158 can be chosen in such a way that @xmath162 ( see @xcite , proposition 0.3 ) . it is easy to show that the weak unstable foliation of @xmath158 is not absolutely continuous . namely , let @xmath163 be a segment of a weak unstable leaf of @xmath16 . then by lemma 2 @xmath164 is a piece of a weak unstable leaf of @xmath158 . we show that lebesgue measure of @xmath165 is equal to zero . for any @xmath166 @xmath167 and ( [ angles ] ) guarantees that @xmath168 can be viewed as a graph of a lipschitz function over a leaf of the weak unstable foliation of @xmath16 . hence @xmath169 suppose that @xmath170 then @xmath171 which contradicts the previous inequality . this construction follows the lines of pesin - sinai @xcite construction of @xmath173-gibbs measures . in our setup the construction is simpler so for the sake of completeness we present it here . measure @xmath172 has full support . thus ergodicity of @xmath172 would imply that almost every point is transitive and hence by step 1 @xmath5 would be @xmath46-differentiable . we do not know how to show ergodicity of @xmath172 . instead we do the condition about orientation ensures that @xmath182 has only one connected component . the set @xmath182 is a small neighborhood of @xmath183 because of the continuity of @xmath128 ( d4 ) . the size of @xmath180 must be chosen in such a way that since @xmath7 is transitive there is an arbitrarily large @xmath153 such that @xmath189 . choose @xmath187 on @xmath190 such that @xmath191 so that @xmath192 by the definition . we choose @xmath153 big enough so that @xmath193 * step 2 . * let @xmath202 be a fixed point for @xmath3 and let @xmath203 be an open bounded neighborhood of @xmath202 in @xmath204 . consider a probability measure @xmath205 supported on @xmath203 with density proportional to @xmath206 . for @xmath26 define @xmath207 so that @xmath208 is supported on @xmath209 and has density proportional to @xmath206 by @xmath210 . . by the krylov - bogoljubov theorem @xmath212 is weakly compact and any of its limits is @xmath3-invariant . let @xmath172 be a one of those limits along a subsequence @xmath213 . we would like to prove that @xmath172 has absolutely continuous conditional measures on the pieces of weak unstable foliation . let us be more precise . consider a small open set @xmath214 which can be decomposed in the following way @xmath215 here @xmath216 is a two dimensional transversal . to simplify the notation let @xmath217 . denote by @xmath218 the transverse measure on @xmath216 : for @xmath219 @xmath220 . similary define @xmath221 and @xmath222 . obviously @xmath223 weakly as @xmath224 . we show that for @xmath218 almost every @xmath106 , @xmath225 the conditional measure @xmath226 on the local leaf @xmath227 is absolutely continuous with respect to lebesgue measure @xmath228 on @xmath227 . first we look at conditional measures of @xmath208 . we fix @xmath230 and @xmath216 as above and we assume that the end points of @xmath209 lie outside of @xmath230 . let @xmath231 . then the formulas for the transverse measure and conditional measures are obvious : @xmath232 notice that @xmath233 actually do not depend on @xmath234 . the goal now is to show that @xmath235 for almost every @xmath106 . it could happen that the end points of @xmath209 lie inside of @xmath230 . support @xmath236 of @xmath221 consists of finitely many points . some of these points correspond to the end points of @xmath209 . denote the set of these points by @xmath237 , @xmath238 . let @xmath239 then there is a natural decomposition of the transverse measure @xmath221 * step 3 . * to prove that @xmath172 a. e. point is transitive we fix a ball in @xmath17 and show that a. e. point visits the ball infinitely many times . then to conclude transitivity we only need to cover @xmath17 by a countable collection of balls such that every point is contained in an arbitrarily small ball . so let us fix a ball @xmath182 and a slightly smaller ball @xmath180 , @xmath255 . let @xmath256 be a non - negative continuous function supported on @xmath182 and equal to @xmath257 on @xmath180 . by birkhoff ergodic theorem applying the standard hopf argument we get that for @xmath172 a. e. @xmath7 the function @xmath267 is constant on @xmath268 . now absolute continuity of @xmath56 together with above observations shows that @xmath269 for @xmath172 a. e. @xmath7 which means according to ( [ ergodictheorem ] ) that a. e. @xmath7 visits @xmath182 infinitely many times . we will be working on two dimensional leaves of @xmath41 . we know that each of these leaves is subfoliated by @xmath56 as well as by @xmath42 . the goal is to prove that @xmath61 so we consider the foliation @xmath270 . as for usual foliations @xmath271 stands for the leaf of @xmath272 passing through @xmath7 and @xmath273 stands for the local leaf of size @xmath48 . obviously @xmath272 subfoliate @xmath41 . a priori the leaves of @xmath272 are just hlder continuous curves . since weak unstable foliations match we see that a leaf @xmath271 intersects each @xmath274 , @xmath275 exactly once . assume that @xmath283 . for the sake of concreteness we also assume that @xmath284 lies between @xmath97 and @xmath285 . we look at configurations @xmath286 , @xmath287 , @xmath288,@xmath289,@xmath290 @xmath291 and study their evolution under @xmath292 and @xmath293 respectively . since under the action of @xmath294 strong unstable leaves contract exponentially faster then weak unstable leaves we get that recall that @xmath57 . the leaves @xmath309 and @xmath310 are parallel lines in @xmath311 that are fixed distance apart . hence the estimate from below is a direct consequence of uniform continuity of @xmath312 with respect to metrics @xmath313 and @xmath314 . now we prove the estimate from above . we need to show that the strip between @xmath302 and @xmath315 can not contain arbitrarily long pieces of strong unstable leaves . the reason for this is uniform transversality of weak unstable and strong unstable foliation . for any positive number @xmath196 we can choose a finite number of points @xmath316 between @xmath96 and @xmath97 on @xmath317 in such a way that @xmath318 is contained in @xmath196-neighborhood of @xmath319 and vice versa , @xmath320 . again this is possible because @xmath321 , @xmath322 are parallel lines and @xmath323 is uniformly continuous . let @xmath324 . choose a small @xmath297 such that in any ball @xmath180 of size @xmath325 @xmath326 in such a ball the direction of @xmath55 is almost constant comparing to the angle between @xmath55 and @xmath39 . clearly it is possible to choose a small @xmath327 and correspondingly the points @xmath328 as above such that any strong unstable leave crosses the strip between @xmath319 and @xmath318 in a ball of size @xmath325 , @xmath320 . this gives us uniform estimates on the lengths of pieces of strong unstable leaves in the strips between @xmath319 and @xmath318 , @xmath329 . the sum of these estimates gives us the desired uniform estimate from above . consider a point @xmath279 then applying claim 1 to the points @xmath333 we get that @xmath334 such that @xmath335 . moreover by claim 2 numbers @xmath336 , @xmath279 are uniformly bounded away from zero . now the statement follows from denseness of @xmath302 in @xmath337 . notice that the property of having a transverse intersection is stable if @xmath338 and @xmath271 intersect transversally then there is a neighborhood @xmath345 of @xmath7 such that @xmath346 @xmath347 and @xmath348 intersect transversally . periodic points are dense therefore absence of transverse intersections at periodic points leads to absence of transverse intersections at all points . let @xmath202 to be a fixed point of @xmath3 . for each @xmath350 the leaf @xmath342 intersects @xmath344 only at @xmath106 . thus we are able to build a ladder of rectangles in @xmath351 as shown on the figure [ fig4 ] . the sides of the rectangles are pieces of weak unstable and strong unstable leaves . the rectangles are subject to condition @xmath352 this guarantees that after the choice of @xmath353 ( there are two choices ) the sequence of rectangles is defined uniquely . let @xmath354 and let @xmath355 be midpoints on the sides of rectangles as shown on the picture . this means that in any fixed bounded neighborhood of @xmath202 the leaf @xmath344 is arbitrarily close to @xmath204 . in particular we have that @xmath359 is arbitrarily close to @xmath360 while we know that they are some fixed distance apart . to make this argument completely rigorous one needs to carry out an estimate on the distance between @xmath360 and @xmath359 using regularity of holonomies along @xmath56 and @xmath42 inside of the leaf @xmath351 . we conclude that @xmath361 . then choose a subsequence @xmath362 such that corresponding rectangles have width going to zero as @xmath94 tend to infinity . each of these rectangles contains a piece of @xmath344 inside of it . let @xmath127 be an accumulation point of @xmath362 considered as a sequence of points in @xmath17 rather than on @xmath351 . since the width of the rectangles is shrinking and the foliations are continuous we get that @xmath363 . hence @xmath332 by claim 3 and we move on to the second case . without loss of generality we can assume that @xmath202 is a fixed point . we chose a sequence @xmath365 such that @xmath366 , @xmath367 . here and afterwards we speak about convergence on the torus , not in the leaf @xmath351 . by claim 1 we know that for any @xmath368 the leaves @xmath369 , @xmath370 and @xmath371 intersect at one point @xmath372 . up to the choice of a subsequence we have that @xmath373 , @xmath367 , where @xmath374 is some point on @xmath375 . since the foliation @xmath272 is continuous we have that @xmath376 as well . the strong unstable foliation is orientable and the pairs @xmath377 have the same orientation i. e. @xmath353 lies between @xmath202 and @xmath374 . now we would like to repeat the procedure . consider another sequence @xmath378 , @xmath379 as @xmath367 and corresponding sequence @xmath380 . then @xmath381 as @xmath367 , @xmath382 . in this way by induction we obtain a sequence of points @xmath383 . these points are ordered on @xmath343 for any positive @xmath368 point @xmath384 lies between @xmath202 and @xmath385 . by claim 2 we know that there are constants @xmath386 and @xmath387 which depend only on the initial choice of @xmath202 and @xmath353 such that @xmath388 . this guarantees that the set @xmath389 is dense and hence applying claim 3 one more time we get that @xmath332 . we did not discuss the proofs of lemmas 3 and 7 . they can be carried out in the same way as the proof of lemma 5 . the technical difficulty with constructing special measure is not present . one can use srb measures instead ( as a matter of fact the construction in step 2 applied to @xmath40 and @xmath42 will produce srb measures ) . notice that we used the assumption that @xmath390 only to prove lemmas 1 and 2 . so for theorem 2 we only need to reprove these two lemmas in the new setting . we use a result from @xcite that states the following . the bootstrap of regularity of @xmath5 to the regularity of @xmath3 and @xmath4 can not be done straightforwardly . the reason is the lack of smoothness of weak unstable foliation . let @xmath391 $ ] . it is known @xcite that given @xmath3 sufficiently @xmath1-close to @xmath16 the individual leaves of weak unstable foliation are @xmath392 immersed curves . in general the the leaves of weak untable foliation can not be more than @xmath392 smooth . an example was constructed in @xcite . hence our method can not lead to smoothness higher than @xmath392 . textll a.t . baraviera , ch . bonatti . removing zero lyapunov exponents . ergodic theory dynam . systems , 23 ( 2003 ) , no . 6 , 1655 - 1670 . d. burago , s. ivanov . partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups . preprint , 2007 . j. franks . anosov diffeomorphisms . global analysis , proceedings of synposia in pure mathematics , 14 , ams , providence , ri 1970 , 61 - 93 . m. hirayama , ya . non - absolutely continuous foliations . israel j. math . , to appear in 2007 . m. jiang , ya . pesin , r. de la llave . on the integrability of intermediate distributions for anosov diffeomorphisms . ergodic theory dynam . systems , 15 ( 1995 ) , no . 2 , 317 - 331 . a regularity lemma for functions of several variables . rev . mat . iberoamericana , 4 ( 1988 ) , no . 2 , 187 - 193 . r. de la llave . smooth conjugacy and s - r - b measures for uniformly and non - uniformly hyperbolic systems . phys . , 150 ( 1992 ) , 289 - 320 . r. de la llave , j.m . marco , r. moriyn . invariants for smooth conjugacy of hyperbolic dynamical systems , i - iv . phys . , 109 , 112 , 116 ( 1987 , 1988 ) . r. de la llave , c.e . wayne . on irwin s proof of the pseudostable manifold theorem . z. , 219 ( 1995 ) , no . 2 , 301 - 321 . a. manning . there are no new anosov diffeomorphisms on tori . , 96(1974 ) , 422 - 429 . sh . newhouse . on codimension one anosov diffeomorphisms . , 92(1970 ) , 761 - 770 . pesin , ya . gibbs measures for partially hyperbolic attractors . ergodic theory dynam . systems , 2 ( 1983 ) , no . 3 - 4 , 417 - 438 .

we consider two @xmath0 anosov diffeomorphisms in a @xmath1 neighborhood of a linear hyperbolic automorphism of three dimensional torus with real spectrum . we prove that they are @xmath2 conjugate if and only if the differentials of the return maps at corresponding periodic points have the same eigenvalues .

high latitude hi in disk galaxies is increasingly being recognized as an important tracer of disk activity and dynamics . the gas scale height is a measure of time integrated activity in the disk and discrete features trace the level of local activity . the disk - halo interface represents a severe transition between two very different environments : the high density , low velocity dispersion disk and the low density , high velocity dispersion halo . globally , it may provide information on the shape of the dark matter halo ( e.g. olling & merrifield 2000 ) . in a few recent cases , lagging hi halos have also been detected , such that the high latitude gas is rotating @xmath4 20 - 50 km s@xmath7 more slowly than the underlying disk ( schaap , sancisi , & swaters 2000 , fraternali et al . 2001 , lee et al . 2001 , rand 2000 and tllmann et al . explanations for such lags have focussed mainly on galactic fountains ( cf . bregman 1980 , swaters et al . 1997 , schaap et al . 2000 , collins et al . 2002 ) . edge - on galaxies are particularly useful in the study of the disk - halo region and , in this paper , we especially wish to investigate how discrete disk - halo features can be used as probes of the environment into which they are emerging . the edge - on galaxy , ngc2613 ( d = 25.9 mpc ) is a good candidate for such a study . in paper i ( chaves & irwin 2001 ) , we presented very large array ( vla ) c array observations of this galaxy . the resulting c array zeroth moment map over an optical image is shown in fig . [ carray ] , illustrating the presence of six symmetrically placed vertical extensions in this galaxy . in this paper , we present new vla d array hi data of ngc 2613 and examine them separately as well as in combination with the previously obtained c array data . the observations are presented in sect . [ observations ] , the results , including a description of extremely large high latitude hi features and a model of the global hi distribution are given in sect . [ results ] , in sect . [ discussion ] , we present the discussion , including a feasibility study of a buoyancy model , and a summary is given in sect . [ conclusions ] . details of the c array data acquisition and mapping have been given in paper i. the d array data were taken on sept . 15 , 2000 . the flux calibrator was 3c 48 and the phase calibrator was 0837 - 198 . the flux calibrator was also used as a bandpass calibrator and was observed once for 17 minutes at the beginning of the observing period . the phase calibrator was observed in 3 minute scans alternating with 25 minute scans on ngc 2613 . the total on - source observing time was 2.4 hours and on - line hanning velocity smoothing was applied . the data were edited , calibrated , and fourier transformed using natural weighting , cleaned , and continuum subtracted in the usual way using standard programs in the astronomical image processing system ( aips ) of the national radio astronomy observatory . the resulting ( ra , dec , velocity ) cube was then examined for broad scale structure . the calibrated d array uv data were then combined with the previously obtained c array uv data ( on - source observing time = 15.3 hours , see paper i ) to improve the signal to noise ( s / n ) ratio . since the high resolution c array data have already been presented , we here concentrate on the broader scale structure by re - mapping the c+d array data , using natural weighting , over a uv range corresponding to the d array data only , i.e. 5 k@xmath8 . that is , we show the lower resolution results ( at a typical d array resolution ) , but with the highest s / n possible . table 1 lists observing and map parameters for d array and for the combined c+d array data . note that , although we show images for the combined data set , we have inspected and analyzed all data sets both independently and in combination . note that a full description of the c array results and relevant images have been given in paper i and we have reproduced the zeroth moment c array map in fig . [ carray ] . the remaining the zeroth and first moment maps from the combined data set are shown in fig . [ moments ] , both rotated so that the x axis is parallel to the major axis of the galaxy . vertical slices have been taken perpendicular to the major axis at positions numbered 1 through 10 in fig . [ moments]a and these are shown as position - velocity ( pv ) plots in fig . [ pv ] . the pv slices in the last two panels of fig . [ pv ] show averages over the receding and advancing sides of the galaxy . for the first time , we see that there is a tail of emission on the south - eastern ( left , fig . [ moments]a ) side of the galaxy which resembles tidal tails seen in galaxy interaction simulations . this tail is likely produced via an interaction with the companion , eso 495-g017 , to the north - west , whose systemic velocity is separated from that of ngc 2613 by only 143 km s@xmath7 . the tail is at negative velocities with respect to the receding side of the galaxy from which it originates and shows a strong velocity gradient along its length , becoming negative with respect to the galaxy s systemic velocity as well , i.e. the velocities become forbidden . this can occur in the case of interactions but is difficult to explain any other way . given this velocity structure , the tail must be trailing from the eastern , receding side of ngc 2613 and is therefore in front of the galaxy . the anomalous emission seen around -200 km s@xmath7 in slice 2 , as well as the average over the receding side of the galaxy ( second last panel ) of fig . [ pv ] belongs to this tail as well . there is excess emission on the north - west ( right , fig . [ moments]a ) side of the galaxy in the vicinity of the companion , but we have found no systematic velocities associated with that emission or any other sign of a corresponding redshifted feature . in paper i , using the c array data alone , we identified and characterized the properties of 6 , symmetrically placed extensions labelled f1 to f6 in fig . [ carray ] . [ moments]a , which includes the lower d array resolution data , now also reveal these same features on broader scales , some of which are blended with other features . since a single cutoff level must be applied over the whole galaxy to create this image , this moment map is best used to see a global view of the disk - halo features and identify where new features might be present . the full _ z _ extents and , of course , the velocities of the individual features are best measured from pv slices perpendicular to the major axis . we have inspected many pv slices over the map and show selected slices ( labelled 1 through 10 in fig . [ moments]a ) in fig . [ pv ] . in the new data , the previously seen features f1 through f6 occur at the same velocities and the same positions as shown in fig . [ carray ] , after taking into account the different beam sizes and the addition of broader scale emission . please refer to both figs . [ moments]a and [ pv ] for a more thorough discussion of these features , presented below . f1 appears similar to what has previously been seen ( fig . [ carray ] ) , but at lower resolution . the pv plot for f1 can be seen in slice 2 which reveals it as a distinct feature at a velocity of 250 km s@xmath7 but it can also be seen in slice 1 as well , extending to a projected _ z _ height of at least 100@xmath9 ( 12.6 kpc ) `` above '' ( i.e. positive _ z _ ) mid - plane . a newly seen extension immediately to the west of f1 , sampled by slice 3 , may actually be associated with or part of f1 , since it is seen in pv space as a small arc at about the same velocity as f1 . f2 ( see slice 1 ) shows several velocity components and is more complex than previously seen . like f1 , f2 extends to at least 100@xmath9 at a velocity of 210 km s@xmath7 . it is not clear whether the other components may be blended with emission from the tidal tail . f3 reveals more structure than the extension in fig . [ carray ] and is sampled by slice 4 which reveals it as a 2-pronged arc above the disk centered at a velocity of 100 km s@xmath7 . such double peaks in velocity space are typical of such features and have been seen in other galaxies ( see , e.g. lee & irwin 1997 , lee et al . 2001 ) . the extension to the left of the f4 label in fig . [ moments]a ) extends to a height 225@xmath9 ( 28 kpc ) below ( i.e. negative _ z _ ) mid - plane in this map . as can be seen in slice 3 of the pv map , this feature occurs at a velocity of 50 km s@xmath7 and shows no velocity gradient with height . f4 reaches the highest _ z _ of all the extensions and , to our knowledge , reaches higher latitudes than any previously seen in an edge - on galaxy , exceeding even the starburst galaxy , ngc 5775 , whose known hi features extend to @xmath4 7 kpc from the plane ( lee et al . this is remarkable , given that ngc 2613 ( log l@xmath10 = 10.00 ) has a massive ( 0.1 @xmath11 100 m@xmath3 ) star formation rate that is a factor of 3 lower than that of ngc 5775 and a supernova input energy rate per unit area that is a factor of 7 less than ngc 5775 ( irwin et al . 1999 ) . feature f5 is sampled by slices 8 and 9 and can most clearly be seen in slice 9 as a vertical feature above the plane in pv space at a velocity of -300 km s@xmath7 . whereas it appeared as a single extension in fig . [ carray ] , it can now be seen to have a counterpart 55@xmath9 ( 7 kpc ) in projection farther out along the disk which is sampled by slice 10 . the counterpart clearly occurs at the same velocity and also shows no velocity gradient with height . these two features may form the base of a large loop or cylinder , similar to what has been seen in galaxies like ngc 5775 ( lee et al . 2001 ) . the most interesting and well - defined feature , however , is associated with f6 . for example , slices 6 , 7 , 8 and 9 pass through a feature below the major axis which extends 22 kpc ( 175@xmath9 ) in projection , from the plane and centers at -307 km individual channel maps of this feature are shown in fig . [ channels ] . we will refer to this as the -307 km s@xmath7 feature , rather than f6 since , as revealed by the new data , the feature is more extensive than what was originally seen in fig . [ carray ] . this feature will be discussed extensively in sect . [ 307feature ] . this disk - halo emission is faint ( 3 - 4 @xmath12 typically , note contour spacings in fig . [ pv ] ) but we consider the above - discussed features to be real because , _ 1 ) _ their spatial morphology is typical of the kinds of extensions and loops seen in other edge - on systems , _ 2 ) _ even though the feature might be faint far from the disk , there is evidence for disturbed contours ` beneath them ' even at highly significant contour levels ( see the 5th and 6th contours of fig . [ moments]a , for example ) , _ 3 ) _ at least the 6 major features , f1 to f6 can be seen in both independent data sets , _ 4 ) _ the features generally occur over several independent beams and/or velocity channels , and _ 5 ) _ in pv space , the features are distinct , yet smooth and connect continuously to the disk , unlike artifacts produced by a badly cleaned beam ( see , e.g. the arc above the disk seen in slice 3 of fig . [ pv ] at @xmath4 270 km s@xmath7 or the larger structures in slices 9 and 10 ) . in addition , _ 6 ) _ the receding and advancing panels of panels of fig . [ pv ] ) show _ averages _ over the entire advancing and receding sides of the disk , rather than just selected individual slices as shown elsewhere in the figure . in these panels , therefore , we have basically averaged together signal with noise and weak extensions would be diluted ( i.e. `` beam - averaged '' , where the beam is the size of half the galaxy ) . yet these panels still show the extensions quite clearly , especially on the advancing side where the disk - halo emission is dominated almost entirely by the -307 km s@xmath7 feature . this feature is seen up to the 4th contour which is a 6@xmath12 detection in the diluted averaged panel . finally , _ 7 ) _ for at least 4 of the 6 features , there is evidence from independent observations that extraplanar emission is present . for example , our radio continuum image ( see fig . 9 of paper i ) shows extended emission at the positions of f3 , f4 , f5 , and f6 . also , an early hi image by bottema ( 1989 ) at lower resolution shows extended hi emission at roughly the same locations . the zeroth moment map of fig . [ moments]a also shows a number of low intensity disconnected " features at high latitudes ( for example , the emission around -300@xmath9 , -200@xmath9 ) . we do _ not _ claim that all of these are real , but we have chosen to display them for the sake of comparison with future observations and because some of the emission shown in independent lower resolution images ( e.g. bottema 1989 , irwin et al . 1999 ) is so extended . an example would be the ` knot ' of emission between f3 and f5 which is sampled by slice 6 . for this feature , the velocities are discordent with respect to the underlying disk ( see fig . [ pv ] ) and therefore do not satisfy the criteria listed above . particularly interesting about these very high latitude features is that there is little or no evidence for any lag in velocity with _ z _ , even to very high distances from the place . the features seen below the plane in slices 3 , 8 , and 9 and features seen above the plane in slices 9 and 10 show no velocity gradient . the early data of bottema ( 1989 ) also showed that the high latitude hi was co - rotating . there is some evidence , in our data , for curvature in the smaller features ( cf . slices 3 & 4 above the plane ) , but one could not make any case for a lag . the only slice which shows some evidence for lagging gas is slice 1 , but this slice is also closest to the tidal tail , confusing the interpretation . the averages of the advancing and receding sides of the galaxy also do not support the presence of a lagging halo . here , we see an asymmetry in the sense of many disk - halo features on the receding side , but the advancing side is dominated by the single -307 km s@xmath7 feature . none of the features , however , show a decline with _ z_. we investigate this further in the next section . we have modeled the global hi density and velocity distributions for ngc 2613 following irwin & seaquist ( 1991 ) and irwin ( 1994 ) . this approach models all spectra in the hi cube by adopting a volume density distribution in and perpendicular to the plane , a rotation curve and velocity dispersion ( if required ) , and a position and orientation on the sky . given trial input parameters describing these distributions , the routine then varies the parameters , examining the residuals , until the best fit solution is found . various sections of the galaxy can be modeled , including galactocentric rings in the galaxy s plane . in the current version , we can also isolate slices parallel to the galaxy s plane . this permits modeling the halo , independent of the disk , which is important in this study in the event that a lagging hi halo might be present . while there is no evidence for lags in the discrete features shown in fig . [ pv ] , it is possible that the smooth , low intensity halo might be better fit with a lagging velocity distribution , similar to what has been found for ngc 2403 ( fraternali et al . the routine , called cubit , interfaces to classic aips ; potential users are requested to contact the first author for the code . for ngc 2613 , the adopted in - plane density distribution is a gaussian ring , described by : @xmath13 , where @xmath14 is the galactocentric radius , @xmath15 is the galactocentric distance of the center of the ring , @xmath16 is the in - plane density at the center of the ring , and @xmath17 is the scale length . @xmath17 can have different values for points @xmath18 ( @xmath19 ) and for points @xmath20 ( @xmath21 ) . the distribution perpendicular to the plane is given by the exponential , @xmath22 , where @xmath1 is the perpendicular distance from midplane and @xmath23 is the exponential scale height . the rotation curve is described by the brandt curve , @xmath24 , where @xmath25 is the galaxy s systemic velocity , @xmath26 is the peak of the rotation curve occurring at a galactocentric radius of @xmath27 and @xmath28 is the brandt index which is a shape parameter . a gaussian velocity dispersion , @xmath29 can also be applied , if needed . note that a velocity dispersion will result from all contributions to line widening along the entire line - of - sight direction , for example , any non - circular motions along the plane , streaming motions due to spiral arms and local turbulence . thus , @xmath29 may be larger than what might be expected from a local value . the above distributions result in 10 free parameters : @xmath16 , @xmath15 , @xmath19 , @xmath21 , @xmath23 , @xmath25 , @xmath26 , @xmath27 , @xmath28 , and @xmath29 . in addition , there are 4 more orientation parameters : the position of the galaxy center , ra(0 ) , dec(0 ) , the major axis position angle , pa , and the inclination , i. over the region of the cube occupied by ngc 2613 , there are 552 independent data points ( beam / velocity - channel resolution elements ) above a 2 @xmath12 intensity level in the combined c+d cube , so the 14 parameters are well constrained , over all . certain parameters may show a larger variation , however , depending on the nature of the distribution ; for example , @xmath27 tends to have a larger error since the point at which the rotation curve changes from rising to flat is not as well constrained by the data as some of the other parameters . table 2 lists the best fit modeled values for ngc 2613 for the combined c+d array data . we also modeled the c array data and d array data alone ; the error bars of table 2 reflect the variations in the parameters over the different arrays that result . given the range of resolution over these arrays , from 26.1@xmath9 @xmath30 20.2@xmath9 ( 3.3 @xmath30 2.5 kpc ) to 96.5@xmath9 @xmath30 42.9@xmath9 ( 12.1 @xmath30 5.4 kpc ) the error bars on the modeled parameters are very small . the best fit model is shown as dashed curves superimposed on the data in fig . [ pv ] . taking only residuals ( data cube - best fit model cube ) greater than 3@xmath12 , the root - mean - square value is 1.77 mjy beam@xmath7 and the average relative error is @xmath4 10% . a zeroth moment map made from the cube of the residuals of the best fit is shown in fig . [ residuals ] . to determine whether there are changes in i , pa , and @xmath23 with galactocentric radius , r , we also modeled the galaxy in concentric rings . for example , either a lower value of i with increasing r ( a warp ) or a higher value of @xmath23 with increasing r ( i.e. a flare ) could conspire to make the halo of a galaxy appear to lag . we find that the inclination does not decrease with radius ( there is a slight increase ) . there is indeed evidence for flaring such that , in each data set , @xmath23 systematically increases with r. the most significant change is measured for the d array data alone such that the mean scale height between a radius of 140@xmath9 and 300@xmath9 ( 17.6 to 38 kpc ) is 5.5@xmath9 ( 693 pc ) compared to a value of 1.7@xmath9 ( 214 pc ) over the whole galaxy . thus , the disk is quite thin with a modest flare at large galactocentric radii . in order to model the halo alone , a variety of lower and upper cutoffs to _ z _ were applied and these data were modeled in a similar fashion . ( note that _ z _ refers to the actual distance from the galaxy s midplane , not just the projected distance from the major axis . ) the number of independent data points is reduced typically by about a factor of 3 , depending on which _ z _ limits are chosen , but we have also held some of the parameters ( the central position , position angle and systemic velocity ) fixed . for the halo models , we find _ no _ evidence that @xmath26 could be lower at higher _ z_. the same conclusion is reached if the c array and d array data are modeled independently . the absence of a lagging hi halo in ngc 2613 will be discussed in sect . [ no - lag ] . a distinct feature occurring at -307 km s@xmath7 ( see the advancing panel of fig . [ pv ] and fig . [ channels ] ) has been mentioned in sect . [ disk - halo ] and extends to a ( negative ) _ z _ height of 22 kpc . ( this feature is also distinguishable on the major axis pv slice of paper i ) . the channel maps show a complex structure below the plane , although two spurs dominate . the feature is very well defined in pv space and contributions to the feature can be seen in slices 6 through 9 which covers a range of 13 kpc in projected distance along the disk of ngc 2613 ( fig . [ moments]a ) . there is also a distinct height ( _ z _ @xmath4 90@xmath9 = 11 kpc ) at which the feature begins to widen abruptly in velocity ( fig . [ pv ] ) and also ( approximately ) spatially ( fig . [ channels ] ) , a point we return to in the model of sect . [ buoyancy ] . further support for hi at high _ z _ at this position comes from early hi observations by bottema ( 1989 ) which show an hi blob " extending to @xmath4 25 kpc from the plane and sharing the rotation of the underlying disk . in addition , our low resolution 20 cm continuum map also shows a single large feature at this position extending 21 kpc from midplane ( irwin et al . thus , we will treat the -307 km s@xmath7 feature to be a single , coherent structure , as implied by its appearance in pv space , centered at a projected galactocentric distance of 15.5 kpc . this feature occurs over a restricted region of observed radial velocity ( @xmath31 = -307 @xmath32 30 km s@xmath7 ) . assuming that the feature originates in the disk ( see sect . [ very - high ] ) , this should allow us to place limits on the position along the line of sight from which it originates since the rotation curve of the galaxy is known . the galactocentric radius is given by @xmath33 ( ignoring sin @xmath34 = 0.98 ) where @xmath35 is the projected galactocentric radius and @xmath36 is the circular velocity at @xmath14 which is approximately constant in this part of the galaxy ( table 2 ) . allowing a @xmath3230 km s@xmath7 range on @xmath31 , the feature could therefore have originated from @xmath37 kpc or anywhere within @xmath32 7 kpc from this value along the line of sight , corresponding to a range of @xmath14 between 15.5 and 17 kpc . since the projected size of the feature is larger than this range ( fig . [ channels ] ) , it is likely that there has been some expansion with distance from the plane . we can also put a limit on the age of the feature . gas which leaves the disk at an earlier time should have a component of line of sight velocity corresponding to the motion of the underlying disk at that time . as the disk rotates , gas which leaves later may have a different line of sight velocity component . consequently , there may be a gradual change in observed velocity with height ( increasing or decreasing ) depending on geometry . since we see no such change , we can use the velocity half - width of the feature , 30 km s@xmath7 , to place a limit on the angle over which the underlying disk has rotated over the age of the feature . we find that the feature must have formed over a timescale which is less than the time required for the disk to rotate @xmath32 24@xmath38 at galactocentric radii between 15.5 and 17 kpc . from the circular velocity in this region , the corresponding timescale is 4.2 @xmath30 10@xmath39 yr . these arguments will depend on how the ejected gas may or may not be coupled to the disk and should be considered order of magnitude only . nevertheless , the result is similar to the kinematic ages found for expanding supershells in other galaxies ( cf . king & irwin 1997 , lee & irwin 1997 , lee et al . basically the `` straightness '' of the feature in pv space requires that the feature have an age @xmath40 the rotation period . the mass of this feature , as measured in pv space to the level at which the feature blends with the disk , is ( @xmath41 m@xmath3 . approximating the feature as a cylinder of height , 22 kpc , width and depth , 13 kpc , the mean density is @xmath42 @xmath43 . ( reducing the diameter of the cylinder to 7.3 kpc ( 150@xmath9 ) results in a mean density of @xmath44 @xmath43 . ) if the feature has reached a _ z _ height of 22 kpc in 4.2 @xmath30 10@xmath39 yr , the implied mean outflow velocity is 512 km s@xmath7 . the corresponding hi mass outflow rate would be 1.9 m@xmath3 yr@xmath7 . we can also compute the potential energy of this feature ( see lee et al . 2001 ) which requires a knowledge of the mid - plane stellar density and scale height . comparing the rotation curve of ngc 2613 ( chaves & irwin 2001 ) to that of the milky way ( sofue & rubin 2001 ) , ngc 2613 is a factor of 4 more massive , and factor of 2 larger in radius . ngc 2613 also has a thinner disk , if we take the thickness of the hi disk ( 188 pc , table 2 ) to be representative of the stellar thickness . scaling the mid - plane stellar density from the galactic value ( 0.185 m@xmath3 pc@xmath45 ) yields 0.74 m@xmath3 pc@xmath45 for ngc 2613 and adopting 100@xmath9 ( 12.6 kpc ) ( about half the full _ z _ extent ) to be representative of the height of the feature above the plane , we find a potential energy of 1.5 @xmath30 10@xmath5 ergs for the -307 km s@xmath7 feature . thus , energy of this order is required to transport the cool gas to the heights observed . although very large , this value is not unlike that determined for the largest supershells in other edge - on galaxies ( cf . an input energy of @xmath4 3 @xmath30 10@xmath5 ergs for ngc 3556 , king & irwin 1997 ) . we return to this issue in sect . [ buoyancy ] . there are now several galaxies for which there is evidence for lagging halos . in ngc 891 ( i @xmath46 88.6@xmath38 ) , the rotation curve of the hi halo between _ z _ = 1.4 to 2.8 kpc reveals a peak velocity @xmath4 25 km s@xmath7 lower than in the disk ( swaters et al . 1997 ) . in ngc 5775 ( i = 86@xmath38 ) , the ionized gas velocities decrease with _ z _ to heights of 5 - 6 kpc above which the velocity remains roughly constant ( rand 2000 ) . the hi in this galaxy shows more complex structure , though lagging gas tends to dominate at _ z _ heights up to 5 to 6 kpc . above this region , the velocities are more nearly constant or the features break up into clumps which are seen over a wide range in velocity ( lee et al . models of ngc 2403 ( inclination of 61@xmath38 ) suggest that a lagging halo extends to _ z _ @xmath4 3 kpc ( schaap et al . 2000 ) . thus , from the sparse data available , the region of the lag extends to typically 3 to 6 kpc above the plane above which ( if gas is detected at all ) there is nt strong evidence for a global lag . given the presence of lagging halos in these other galaxies and also some theoretical expectation of lags ( cf . bregman 1980 , benjamin 2002 , collins et al . 2002 ) , the absence of a lagging velocity gradient along the discrete features ( sect . [ disk - halo ] ) and the absence of a global lag ( sect . [ models ] ) in ngc 2613 require comment . firstly , given the inclination of this galaxy ( 79@xmath38 , table 2 ) and its hi radius of @xmath4 35 kpc , the discrete features must reach a height of _ z _ @xmath47 6.5 kpc before they will be seen beyond the projection of the disk . if it is true that lags only exist up to 3 to 6 kpc and above this there is no longer a lag ( see above ) , then lags ( if they exist ) along the discrete features would be projected against the background disk . this would make a velocity gradient ( if present ) impossible to detect along the discrete features . the ` straightness ' of the -307 km s@xmath7 feature in velocity ( fig . [ pv ] ) to such high latitudes beyond the projection of the disk is , however , quite remarkable . if this feature emerges into a hot x - ray corona , as suggested in sect . [ buoyancy ] , then either the velocity of the broad - scale x - ray corona is not appreciably different from the underlying disk ( although the velocity dispersion is likely much higher ) , or the timescale over which it could appreciably affect this gas column is greater than the age of the feature . as for a globally lagging halo , our kinematical model is indeed capable of detecting such a lag , even if it is projected against the disk ( see sect . [ models ] ) provided there is sufficient high latitude emission that halo gas can be detected at all . what we have found , however , is that ngc 2613 does not have an hi thick disk or halo . the vertical exponential scale height ( 188 pc , table 2 ) indicates that the disk is , in fact , thin . if we create a model galaxy using the parameters of table 2 , it is straightforward to show that emission from all high latitude ( _ z _ @xmath47 1 kpc ) hi in fact falls below the map noise . thus , ngc 2613 does not have a detectable lagging halo because it does not have a halo at all . as a further comparative test , we consider the hi halo of another galaxy at the same distance , ngc 5775 ( i = 86@xmath38 , d = 25 mpc ) , for which we detected an hi halo with exponential scale height of 9.14@xmath9 ( 1.1 kpc ) using the same technique ( irwin 1994 ) . if we create a model for the lower inclination ( i = 79@xmath38 ) ngc 2613 in which the vertical density distribution declines exponentially from its in - disk value with the same scale height as ngc 5775 , we find that it could easily have been detected , even at the lower inclination of ngc 2613 . this result illustrates the importance of using a model , rather than pv slices alone , in drawing conclusions about the possible existence of lagging halos . for example , the model can disentangle inclination effects from the effects of a thick disk . for ngc 2613 , since the modeled inclination is now known to be on the low end of the range quoted in chaves & irwin ( 2000 ) , our earlier conclusion of a lag with _ z _ can now be largely explained by projection against the background disk . this is succintly illustrated via the dashed curves in fig . [ pv ] which show that our thin disk model provides an almost perfect fit to the data . we have argued elsewhere ( paper i ) , largely on the basis of above / below plane symmetry ( fig . [ carray ] ) , that the observed features are internally generated and represent outflows . while these new data show that the galaxy is indeed interacting , it is unlikely that most of the observed features are produced via cloud impacts since the cloud would have to pass completely through the disk , forming similar structures on both sides , in contrast to what is expected theoretically ( see santilln et al . anomalous velocities are also more likely in the case of impacts , whereas the velocities observed in ngc 2613 are typical of the underlying disk . thus , the new data are consistent with the outflow interpretation . we can not , however , rule out the possibility of some impacting clouds in this system and it is also possible that the interaction may assist in the process of disk - halo dynamics , for example via stimulating a starburst or instabilities such as the parker instability . if the features are internally generated , their very large sizes ( _ z _ up to 28 kpc , sect . [ disk - halo ] imply exceptional energies ( @xmath4 10@xmath5 ergs ) which are difficult to conceive of , especially at large distances from the nuclear vicinity and in a galaxy which is not a starburst . this energy problem has been known since the first detections of heiles shells in the milky way ( heiles 1979 , 1984 ) and has generated suggestions as far ranging as multiple supernovae and stellar winds ( heckman 2001 ) , gamma ray bursters ( efremov et al . 1998 , loeb & perna 1998 ) , and jet bubbles ( gopal - krishna & irwin 2000 ) to help explain the high energies . in the following section , we suggest that the exceptional heights achieved by the features in ngc 2613 may be as much a result of the environment into which the features emerge , as to the input source itself . the -307 km s@xmath7 feature ( fig . [ pv ] , advancing ) is remarkable in its well defined structure , in the very high latitude ( 22 kpc ) that it achieves , and in its sudden and dramatic widening and break up in velocity at a height of @xmath4 11 kpc . in individual slices ( see slices 8 and 9 , in particular ) , this velocity widening gives the feature a mushroom - like appearance and is reminiscent of buoyant gas rising through a higher density medium . a precedent for this kind of behaviour is the 350 pc galactic mushroom discovered in canadian galactic plane survey data and which has been interpreted in terms of buoyant outflow ( english et al . 2000 ) . given this similarity , we here consider whether buoyant outflow could explain the observed high latitude discrete hi features in ngc 2613 . we present the following as a feasibility study only to see whether the results provide a reasonable match to the data using realistic parameters . we consider a match to the -307 km s@xmath7 feature only , at this time , since it is the clearest case amongst the disk halo features in ngc 2613 ; however , if the model is correct , it must clearly apply to the other features as well . a similar development has been presented by avillez & mac low ( 2001 ) for smaller features like the galactic mushroom , but we here consider much larger scales ( many kpc ) , and also include the effects of drag . the scenario envisioned is one in which a hot , x - ray emitting corona already exists around the galaxy , possibly set up via venting through previous fountain or chimney activity . some event or events occur within the disk which are sufficiently energetic to produce blow - out . the hi is already entrained or swept up in some way by the time the outflow emerges into the halo . an initial velocity ( i.e. the velocity at blow - out ) could be present but is not included here . thus , the velocities achieved in the plume are a result only of buoyant forces . the attraction of this model is that , rather than requiring that all of the energy in the hi plume be supplied by the instigating event in the disk ( @xmath48 ergs , sect . [ 307feature ] ) , we require only enough energy to produce the blow - out condition ( e.g. @xmath49 ergs , tomisaka 1998 ) . the remaining energy is extracted from the pressure gradient in the hot coronal gas . ultimately , the galaxy s potential itself is providing the energy source . although we do not investigate the details of the entrainment or sweeping up of hi , we assume that the hot gas inside the plume carries the cool hi with it and that the density of the hi declines with _ z _ in a fashion similar to the hot buoyant plume material . in this feasibility study , we do not consider structure in the corona or outflow plume , other than the cylindrical geometry chosen for the plume , and also neglect the effects of shocks . the integrity of the hi in the presence of hot gas will be considered in the next section . the stellar / mass density distribution of the thin disk is described by : @xmath50 where @xmath51 is the scale height of the stellar disk , and @xmath52 is the stellar density at mid - plane . integration of poisson s equation : @xmath53 over the regime , @xmath54 , together with the above density distribution , results in a gravitational acceleration which is constant with @xmath1 and has a magnitude : g= 8g_*(0)z _ * the hot coronal gas ( subscripted , @xmath55 ) is taken to be isothermal at temperature , @xmath56 , with an exponential fall - off in both pressure and density , respectively : p_c(z)=p_c(z_*)e^-z / h _ c(z)=_c(z_*)e^-z / h where @xmath57 is the scale height . this hot coronal distribution starts at the top of the ( hi + stellar ) thin disk , i.e. at @xmath51 where the outflow just blows out of the thin layer . the scale height is given by h = kt_c/(gm_p ) where @xmath58 is boltzmann s constant , @xmath59 is the mean molecular weight and @xmath60 is the mass of the proton . we consider a pure hydrogen gas ( @xmath59 = 1 ) for simplicity . inside the plume ( subscripted , @xmath34 ) , we consider the gas to be adiabatic . the temperature gradient is : = ( 1 - 1 ) t_idp_i for a plume in pressure equilibrium with its surroundings , p_i(z)=p_c(z_*)e^-z / h where , again , we take the plume base to occur at @xmath51 . substituting eqn . and its derivative into eqn . yields : t_i(z)=t_i(z_*)e^((1-)zh ) and from eqn . , eqn . and the perfect gas law : _ i(z)=_i(z_*)e^ - ( z ) the equation of motion ( force per unit volume ) of the plume material is : _ i(z)dv = _ c(z ) g- _ i(z)g - c_d_c(z)v^2z - 2v the first term on the right hand side denotes the buoyancy force , the second term gives the weight of the plume material , the third term specifies the drag force against the upper surface ( assuming a case in which the effects of this force can propagate through the column ) where @xmath61 is the dimensionless drag coefficient , and the fourth term represents the viscous drag on the cylinder sides , where r is the cylinder radius and @xmath62 is the coefficient of viscosity ( @xmath63 ) . both @xmath61 and @xmath62 depend on the reynolds number : @xmath64 assuming a pure ionized hydrogen gas , where l is a scale length above which motion is damped by viscous effects and @xmath65 where @xmath66 is the coronal electron density . taking @xmath67 km s@xmath7 ( sect . [ 307feature ] ) and @xmath68 k , we find for size scales , @xmath69 , of order several kpc , @xmath70 is in the range @xmath71 . in this range of @xmath70 , @xmath72 for an incompressible fluid with cylindrical geometry . the viscosity coefficient includes both turbulent and molecular terms , i.e. @xmath73 , where @xmath74 = @xmath75 and @xmath76 , where @xmath77 is the critical reynolds number which designates the value of @xmath70 at which the flow becomes turbulent . @xmath78 is not known and depends on the geometry and nature of the interface , but typically has values between 1 and 100 . we can consider whether the drag force on the cylinder sides ( 4th term of eqn . ) is appreciable in comparison to the drag on the top of the cylinder ( 3rd term of eqn . ) . taking @xmath79 , the ratio of the 3rd to 4th drag terms becomes [ @xmath80 . since @xmath81 is of order @xmath1 , then for @xmath82 , even @xmath78 up to 100 ensures that the drag at the top will dominate over that at the sides . therefore the 4th term in eqn . is small and will be neglected . substituting @xmath83 , @xmath84 , and @xmath85 into eqn . and rearranging yields : = ( z _ * ) e^ -0.4zh gv- gv- ( z _ * ) e^-0.4zh vz this equation was integrated numerically for the input parameters shown in table 3 , yielding the curves of @xmath86 and @xmath87 shown in fig . [ vz ] and fig . [ zt ] , respectively . the stellar scale height , @xmath51 is set to 188 pc ( table 2 ) and we consider a coronal temperature , @xmath88 k ( models 1 to 4 ) which is comparable to that found from x - ray observations of ngc 253 ( pietsch et al . 2000 ) as well as a temperature which is a factor of 2 higher ( model 5 ) . since the midplane stellar density of 0.74 m@xmath3 pc@xmath45 is a value which has been scaled from galactic values , given the size and mass of ngc 2613 ( see sect . [ 307feature ] ) , we also consider a slightly lower value ( models 3 and 4 ) . the behaviour of the curves depends only on the density ratio at the base of the corona , @xmath89 , rather than the individual densities , but we can additionally fix the density at the bottom of the corona , @xmath90 to be equal to the hi density at the top of the thin disk at the position of the plume ( see table 2 ) , providing constraints upon the density within and outside of the plume . the peaks of the velocity curves ( fig . [ vz ] ) indicate where the acceleration of the plume material goes to zero ; we define the @xmath1 height at which this occurs to be @xmath91 . this position should correspond to the point at which the plume widens in velocity space , observationally determined to be @xmath4 11 kpc . models 1 and 2 ( fig . [ vz ] ) show the effect of changing the initial density ratio , such that the lower ratio ( model 2 ) results in a lower maximum velocity and a lower @xmath91 ( 7.4 kpc as compared to 11.9 kpc ) . a comparison of models 1 and 3 or of models 2 and 4 show the effect of decreasing the midplane mass density . this lowers the gravitational acceleration ( eqn . ) which lowers both the buoyancy and the weight of the plume material . it also increases the scale height ( eqn . ) which increases both the buoyancy and the drag . the net effect is higher @xmath91 at lower @xmath92 for the range of parameters given here . a comparison of models 1 and 5 shows the effect of increasing the coronal gas temperature . the higher temperature increases the scale height alone ( eqn . ) which , again , increases both buoyancy and drag . since drag is velocity dependent , the net effect is that the peak velocities achieved are lower for higher coronal temperatures . clearly , models 1 , 4 , and 5 provide an adequate match to the observed @xmath91 . model 1 , however , results in a mean plume density ( between @xmath51 and @xmath91 ) of only @xmath93 @xmath43 whereas the hi density alone is @xmath4 @xmath94 ( sect . [ 307feature ] ) , and is therefore not realistic . models 4 and 5 both result in reasonable fits to the known observational parameters , with model 5 slightly preferred because of its higher internal plume density ( higher than that of the hi alone ) and lower internal temperature . note that the mean internal temperature derived here ( @xmath95 k ) is comparable to the hot ( t = 1.2 kev = @xmath96 k ) component of the outflow in ngc 253 ( pietsch et al . the peak velocities derived here ( @xmath4 250 to 300 km s@xmath7 ) are lower than the gross estimate of 500 km s@xmath7 computed in sect . [ 307feature ] since we only model the feature up to the stall height ( 11 kpc ) rather than over its total length . the timescales of @xmath4 5 to 6 @xmath30 10@xmath39 yr ( fig . [ zt ] ) are in good agreement with the estimate found from the kinematical structure of the plume ( 4@xmath9710@xmath39 yrs ) . the mass outflow rate to the stall point is @xmath4 1 m@xmath3 yr@xmath7 . mass flow continuity requires that the plume of model 5 should increase in radius by a factor of 1.8 between _ z _ heights of 4.7 and 11.1 kpc . it is unlikely that the hi will be uniformly distributed at its mean density of @xmath4 @xmath98 @xmath43 but may exist in denser clouds or clumps whose sizes and distribution are not described by this simple model . in general , however , the situation will be not unlike that of those high velocity clouds ( hvcs ) which are in the hot halo of the milky way and the same issues regarding whether or not the clouds can remain neutral must be considered . photoionization of the hi by starlight should be negligible , considering the high galactic latitudes achieved . as pointed out by murali ( 2000 ) , the relevant interactions are hi - proton and hi - electron interactions for which the interaction cross - sections are in the range @xmath99 to @xmath100 @xmath101 for a relative velocity of order 200 km s@xmath7 as is appropriate here . the mean free path into the hi cloud is then l = 1/(@xmath102 ) = 0.3 to 3 pc or smaller if the hi is clumped . this is considerably smaller than the size scale of the plume and therefore these ionizing interactions should be minor . the most important interaction will be heating due to thermal conductivity leading to the evaporation of clouds . the classical mass evaporation rate , applicable to the case in which the mean free path is small in comparison to cloud size , is given , for approximately spherical clouds , by @xmath103 = 2.75 @xmath30 10@xmath104 t@xmath105 r@xmath106 @xmath107 g s@xmath7 , where t is the temperature of the external medium , r@xmath106 is 1/2 of the largest dimension of the cloud in parsecs , and @xmath107 is a parameter which measures the inhibition of heat flux due to the magnetic field and cloud geometry ( cowie & mckee 1979 , cowie et al . 1981 ) . we will assume @xmath107 = 1 ( no inhibition ) which maximizes the evaporation rate . using t = 2.5 @xmath30 10@xmath108 k and r = 5 @xmath30 10@xmath109 pc we find @xmath103 = 0.021 m@xmath110 yr@xmath7 ( to within factors of a few , given the different geometry ) . the timescale for complete evaporation of the 8@xmath9710@xmath39 m@xmath110 hi plume at a constant rate is then m/ @xmath103 = 4 @xmath30 10@xmath111 yr . since this is two orders of magnitude larger than the age of the plume , we expect that the plume will not evaporate over its lifetime . a caveat , however , is that since m/ @xmath103 @xmath112 r@xmath113 , if hi is distributed in many smaller clouds , then the evaporation time could approach the age of the plume . if the hi is indeed reaching such high values of _ z _ because of buoyancy , then hi disk - halo features probe the parameters of the halo gas , as suggested in sect . [ introduction ] . we have so far considered only whether buoyancy is feasible . there may be other dynamics at work as well ( for example , an initial velocity at blow - out or magnetic fields ) and the proposed hot corona is also unlikely to have the smooth distributions postulated here . nevertheless , in the context of the model , it is interesting to predict the x - ray luminosity of the corona in the vicinity of the plume . the results of table 3 show that it is feasible to transfer large masses to high galactic latitudes via buoyancy in a postulated x - ray corona . this drastically reduces the computed input energy requirements since it is no longer necessary to eject large masses to high altitudes . it is only necessary to achieve blow - out through the thin hi disk of ngc 2613 . the conditions required for blow - out have been investigated by a variety of authors ( see tomisaka 1998 , for example ) but energy requirements are typically of order 10@xmath127 ergs , rather than the 10@xmath5 ergs that would normally be required for the -307 km s@xmath7 feature . while the details of the interaction between the hi and hot outflowing gas are beyond the scope of this paper , it is important to ask whether there would originally have been sufficient hi in the disk from which the 8 @xmath30 10@xmath39 m@xmath3 in the -307 km s@xmath7 feature could have been swept up . assuming cylinderical geometry , a disk region of radius , 6.5 kpc ( sect . [ 307feature ] ) and using the modeled density and thickness of the disk ( table 2 ) , the available hi mass is @xmath4 3 @xmath30 10@xmath128 m@xmath3 suggesting that @xmath4 26% of the disk mass is swept up . this fraction would be higher for cone - like outflow . we should also consider whether the mass outflow rate is consistent with that of a galactic fountain . the estimated hi mass outflow rate ( sect . [ 307feature ] , sect . [ buoyancy ] ) is @xmath4 1 to 2 m@xmath3 yr@xmath7 . thus , the combined hi + hot gas outflow rate will be of order several m@xmath3 yr@xmath7 . collins et al . ( 2002 ) have estimated the mass flow rate for the diffuse ionized gas ( dig ) component in a galactic fountain , assuming a ballistic model of gas clouds . they find global values of @xmath129 = 22 @xmath130 m@xmath3 yr@xmath7 for ngc 891 and @xmath129 = 13 @xmath130 m@xmath3 yr@xmath7 for ngc 5775 , where @xmath131 is the filling factor . if these global values apply to a 20 kpc radius disk , taking @xmath131 = 0.2 and scaling to the disk area of the -370 km s@xmath7 feature yields mass outflow rates of 1.4 and 2.3 m@xmath3 yr@xmath7 which are comparable to what we estimate , above . it is not yet clear , however , whether these galactic fountain values can be directly scaled to the relevant regions of ngc 2613 . it may be that some additional source of pressure is still required , for example , magnetic fields in the form of a parker instability . if such fields continue to rise into the corona , the inclusion of this magnetic pressure would relax ( i.e. lower ) the density or temperature requirements internal to the plume ( table 3 ) . new vla d array data of ngc 2613 have been combined with previous higher resolution observations ( chaves & irwin 2001 ) to show a more extensive hi distribution than previously observed . the galaxy is now seen to have a tidal tail on its eastern side due to an interaction with its companion , eso 495-g017 , to the north - west . the three - dimensional hi distribution in ngc 2613 has been modeled following irwin & seaquist ( 1991 ) and irwin ( 1994 ) , a method which allows the volume density distribution to be determined as well as the scale height and inclination to be disentangled . we find that the inclination of the galaxy ( 79@xmath38 ) is on the low end of the range given in chaves & irwin ( 2001 ) and the model now shows that there is no hi halo in ngc 2613 . rather , the global hi distribution is well fit by a thin disk of exponential scale height , @xmath132 = 188 pc . the use of such a model is very important in drawing conclusions about the presence or absence of a global halo in a galaxy of this inclination . previous reports of a lagging halo from pv slices alone can largely be attributed to projection against the background disk . while there is no significant global hi halo in ngc 2613 , there are more discrete disk - halo hi features than previously detected and these hi features achieve extremely high latitudes . even though a tidal interaction is occurring , we suggest that most of the discrete kpc - scale features have been produced internally rather than from impacting clouds , although we do not rule out the possiblity of the companion galaxy having some indirect effect ( e.g. triggering instabilities ) . the presence of many discrete features may be related to the fact that the global hi disk is thin , favouring blow - out . the observed _ z _ heights are quite remarkable ( e.g. up to 28 kpc ) . the -307 km s@xmath7 feature , in particular , below the plane on the advancing side , reaching 22 kpc in @xmath133 height and of total mass , ( 8 @xmath32 2 ) @xmath30 10@xmath39 m@xmath3 , is very obvious and well - defined in pv space . its center is likely close to its projected radius of 15.5 kpc and it extends over a large ( @xmath32 7 kpc radius ) projected galactocentric radius . if this feature has achieved its @xmath133 height as a result of internal processes , then extremely large energies are required , @xmath4 10@xmath5 ergs . given the very high input energies required for the -307 km s@xmath7 feature , its resemblance to smaller buoyant features ( cf . the galactic mushroom , english et al . 2000 ) , and the fact that x - ray halos are being found around an increasing number of star forming spiral galaxies , we have carried out a feasibility study as to whether this feature can be interpreted as an adiabatic buoyant plume . the observed hi would be carried out by a hot , low density outflowing gas and , after having achieved blowout , would rise through a hot pre - existing x - ray corona . a reasonable example ( model 5 ) , has a mean plume temperature and density of @xmath134 k and 5.5 @xmath30 10@xmath45 @xmath43 rising into a hot isothermal corona of temperature and mean density , @xmath135 k and 0.035 @xmath43 , respectively . these conditions produce a stall height of 11 kpc which is where the observed plume widens in velocity and position space . the coronal density at the stall height is 2.8 @xmath30 10@xmath45 @xmath43 . the maximum outflow velocity in this model is 290 km s@xmath7 and it reaches the stall height in 5.4@xmath9710@xmath39 yrs . this model shows that , even with buoyancy alone ( and there may be additional sources of pressure such as magnetic fields ) , hi can reach these extreme _ z _ heights . the advantage of such a model is that the energy requirements from the initial event are drastically reduced to being only what is required for blowout , a reduction of several orders of magnitude . the energy is largely extracted from the gravitational potential of the galaxy rather than the initial event within the disk . the behaviour of the plume should sample the parameters of the x - ray corona . the predicted x - ray luminosity suggests that the corona should be observable at heights below the stall height although we expect that the distribution of x - ray emission may not be as smooth as assumed by the model . ji wishes to thank the natural sciences and engineering research council of canada for a research grant . we are grateful to dr . r. n. henriksen for fruitful and envigorating discussions . thanks also to mustapha ishak for assistance with maple . benjamin , r. 2002 , in seeing through the dust " , asp conf series , ed . a. r. taylor , t. l. landecker , & t. willis bottema r. 1989 , , 225 , 358 bregman j. n. 1980 , , 236 , 577 chaves t. a. , & irwin j. a. 2001 , , 557 , 646 ( paper i ) collins j. a. , rand r. j. , duric n. , & walterbos r. a. m. 2000 , , 536 , 645 collins j. a. , benjamin r. a. , & rand r. j. 2002 , in press cowie l. l. , & mckee c. f. 1977 , , 211 , 135 cowie l. l. , mckee c. f. , & ostriker j. p. 1981 , , 247 , 908 de avillez m. a. , & berry d. l. 2001 , , 328 , 708 duric n. , irwin j. , & bloemen h. 1998 , , 331 , 428 efremov y. n. , elmegreen b. g. , & hodge p. w. 1998 , , 501 , l163 english j. , taylor a. r. , mashchenko s. y. , irwin j. a. , basu s. , & johnstone d. 2000 , , 533 , l25 fraternali f. , oosterloo t. , sancisi r. , & van moorsel g. 2001 , astro - ph 0110369 gopal - krishna , & irwin j. a. 2000 , , 361 , 888 heckman t. 2001 , in gas and galaxy evolution , asp conf . series , vol . 240 , ( san francisco : asp ) , ed . j. e. hibbard , m. p. rupen , & j. h. van gorkom , 345 heiles c. 1979 , , 229 , 533 heiles c. 1984 , , 55,585 irwin j. a. 1994 , , 429 , 618 irwin j. a. , english j. , & sorathia b. 1999 , , 117 , 2102 irwin j. a. & seaquist e. r. 1991 , , 371 , 111 , erratum , 415,415 king d. l. , & irwin j. a. 1997 , newa , 2 , 251 lang k. 1999 , astrophysical formulae ( springer : berlin ) lee s .- w , & irwin j. a. 1997 , , 490 , 247 lee s .- w . , irwin j. a. , dettmar r .- j . , cunningham c. t. , golla g. , & wang q. d. 2001 , , 377 , 759 loeb a. , & perna r. 1998 , , 503 , l135 lotz w. 1967 , , 14 , 207 murali c. 2000 , , 529 , l81 olling r. , & merrifield m. r. 2000 , , 311 , 361 pietsch w. , vogler a. , klein u. , & zinnecker h. 2000 , , 360 , 24 rand r. j. 2000 , , 494 , l45 santilln a. , franco j. , martos m. , & kim j. 1999 , , 515 , 657 schaap w. e. , sancisi r. , & swaters r. a. 2000 , , 356 , l49 sofue y. & rubin v. 2001 , , 39 , 137 swaters r. a. , sancisi r. , & van der hulst j. m. 1997 , , 491 , 140 tomisaka k. 1998 , , 298 , 797 tllmann r. , dettmar r .- j . , soida m. , urbanik m. , & rossa j. 2000 , , 364 , l36 lcc no . velocity channels & 63 & 63 + velocity resolution ( km s@xmath7 ) & 20.84 & 20.84 + total bandwidth ( mhz ) & 6.25 & 6.25 + synthesized beam & & + major @xmath30 minor axis ( @xmath9 @xmath30 @xmath9 ) @ pa ( @xmath38 ) & 96.5 @xmath30 42.9 @ -21.6 & 47.1 @xmath30 32.1 @ -8.2 + rms noise / channel ( mjy beam@xmath7 ) & 0.93 & 0.45 + rms noise / channel ( k ) & 0.14 & 0.18 + ra ( j2000 ) ( h m s ) & 08 33 22.8 @xmath32 0.3 + dec ( j2000 ) ( @xmath38 @xmath136 @xmath9 ) & -22 58 29 @xmath137 + pa ( @xmath38 ) & 114.2 @xmath138 + i ( @xmath38 ) & 79.2 @xmath139 + v@xmath140 ( km s@xmath7 ) & 1663 @xmath141 + v@xmath142 ( km s@xmath7 ) & 304 @xmath32 4 + r@xmath142 ( @xmath9 ) & 112 @xmath143 + m & 0.51 @xmath144 + @xmath29 ( km s@xmath7 ) & 17 @xmath32 5 + n@xmath145 ( @xmath43)@xmath146 & 0.43 @xmath147 + r@xmath145 ( @xmath9 ) & 140 @xmath32 5 + d@xmath148 ( @xmath9 ) & 95 @xmath149 + d@xmath150 ( @xmath9 ) & 81 @xmath151 + h@xmath152 ( @xmath9 ) & 1.5 @xmath153 + 5 + @xmath56 ( @xmath154 k ) & 2.5 & 2.5 & 2.5 & 2.5 & 5.0 + @xmath52 ( @xmath155 ) & 0.74 & 0.74 & 0.50 & 0.50 & 0.74 + h@xmath156 & 1.4 & 1.4 & 2.1 & 2.1 & 2.8 + @xmath89 & 100 & 20 & 100 & 20 & 10 + @xmath157 ( @xmath43 ) & 0.15 & 0.15 & 0.15 & 0.15 & 0.15 + @xmath158 ( @xmath43 ) & 0.0015 & 0.0075 & 0.0015 & 0.0075 & 0.015 + @xmath91 ( kpc ) & 11.9 & 7.4 & 17.5 & 11.0 & 11.1 + @xmath159 ( @xmath43 ) & 0.015 & 0.025 & 0.016 & 0.026 & 0.035 + @xmath160 ( @xmath98 @xmath43 ) & 0.031 & 0.76 & 0.036 & 0.80 & 2.8 + @xmath161 ( @xmath98 @xmath43 ) & 0.27 & 2.1 & 0.28 & 2.2 & 5.5 + @xmath162 ( @xmath163 k ) & 6.8 & 2.0 & 6.9 & 2.0 & 1.2 +

we combine new vla d array hi data of ngc 2613 with previous high resolution data to show new disk - halo features in this galaxy . the global hi distribution is modeled in detail using a technique which can disentangle the effects of inclination from scale height and can also solve for the average volume density distribution in and perpendicular to the disk . the model shows that the galaxy s inclination is on the low end of the range given by chaves & irwin ( 2001 ) and that the hi disk is thin ( @xmath0 = 188 pc ) , showing no evidence for halo . numerous discrete disk - halo features are observed , however , achieving @xmath1 heights up to 28 kpc from mid - plane . one prominent feature in particular , of mass , @xmath2 m@xmath3 and height , 22 kpc , is seen on the advancing side of the galaxy at a projected galactocentric radius of 15.5 kpc . if this feature achieves such high latitudes because of events in the disk alone , then input energies of order @xmath4 10@xmath5 ergs are required . we have instead investigated the feasibility of such a large feature being produced via buoyancy ( with drag ) within a hot , pre - existing x - ray corona . reasonable plume densities , temperatures , stall height ( @xmath4 11 kpc ) , outflow velocities and ages can indeed be achieved in this way . the advantage of this scenario is that the input energy need only be sufficient to produce blow - out , a condition which requires a reduction of three orders of magnitude in energy . if this is correct , there should be an observable x - ray halo around ngc 2613 . # 1@xmath6 # 1([eq:#1 ] )

the solar spectrum observed close to the solar limb is linearly polarized . the polarization of the continuum , first observed by @xcite , is mainly produced by scattering at neutral hydrogen ( rayleigh scattering ) and free electrons ( thomson scattering ) . but spectral lines , linearly polarized by scattering processes , show incredibly rich and complex polarizations patterns ( e.g. , @xcite ) . second solar spectrum _ ( @xcite ) has been the subject of many theoretical investigations because of its diagnostic potential for the magnetism ( and thermodynamics ) of the solar atmosphere ; but some polarization patterns are not yet well understood ( see @xcite for a review ) . scattering line polarization is usually modeled independently of the continuum polarization . the continuum polarization is modeled as a coherent scattering process ( @xcite ) , which is a suitable approximation far from spectral lines . thomson and rayleigh scattering are coherent in the scatterer s frame ( e.g. , @xcite ) . however , the doppler broadening corresponding to the thermal velocity of electrons and hydrogen atoms is several times the width of most spectral lines , which may lead to redistribution between the polarization of the spectral line and the nearby continuum ( e.g. , @xcite , henceforth ll04 ) . the effect of non - coherent continuum scattering in radiative transfer was considered by @xcite , but his work did not include light polarization . he showed that the effect of the non - coherence on the intensity line spectrum is to broaden the profile and to make it shallower . the treatment of the rayleigh and thomson scattering was first extended to the non - coherent and polarized case by @xcite , who studied some of the effects that this phenomenon may have on the emergent spectral line radiation . these initial steps were later continued by other researchers ( e.g. , @xcite ) who studied the problem of non - coherent electron scattering and partial frequency redistribution on the polarization of resonance lines , pointing out the significance of electron scattering redistribution in the far wings of the line polarization profile . this result has been recently confirmed by @xcite after solving the same type of problem through the application of more efficient numerical radiative transfer methods . in this paper we treat the radiation transfer problem of resonance line polarization taking into account its interaction with non - coherent scattering in the continuum . we treat the rayleigh and thomson redistribution as angle independent ( angle averaged redistribution ) , and the line emission and absorption using the two - level atom model with unpolarized lower level in the limit of complete frequency redistribution ( crd ) . to solve the relevant equations , formulated within the framework of the density matrix theory ( see ll04 ) , we develop an efficient jacobian iterative method , which can be considered as a generalization of that proposed by @xcite for the crd line transfer case . we apply this numerical method to solve the radiation transfer problem in a milne - eddington atmosphere and in a stratified model atmosphere with a temperature minimum and a chromospheric temperature rise . we study the effects of the non - coherence of the continuum scattering on intrinsically unpolarizable ( transition between upper and lower levels with angular momentum @xmath0 ) and polarizable ( @xmath1 and @xmath2 ) lines . in particular , we show the possibility of generating emission " fractional linear polarization features ( i.e. , with larger polarization than in the adjacent continuum ) in the core of intrinsically unpolarizable spectral lines . we consider resonance line polarization ( assuming the crd and two - level atom model without stimulated emission ) in the presence of a polarized continuum in a plane - parallel , static and non - magnetic atmosphere . due to the symmetry of the problem , the radiation field is rotationally invariant with respect to the vertical direction ( which we choose to be the @xmath3 axis ) and it is thus linearly polarized along a direction either parallel or perpendicular to the projected limb . using the reference system for polarization of fig . [ figaxis ] the radiation field is characterized by just the stokes parameters @xmath4 and @xmath5 . assuming that the lower level of the transition is unpolarized ( either its total angular momentum is @xmath6 or @xmath7 , or collisions dominate its excitation ) , the absorption process is isotropic and the radiative transfer equations for @xmath4 and @xmath5 at frequency @xmath8 and propagation direction @xmath9 are and @xmath10 are the polar and the azimutal angles of the ray under consideration , respectively . @xmath9 is the propagation direction , @xmath11 is perpendicular to @xmath9 and is on the meridian plane , and @xmath12 is perpendicular to @xmath9 and @xmath11 . in all the equations , the direction of positive stokes @xmath5 is taken along @xmath11 , i.e. , perpendicular to the projected limb.,width=325 ] [ eqrt]@xmath13 where @xmath14 is the element of optical distance ( where @xmath15 is the geometrical distance measured along the ray direction ) , @xmath16 is the total absorption coefficient ; @xmath17 and @xmath18 are the integrated line and total continuum absorption coefficients , respectively ; @xmath19 and @xmath20 are the thermal and scattering continuum absorption coefficients ; @xmath21 is the line absorption profile , and @xmath22 is the frequency separation from the resonance frequency @xmath23 in units of the doppler width @xmath24 . @xmath25 and @xmath26 are the source functions , which for a two - level atom with polarized continuum are [ eqs]@xmath27 where @xmath28 . the line source functions are expressed in terms of the excitation state of the upper level of the transition . in this case , due to symmetry , the only non - zero spherical components are @xmath29 ( @xmath30 times the total population ) and @xmath31 ( alignment coefficient ) of the density matrix ( @xcite ) of the upper level , and the line source functions are ( e.g. , @xcite ) [ eqsline]@xmath32 , \label{eqsiline } \\ s_{q}^{l } = & \frac{2 h \nu^{3}}{c^{2}}\frac{2 j_{\ell } + 1}{\sqrt{2 j_{u } + 1 } } \frac{3 w^{\left(2\right)}_{j_{u } j_{\ell}}}{2 \sqrt{2}}\left(\mu^{2 } - 1\right ) \rho^{2}_{0 } , \label{eqsqline}\end{aligned}\ ] ] where @xmath33 is a numerical coefficient which depends on the total angular momentum of the levels involved in the transition ( table 10.1 in ll04 ; e.g. , @xmath34 , @xmath35 ) . @xmath36 , where @xmath37 is the angle of the line of sight ( los ) to @xmath3 ( see fig . [ figaxis ] ) . the density matrix elements are obtained from the following statistical equilibrium equations ( @xcite ) : [ eqrho]@xmath38 where @xmath39 is the planck function , @xmath40 is the collisional destruction probability due to inelastic collisions ( @xmath41 and @xmath42 are the collisional de - excitation rate and einstein coefficient for spontaneous emission , respectively ) and @xmath43 ( @xmath44 is the depolarizing rate of the level due to elastic collisions with neutral hydrogen ) . the radiation field tensors in eqs . are given by [ eqjbar]@xmath45 where @xmath46 and @xmath47 are the frequency - dependent radiation field tensors defined as ( ll04 ) [ eqjx]@xmath48 .\label{eqj20x } \end{split}\end{aligned}\ ] ] the source functions for the background continuum in eqs . , taking into account thermal emission and scattering , can be expressed as ( e.g. , @xcite ) [ eqscont]@xmath49 , \label{eqsicont } \\ s_{q}^{c}\left(x\right ) = & \left(1 - s\right)\frac{3}{2 \sqrt{2 } } \left(\mu^{2 } - 1\right ) \breve{j}^{2}_{0}\left(x\right ) , \label{eqsqcont}\end{aligned}\ ] ] where @xmath50 , with the convolved radiation field tensors [ eqjconv]@xmath51 where @xmath52 and @xmath53 are the frequencies of the incident and scattered photons , respectively . the convolution profile @xmath54 accounts for the frequency redistribution caused by the doppler effect , due to the velocity distribution of the scatterers ( electrons for thomson scattering ; hydrogen and helium for rayleigh scattering ) . thomson scattering is coherent in the scatterer s reference system . we take into account the doppler shifts due to the motions of the electrons relative to the laboratory frame by averaging over their velocity distribution , which we assume to be maxwellian . we also take the average over the solid angle ( greatly reducing the computational cost ) because the angular distribution is less important than the frequency distribution ( @xcite ) and the difference with the angle - dependent distribution function is small for optically thick atmospheres ( @xcite ) . the final expression for the angle averaged convolution profile is ( @xcite ) @xmath55 , \label{eqconvprof}\ ] ] with @xmath56 where @xmath57 is the ratio between the doppler widths of the perturbers and the atom of interest . rayleigh scattering is produced in the far wings of the lyman lines of neutral hydrogen and helium . we may consider that the scattering in the very far wings of a resonance line is essentially coherent in the scatterer rest frame ( e.g. , @xcite ) and the above discussion for thomson scattering applies also to rayleigh scattering taking into account the different value of @xmath57 . if we consider the simultaneous contribution of thomson and rayleigh scattering , @xmath58 , different source function terms appear for each convolution kernel ( thomson and rayleigh ) and convolved radiation field tensor . for simplicity , we will not write explicitly such expressions here . to avoid a lengthy expression , we consider explicitly only one of the contributions of the background continuum scattering ; accounting for additional contributions is straightforward . the source functions in eqs . may be expressed in a more simple and symmetric form as [ eqst]@xmath59 here @xmath60 are the frequency - dependent source function tensors : [ eqskqdef]@xmath61 where the @xmath62 tensors are given by [ eqskqldef]@xmath63 and the continuum frequency - dependent tensors by [ eqskqcdef]@xmath64 equations together with eqs.- or , equivalently , - , form a coupled system of integro - differential equations which we solve numerically . we consider an iterative method of solution : if an estimate of the source functions is given , eqs . can be integrated for a given set of boundary conditions ; from the radiation field thus calculated we reevaluate the @xmath65 and @xmath66 tensors which are in turn used to recalculate the new source functions and hence a new radiation field estimate . the formal solution integration ( sect . [ s31 ] ) is based on the short - characteristics ( sc ) method ( @xcite ) ; in order to guarantee convergence , the iterative scheme ( sect . [ s32 ] ) is a generalization of the accelerated lambda iteration ( @xcite ) developed by @xcite , which is based on the jacobi method . if the source functions are given , eqs . can be integrated explicitly between two spatial points @xmath67 and @xmath68 , for a given frequency and angle : @xmath69 and analogously for @xmath5 . in eq . , @xmath70 is the optical distance along the ray between points _ i _ and _ j _ at the reduced frequency @xmath53 . we assume that the source function varies parabolically between three consecutive points m , o and p : o is the point where we want to calculate the stokes parameters , while m and p are respectively the preceding and following points according to the propagation direction . eq . can then be rewritten as ( @xcite ) @xmath71 where @xmath72 and @xmath73 are the intensities at points o and m , @xmath74 is the optical distance between points m and o ; @xmath75 , @xmath76 and @xmath77 are the values of the intensity source function at the points m , o and p , respectively , and @xmath78 , @xmath79 and @xmath80 are three functions that only depend on the optical distance between the local point ( o in this case ) and the preceding and following points ( m and p in this equation ) . equation expresses the intensity at point o as a linear combination of the source function at adjacent points in the atmosphere and the intensity at a _ point m along the ray . the same scheme can in turn be applied to the _ previous _ point and repeated all the way back to the boundary where the incoming radiation is given . therefore , the stokes parameters at a point @xmath67 along a given ray in the atmosphere can be expressed as [ eqrtld]@xmath81 where the @xmath82 coefficients depend on the optical distances between points `` @xmath67 '' and `` @xmath68 '' , @xmath83 are the _ transmitted _ stokes parameters from the boundary , and @xmath84 the number of spatial grid points . averaging these expressions over the angles ( eqs . ) and taking into account the dependence of the source function components on @xmath65 ( eqs . [ eqst ] ) , the radiation field tensors at a point @xmath67 in the atmosphere can be expressed as [ eqjxnum]@xmath85 the explicit expressions for @xmath86 and @xmath87 in terms of @xmath82 and @xmath88 are given in the appendix . it is important to emphasize that we do not need to calculate them explicity ( except for the diagonal elements ) ; they are implicitly evaluated according to the sc algorithm described in the previous section . equations are only convenient to derive the iterative scheme , as we will now show . let @xmath89 , @xmath90 , @xmath91 and @xmath92 be estimates at some iterative step of the atomic and radiation field tensors , and @xmath93 and @xmath94 the corresponding frequency dependent source function tensors derived from them using eqs . - . let @xmath95 and @xmath96 be the values of the radiation field tensors obtained through the formal solution of the radiative transfer equation ( sect . [ s31 ] ) using the above - mentioned old " quantities formally , using @xmath97 on the right hand side of eqs . . if we used @xmath98 to calculate the corresponding @xmath99 and @xmath100 ( eqs . and , respectively ) , and then , eqs . - to obtain new estimates of @xmath65 and @xmath66 , we would have a generalization of the lambda iteration scheme which is known to have very poor convergence properties ( e.g. , @xcite ) . in order to improve the convergence rate , let s consider eqs . . formally , now we shall calculate the radiation field tensors at a given point @xmath67 " from the @xmath101 at all grid points @xmath102 , and the yet unknown new " value @xmath103 at point @xmath67 " . rearranging terms : [ eqjxali]@xmath104 where @xmath105 equations show how to actually compute these new radiation field tensors : @xmath106 is calculated exactly as explained in the previous paragraph ; the diagonal components of the operators @xmath107 can be efficiently computed while performing the formal solution ( see @xcite ) ; finally , the yet - to - be - obtained @xmath66 elements are kept explicitly ; the whole iterative scheme will be obtained from consistently applying these expressions for @xmath108 and finally solving the resulting system of algebraic equations for @xmath66 . it can be demonstrated that in solar - like atmospheres the convergence rate of this iterative scheme is practically unaffected if one retains only the zeroth - order lambda operator @xmath109 while putting @xmath110 in eqs . ( see @xcite ) . therefore , we shall develop this simplified jacobian iterative scheme in the following . we calculate the average over the line profile of the radiation field tensor : @xmath111 substituting this equation for the mean radiation field tensor into eq . for the source function @xmath112 and subtracting @xmath113 , we find @xmath114 , \label{eqmt4 } \end{split}\ ] ] where we have made explicit the height dependence of @xmath21 ( the dependence of @xmath115 , @xmath116 and @xmath117 is kept implicit ) . if we substitute into this equation the expression of the source function @xmath118 of eq . , we obtain @xmath119 . \label{eqmt5 } \end{split}\ ] ] moreover , defining @xmath120 taking into account that @xmath121 where @xmath122 noting also that @xmath123 and using eqs . and , we find that the correction to the line source function is @xmath124 , \label{eqmt7 } \end{split}\ ] ] applying the same reasoning to the continuum source function , from eq . , with @xmath125 @xmath126 taking the variation of the field tensor , @xmath127 and substituting eq . into eq . , after gathering the terms in @xmath128 , we obtain : @xmath129 \\ & \times\delta j^{0}_{0}\left(x';i\right ) = j^{0}_{0}{}^{\dagger}\left(x;i\right ) - j^{0}_{0}{}^{\rm old}\left(x;i\right ) \\ & + \frac{\lambda^{0}_{0}\left(x;i , i\right ) r_{x } \left(1 - \epsilon\right)\left(\bar{j}^{0}_{0}{}^{\dagger}\left(i\right ) - \bar{j}^{0}_{0}{}^{\rm old}\left(i\right)\right)}{1 - \left(1 - \epsilon\right ) \bar{\lambda}^{0}_{0}\left(i , i\right ) } . \label{eqmt11 } \end{split}\ ] ] the discretization of eq . in the frequency domain gives a linear system of @xmath130 equations ( with @xmath130 the number of frequency points ) for @xmath131 . substitution of this solution into eq . completes the iterative scheme for @xmath118 . as pointed out by @xcite and stated above , the solution of standard resonance line polarization problems using methods based on jacobi iteration can simply rely on the diagonal of the @xmath109 operator . the resulting equation for @xmath132 is thus formally equivalent to consider lambda iteration for @xmath26 . however , it is crucial to note that @xmath47 is improved at the rate of @xmath131 , because the anisotropy tensor @xmath47 is dominated by the stokes @xmath4 parameter which is , in turn , basically set by the values of @xmath118 . from eqs . , and @xmath133 where @xmath134 and @xmath135 result from the substitution of @xmath95 and @xmath96 into eqs . and . in summary , at each iterative step we solve the system of equations in order to obtain the correction of the @xmath46 radiation field . then , we use this result to solve eq . , which gives us the correction for the @xmath112 source function . finally , eq . gives us the correction for the @xmath136 source function . the numerical method presented in the last section makes use of jacobi s iterative method both for the line and the continuum part . the simpler alternative of this method is using lambda iteration for the continuum , which converges provided that the continuum opacity is weak enough with respect to that of the line . the method presented can solve both the crd line case without continuum opacity and the coherent continuum problem without line opacity , two problems that have different convergence rates . in order to illustrate this property of the numerical method we show the convergence rate for three different cases : i ) crd line for a @xmath137 transition without continuum , ii ) coherent continuum without line , iii ) non - coherent continuum without line . for the first case , we take a gaussian profile with @xmath138 ( distance between consecutive points in the frequency grid ) and @xmath139 . for the continuum cases , we take @xmath140 and , for the non - coherent case , @xmath141 ( width of the redistribution profile ) . we suppose an isothermal atmosphere and we solve with @xmath142 ( distance between consecutive points in the height grid , in units of the opacity scale height ) and @xmath143 gaussian nodes for angular integration in each hemisphere . we present the corresponding convergence rates in fig . [ figrc ] . ( solid lines ) and @xmath144 ( dashed lines ) at each iterative step for a ( crd ) resonance line without continuum ( black lines ) , and for the continuum case without line ( gray lines ) . we point out that the convergence rates for the coherent and non - coherent cases are indistinguishable . for the continuum case the source function is frequency dependent , but here we take a fixed frequency because the convergence rate is virtually identical for all of them.,width=325 ] to demonstrate the virtue of the method with respect to the continuum treatment , we solve a problem where we include both a weak line and continuum , but using lambda iteration for the continuum part . we take the same parameters used in fig . [ figrc ] and @xmath145 . this is a weak spectral line case , so the final rate of convergence is greatly influenced by the lambda iteration of the continuum , that has a very poor convergence rate ( see fig . [ figrc2 ] ) . ( solid lines ) , @xmath144 ( dashed lines ) and @xmath46 ( coincident with the solid line ) at each iterative step for the crd line transfer problem with continuum using the method described in section [ s32 ] ( black lines ) and using lambda iteration for the continuum part ( gray lines).,width=325 ] the code can also use @xcite acceleration to decrease the total computing time . to show its efficiency we solve the problem of fig . [ figrc2 ] using ng acceleration of third order . the number of iterative steps needed to reach convergence is greatly reduced without increasing significantly the computing time at each iterative step ( see fig . [ figrc3 ] ) . ( solid lines ) , @xmath144 ( dashed lines ) and @xmath46 ( coincident with the solid line ) at each iterative step for the crd line transfer problem with coherent and non - coherent continuum , with ( black lines ) and without ( gray lines ) ng acceleration.,width=325 ] the precision of the numerical method depends on the parameters of the discretization in space , angles and frequencies . in the figures that are shown in section [ s4 ] we take the following discretizations . a spatial height axis from @xmath146 to @xmath147 or @xmath148 ( this is more than needed to have an optically thick atmosphere at the bottom and an optically thin surface ) with @xmath149 or @xmath150 , with the height @xmath3 measured in units of the opacity scale height . we use gaussian quadrature with @xmath151 nodes at each hemisphere and a frequency axis that reaches @xmath152 doppler widths with @xmath153 in the core and with @xmath154 increasing with the distance to the resonance frequency @xmath155 until having @xmath156 in the far wings . in order to demonstrate the reliability of our radiative transfer code , we solve the radiation transfer problem in a plane - parallel homogeneous atmosphere , relying on the fact that the @xmath157-law ( @xcite ; generalized to the polarized case by @xcite ) provides an exact analytical result for the solution of this problem . we solve two of the problems of section [ s33 ] : coherent scattering in the continuum without line and resonance line without continuum . in the far wings of the line , the spectrum can be considered frequency independent . the source function equations are thus simplified as [ eqscoh]@xmath158 with [ eqscoh2]@xmath159 the @xmath157-law gives us the relation @xmath160 . in table [ tblcoh ] we show the relative error between the numerical result and this analytical relation ; the agreement is very satisfactory . ccc + @xmath117 & @xmath161 & error ( % ) + @xmath162 & @xmath163 & @xmath164 + @xmath165 & @xmath166 & @xmath167 + @xmath168 & @xmath169 & @xmath170 + @xmath164 & @xmath171 & @xmath172 + @xmath150 & @xmath173 & @xmath174 + @xmath175 & @xmath176 & @xmath177 + [ tblcoh ] in the absense of continuum the source function equations become [ eqslinepure]@xmath178 with [ eqslinepure2]@xmath179 in table [ tblpul ] we check the @xmath157-law for this line transfer problem with @xmath180 ; the law is satisfied with good agreement . ccc + @xmath115 & @xmath181 & error ( % ) + @xmath162 & @xmath182 & @xmath183 + @xmath165 & @xmath184 & @xmath185 + @xmath168 & @xmath186 & @xmath168 + @xmath164 & @xmath187 & @xmath188 + @xmath150 & @xmath189 & @xmath190 + @xmath175 & @xmath191 & @xmath192 + [ tblpul ] in the next section we apply our radiative transfer code to some particular cases , where we have both line and continuum . we study some of the effects of the non - coherence of the scattering . in this section we present some results of radiative transfer calculations in some model atmospheres . first , we make calculations in milne - eddington atmospheres with constant opacity ratios , because they are suitable for understanding the physics involved . secondly , we suppose some ad - hoc variation with height of the properties of a model atmosphere with a temperature minimum and a chromospheric temperature rise . in both cases we consider line transitions with and without intrinsic polarization , the last case being quite interesting in terms of the emergent fractional polarization profile . we study the interaction between a resonance line and the continuum radiation for two cases where non - coherent scattering in the continuum is taken into account or neglected . we assume a milne - eddington atmosphere with constant ratios between the different opacities involved . the important parameters in this model are the ratio between the opacity of the line and the continuum , @xmath193 , and the relative weight of the thermal part to the total opacity of the continuum , @xmath194 . we assume an intrinsically unpolarizable resonance line ( @xmath195 ) and an intrinsically polarizable line ( @xmath196 , @xmath197 ) . in both cases we solve the radiative transfer problem for a strong line @xmath198 and for a weak line @xmath199 with @xmath139 . for the continuum redistribution width we take @xmath141 ( value that we choose thinking in a forthcoming application to a realistic model ; in particular , this number is the ratio between the doppler widths of barium and hydrogen ) , and different values of @xmath117 . we use a milne - eddington atmosphere with slope @xmath200 . the non - coherent continuum scattering produces changes in the shape of the emergent fractional polarization profile , as has already been demonstrated in previous works ( see sect . when we study a strong ( @xmath201 ) unpolarizable line , the coherent profile gives zero polarization in the core of the line , as expected . it is interesting to note that the redistribution produced by the non - coherent scattering polarizes the core of the line , although its @xmath202 amplitude lies always below the continuum polarization level , i.e. , the line always depolarizes the continuum . thus , aside from being wider , the fractional polarization profile also shows non - zero polarization in the line core . the same happens to the @xmath202 profile in the case of a weak ( @xmath203 ) intrinsically unpolarizable line ( see fig . [ figme ] , left ) . the change in the @xmath202 profile is larger for weaker lines and smaller @xmath117 values ( or , equivalently , the more important is the scattering in the continuum ) . if we consider an intrinsically polarizable line , in order to obtain a noticeable change in the fractional polarization profile due to the non - coherent scattering in the continuum , we need the scattering coefficient @xmath20 to be dominant over the thermal absorption term ( small @xmath117 ) . the smaller @xmath117 , the more the polarization profiles changes . in all the cases shown in the right panels of fig . [ figme ] the intrinsic polarization of the line is dominant in its core and the non - coherence smoothes and broadens the fractional polarization profile in the wings of the line for small enough values of @xmath117 . we assume now a certain height variation of the parameter @xmath117 and of the planck function in order to obtain a more realistic stratification in the model atmosphere . inspired by semi - empirical models of the solar atmosphere , we choose @xmath20 in a way such that @xmath204 tends to unity near the surface and goes to zero at the bottom of the atmosphere ( see fig . [ figvarmo ] , and note that @xmath205 ) . we use two models that differ in the scattering coefficient . the variation with height of the scattering coefficient in the model @xmath206 is larger than in the model @xmath207 and the value of the scattering coefficient is the same at the height where the line integrated optical depth is unity . with these atmospheric models , we solve the two - level atom line transfer problem with @xmath139 and a non - coherent scattering redistribution width @xmath208 , both for an intrinsically unpolarizable line ( @xmath195 ) and for a polarizable one ( @xmath196 , @xmath197 ) and continuum thermal absorption @xmath19 variations , but different behaviors for the continuum scattering coefficient @xmath20 . the upper panel shows the planck function versus the integrated line optical depth . the middle panel shows the scattering coefficient @xmath20 versus the optical depth , with the black line indicating the model @xmath206 and the gray line the model @xmath207 . the bottom panel shows the quantity @xmath205 , and we point out that it has the typical variation that can be found in semi - empirical models , such as those of @xcite.,width=325 ] for a strong and unpolarizable line , the coherent profile shows a strong depolarization in the core ( fig . [ figvarqi ] , dashed lines ) . however , the non - coherent @xmath202 profile is strongly modified because the scattering redistribution produces polarization in the core of the lines ( fig . [ figvarqi ] , solid lines ) . for the model @xmath206 ( see fig . [ figvarmo ] ) , we see that the line does not fully depolarize the continuum , but the core is polarized and the profile is wider . for the model @xmath207 the non - coherent scattering generates an emission @xmath202 profile ( see fig . [ figvarqi ] ) . for a strong polarizable line , the coherent @xmath202 profile shows the expected polarization emission in the core of the line . for the model @xmath206 the polarization in the core of the line does not change , and the main effect of the non - coherent scattering is the smoothing of the peaks in the wings of the line and the broadening of the @xmath202 profile . for the model @xmath207 , the redistribution is able to change even the polarization in the core of the line , while producing a smoother and wider @xmath202 profile . what we want to emphasize with fig . [ figvarqi ] is that _ the non - coherent scattering in the continuum can be important and , under certain conditions , there can be an emission feature in the fractional linear polarization profile even when a total depolarization is expected . _ finally , we study the influence of the mass of the scatterer . to this aim , we take a milne - eddington atmosphere with slope @xmath200 and the @xmath20 variation of the model @xmath207 in fig . [ figvarmo ] . we solve the radiative transfer problem for a @xmath209 transition with gaussian absorption profile , with different widths of the redistribution function ( this width is inversely proportional to the square root of the mass of the scatterer ) . for small widths we approach the coherent case , where the line is depolarized . as we increase the width of the velocity redistribution profile , the linear polarization in the core of the line increases . in fig . [ figvarwpro ] we show some fractional polarization profiles for several values of the widths of the redistribution profile . if we take the center of the line as reference and we plot the fractional polarization at this frequency versus the widths of the redistribution profile , we obtain the curve shown in fig . [ figvarw ] , where we have also indicated the continuum fractional polarization level . in this figure we can see that from a given value of @xmath57 the line - center signal of the @xmath202 profile lies above the continuum level and increases to an asymptotic value . from this figure we can infer that , for a given value of @xmath117 , thomson scattering ( whose associated width is approximately @xmath210 times the doppler width of hydrogen ) produces a greater polarization than rayleigh scattering for an intrinsically unpolarizable line . ) for a @xmath211 transition in a milne - eddington atmosphere , with the @xmath20 and @xmath212 variations given by the model @xmath207 of fig . [ figvarmo ] , for different widths of the scattering redistribution function . the dotted line shows the coherent case.,width=325 ] ) for a @xmath211 transition in a milne - eddington atmosphere ( with @xmath204 given by the model @xmath207 of fig . [ figvarmo ] ) versus the width of the scattering redistribution function . the gray line represents the fractional continuum polarization amplitude.,width=325 ] in this paper we have studied the radiative transfer problem of resonance line polarization taking into account non - coherent continuum scattering , paying particular attention to the fractional linear polarization @xmath202 signals that can be produced around the core of intrinsically unpolarizable lines . we used the two - level atom model with crd and angle - averaged non - coherent scattering in the continuum . to numerically solve this type of radiative transfer problem we developed a jacobian iterative method for the line and continuum source functions , which yields a fast convergence rate even in the case of very small line strengths . the formulation of the numerical method makes it very suitable for a direct generalization to partial frequency redistribution and angle - dependent non - coherent scattering . we have shown that , under certain conditions , the non - coherent continuum scattering can change dramatically the core spectral region of the emergent @xmath202 profile with respect to that calculated assuming coherent continuum scattering . interestingly , @xmath202 polarization signals above the continuum level can be generated in the core of intrinsically unpolarizable @xmath213 lines ( i.e. , in spectral lines that were expected to simply depolarize the continuum polarization level ) . this result is of great potential interest for a better understanding of some enigmatic spectral lines of the second solar spectrum , which showed @xmath202 line - center signals above the continuum polarization level in spite of resulting from transitions between levels that were thought to be intrinsically unpolarizable ( see stenflo et al . 2000 ) . of particular interest for a first application is the d@xmath214 line of ba ii at 4934 , especially because 82% of the barium isotopes have nuclear spin @xmath215 ( i.e. , their d@xmath214 line transition is indeed between an upper and lower level with total angular momentum @xmath0 ) . in fact , our preliminary calculations for the ba ii d@xmath214 line ( neglecting the contribution of the 18% of barium that has hyperfine structure ) suggest that under certain stellar atmospheric conditions the physical mechanism discussed in this paper can produce significant @xmath202 emission features . finally , we point out that the core of strong lines with intrinsic polarization are practically not affected by the non - coherent scattering . therefore , the effects of the non - coherent scattering in the continuum are not always relevant and depend on the spectral line under study . financial support by the spanish ministry of economy and competitiveness through projects ( solar magnetism and astrophysical spectropolarimetry ) and consolider ingenio csd2009 - 00038 ( molecular astrophysics : the herschel and alma era ) is gratefully acknowledged . [ eqlam]@xmath217\lambda\left(x,\mu;i , j\right ) , \label{eqlam22 } \end{split } \\ t^{0}_{0}\left(x;i\right ) = & \frac{1}{2}\int_{-1}^{1}d\mu \ , t_{i}\left(x,\mu;i\right ) , \label{eqt00 } \\ \begin{split } t^{2}_{0}\left(x;i\right ) = & \frac{1}{4\sqrt{2}}\int_{-1}^{1}d\mu\bigg[\left(3\mu^{2 } - 1\right)t_{i}\left(x,\mu;i\right ) + 3\left(\mu^{2 } - 1\right)t_{q}\left(x,\mu;i\right)\bigg ] .\label{eqt20 } \end{split}\end{aligned}\ ] ]

line scattering polarization can be strongly affected by rayleigh scattering by neutral hydrogen and thompson scattering by free electrons . often a continuum depolarization results , but the doppler redistribution produced by the continuum scatterers , which are light ( hence , fast ) , induces more complex interactions between the polarization in spectral lines and in the continuum . here we formulate and solve the radiative transfer problem of scattering line polarization with non - coherent continumm scattering consistently . the problem is formulated within the spherical tensor representation of atomic and light polarization . the numerical method of solution is a generalization of the accelerated lambda iteration that is applied to both , the atomic system and the radiation field . we show that the redistribution of the spectral line radiation due to the non coherence of the continuum scattering may modify significantly the shape of the emergent fractional linear polarization patterns , even yielding polarization signals above the continuum level in intrinsically unpolarizable lines .

few star forming regions surpass the resplendent beauty of ngc2264 , the richly populated galactic cluster in the mon ob1 association lying approximately 760 pc distant in the local spiral arm . other than the orion nebula cluster , no other star forming region within one kpc possesses such a broad mass spectrum and well - defined pre - main sequence population within a relatively confined region on the sky . estimates for the total stellar population of the cluster range up to @xmath01000 members , with most low - mass , pre - main sequence stars having been identified from h@xmath4 emission surveys , x - ray observations by _ rosat _ , _ chandra _ , and _ xmm - newton _ , or by photometric variability programs that have found several hundred periodic and irregular variables . the cluster of stars is seen in projection against an extensive molecular cloud complex spanning more than two degrees north and west of the cluster center . the faint glow of balmer line emission induced by the ionizing flux of the cluster ob stellar population contrasts starkly with the background dark molecular cloud from which the cluster has emerged . the dominant stellar member of ngc2264 is the o7 v star , s monocerotis ( s mon ) , a massive multiple star lying in the northern half of the cluster . approximately 40 south of s mon is the prominent cone nebula , a triangular projection of molecular gas illuminated by s mon and the early b - type cluster members . ngc2264 is exceptionally well - studied at all wavelengths : in the millimeter by crutcher et al . ( 1978 ) , margulis & lada ( 1986 ) , oliver et al . ( 1996 ) , and peretto et al . ( 2006 ) ; in the near infrared ( nir ) by allen ( 1972 ) , pich ( 1992 , 1993 ) , lada et al . ( 1993 ) , rebull et al . ( 2002 ) , and young et al . ( 2006 ) ; in the optical by walker ( 1956 ) , rydgren ( 1977 ) , mendoza & gmez ( 1980 ) , adams et al . ( 1983 ) , sagar & joshi ( 1983 ) , sung et al . ( 1997 ) , flaccomio et al . ( 1999 ) , rebull et al . ( 2002 ) , sung et al . ( 2004 ) , lamm et al . ( 2004 ) , and dahm & simon ( 2005 ) ; and in x - rays by flaccomio et al . ( 2000 ) , ramirez et al . ( 2004 ) , rebull et al . ( 2006 ) , flaccomio et al . ( 2006 ) , and dahm et al . ( 2007 ) . ngc2264 was discovered by friedrich wilhelm herschel in 1784 and listed as h viii.5 in his catalog of nebulae and stellar clusters . the nebulosity associated with ngc2264 was also observed by herschel nearly two years later and assigned the designation : h v.27 . the roman numerals in herschel s catalog are object identifiers , with ` v ' referring to very large nebulae and ` viii ' to coarsely scattered clusters of stars . one of the first appearances of the cluster in professional astronomical journals is wolf s ( 1924 ) reproduction of a photographic plate of the cluster and a list of 20 suspected variables . modern investigations of the cluster begin with herbig ( 1954 ) who used the slitless grating spectrograph on the crossley reflector at lick observatory to identify 84 h@xmath4 emission stars , predominantly t tauri stars ( tts ) , in the cluster region . herbig ( 1954 ) postulated that these stars represented a young stellar population emerging from the dark nebula . walker s ( 1956 ) seminal photometric and spectroscopic study of ngc2264 discovered that a normal main sequence exists from approximately o7 to a0 , but that lower mass stars consistently fall above the main sequence . this observation was in agreement with predictions of early models of gravitational collapse by salpeter and by henyey et al . walker ( 1954 , 1956 ) proposed that these stars represent an extremely young population of cluster members , still undergoing gravitational contraction . walker ( 1956 ) further noted that the ttss within the cluster fall above the main sequence and that they too may be undergoing gravitational collapse . walker ( 1956 ) concluded that the study of ttss would be `` of great importance for our understanding of these early stages of stellar evolution . '' square false - color iras image ( 100 , 60 & 25 @xmath5 m ) of the mon ob1 and mon r1 associations . ngc2264 lies at the center of the image with several nearby iras sources identified , including the reflection nebulae ngc2245 and ngc2247 , and ngc2261 ( hubble s variable nebula ) . south of ngc2264 is the rosette nebula and its embedded cluster ngc2244 , lying 1.7 kpc distant in the perseus arm . [ f1 ] ] the molecular cloud complex associated with ngc2264 was found by crutcher et al . ( 1978 ) to consist of several cloud cores , the most massive of which lies roughly between s mon and the cone nebula . throughout the entire cluster region , oliver et al . ( 1996 ) identified 20 molecular clouds ranging in mass from @xmath6 to @xmath7 m@xmath3 . with ngc2264 these molecular clouds comprise what is generally regarded as the mon ob 1 association . active star formation is ongoing within ngc2264 as evidenced by the presence of numerous embedded protostars and clusters of stars , as well as molecular outflows and herbig - haro objects ( adams et al . 1979 ; fukui 1989 ; hodapp 1994 ; walsh et al . 1992 ; reipurth et al . 2004a ; young et al . 2006 ) . two prominent sites of star formation activity within the cluster are irs1 ( also known as allen s source ) , located several arcminutes north of the tip of the cone nebula , and irs2 , which lies approximately one - third of the distance from the cone nebula to s mon . new star formation activity is also suspected within the northern extension of the molecular cloud based upon the presence of several embedded iras sources and giant herbig - haro flows ( reipurth et al . 2004a , c ) . from 60 and 100 @xmath5 m iras images of ngc2264 , schwartz ( 1987 ) found that the cluster lies on the eastern edge of a ring - like dust structure , 3@xmath8 in diameter . shown in figure 1 is a 125@xmath1125 false - color iras image ( 100 , 60 , and 25 @xmath5 m ) centered near ngc2264 . the reflection nebulae ngc2245 and ngc2247 , members of the mon r1 association , are on the western boundary of this ring ( see the chapter by carpenter & hodapp ) . other components of the mon r1 association include the reflection nebulae ic446 and ic2169 , lkh@xmath4215 , as well as several early type ( b3b7 ) stars . it is generally believed that the mon r1 and mon ob 1 associations are at similar distances and are likely related . the rosette nebula , ngc2237 - 9 , and its embedded young cluster ngc2244 lie 5@xmath8 southwest of ngc2264 , 1.7 kpc distant in the outer perseus arm ( see the chapter by romn - ziga & lada ) . several arcs of dust and co emission have been identified in the region , which are believed to be supernovae remnants or windblown shells . many of these features are apparent in figure 2 , a wide - field h@xmath4 image of ngc2264 , ngc2244 , and the intervening region obtained by t. hallas and reproduced here with his permission . it is possible that star formation in the mon ob1 and r1 associations was triggered by nearby energetic events , but it is difficult to assess the radial distance of the ringlike structures evident in figure 2 , which may lie within the perseus arm or the interarm region . shown in figure 3 is a narrow - band composite image of ngc2264 obtained by t.a . rector and b.a . wolpa using the 0.9 meter telescope at kitt peak . s mon dominates the northern half of the cluster , which lies embedded within the extensive molecular cloud complex . image of ngc2264 ( upper center ) , the rosette nebula and ngc2244 ( lower right ) , and the numerous windblown shells and supernova remnants possibly associated with the mon ob1 or mon ob2 associations . the cone nebula is readily visible just above and left of image center as is s mon . also apparent in the image is the dark molecular cloud complex lying to the west of ngc2264 . this image is a composite of 16 20-minute integrations obtained by t. hallas using a 165 mm lens and an astrodon h@xmath4 filter . [ f2 ] ] to summarize all work completed over the last half - century in ngc2264 would be an overwhelming task and require significantly more pages than alloted for this review chapter . the literature database for ngc2264 and its members has now grown to over 400 refereed journal articles , conference proceedings , or abstracts . here we attempt to highlight large surveys of the cluster at all wavelengths as well as bring attention to more focused studies of the cluster that have broadly impacted our understanding of star formation . the chapter begins with a review of basic cluster properties including distance , reddening , age , and inferred age dispersion . it then examines the ob stellar population of the cluster , the intermediate and low - mass stars , and finally the substellar mass regime . different wavelength regions are examined from the centimeter , millimeter , and submillimeter to the far- , mid- , and near infrared , the optical , and the x - ray regimes . we then review many photometric variability studies of the cluster that have identified several hundred candidate members . finally , we consider future observations of the cluster and what additional science remains to be reaped from ngc2264 . the cluster has remained in the spotlight of star formation studies for more than 50 years , beginning with the h@xmath4 survey of herbig ( 1954 ) . its relative proximity , low foreground extinction , large main sequence and pre - main sequence populations , the lack of intense nebular emission , and the tremendous available archive of observations of the cluster at all wavelengths guarantee its place with the orion nebula cluster and the taurus - auriga molecular clouds as the most accessible and observed galactic star forming region . ( red - orange ) , and [ s ii ] ( blue - violet ) . the field of view is approximately 0.75@xmath8@xmath11@xmath8 . s mon lies just above the image center and is believed to be the ionizing source of the bright rimmed cone nebula . ngc2264 is ideally suited for accurate distance determinations given its lack of significant foreground extinction and the abundant numbers of early - type members . difficulty in establishing the cluster distance , however , arises from the near vertical slope of the zero age main sequence ( zams ) for ob stars in the color - magnitude diagram and from the depth of the cluster along the line of sight , which has not been assessed . the distance of ngc2264 is now widely accepted to be @xmath0760 pc , but estimates found in the early literature vary significantly . walker ( 1954 ) used photoelectric observations of suspected cluster members earlier than a0 to derive a distance modulus of 10.4 mag ( 1200 pc ) by fitting the standard main sequence of johnson & morgan ( 1953 ) . herbig ( 1954 ) adopted a distance of 700 pc , the mean of published values available at the time . for his landmark study of the cluster , walker ( 1956 ) revised his earlier distance estimate to 800 pc using the modified main sequence of johnson & hiltner ( 1956 ) . prez et al . ( 1987 ) redetermined the distances of ngc2264 and ngc2244 ( the rosette nebula cluster ) assuming an anomalous ratio of total - to - selective absorption . their revised distance estimate for ngc2264 was @xmath9 pc . included in their study is an excellent summary of distance determinations for ngc2244 and ngc2264 found in the literature from 1950 to 1985 ( their table xi ) . the mean of these values for ngc2264 is @xmath10 pc , significantly less than their adopted distance . prez ( 1991 ) provides various physical characteristics of the cluster including distance , total mass , radius , mean radial velocity , and age , but later investigations of the cluster have since revised many of these estimates . from modern ccd photometry of ngc2264 , sung et al . ( 1997 ) , determined a mean distance modulus of @xmath11 or 760 pc using 13 b - type cluster members , which have distance moduli in the range from @xmath12 mag . this value is now cited in most investigations of the cluster . the projected linear dimension of the giant molecular cloud associated with ngc2264 including the northern extension is nearly 28 pc . if a similar depth is assumed along the cluster line of sight , an intrinsic uncertainty of nearly 4% is introduced into the distance determination . interstellar reddening toward ngc2264 is recognized to be quite low . walker ( 1956 ) found @xmath13 or @xmath14 assuming the normal ratio of total - to - selective absorption , @xmath15 . prez et al . ( 1987 ) found a similar reddening value with @xmath16 , but used @xmath17 to derive their significantly greater distance to the cluster . in their comprehensive photometric study , sung et al . ( 1997 ) found the mean reddening of 21 ob stars within the cluster to be @xmath18 , in close agreement with the estimates of walker ( 1956 ) and prez et al . no significant deviation from these values has been found using the early - type cluster members . individual extinctions for the suspected low - mass members , however , are noted to be somewhat higher . rebull et al . ( 2002 ) derive a mean @xmath19 or @xmath20 from their spectroscopic sample of more than 400 stars , only 22% of which are earlier than k0 . dahm & simon ( 2005 ) find that for the h@xmath4 emitters within the cluster with established spectral types , a mean a@xmath21 of 0.71 mag follows from @xmath22 . some of these low - mass stars are suffering from local extinction effects ( e.g. circumstellar disks ) or lie within deeply embedded regions of the molecular cloud . the mean extinction value derived for the ob stellar population , which presumably lies on the main sequence and possesses well - established intrinsic colors , should better represent the distance - induced interstellar reddening suffered by cluster members . table 1 summarizes the distance and extinction estimates of the cluster found or adopted in selected surveys of ngc2264 . lcccccc + + authors & age ( myr ) & m@xmath21 range & isochrone & e(@xmath23 ) & distance ( pc ) & notes + walker ( 1956 ) & 3.0 & @xmath24 & henyey et al . ( 1955 ) & 0.082 & 800 & pe and pg + & & & & & & + mendoza & gmez ( 1980 ) & 3.0 & @xmath25 & iben & talbot ( 1966 ) & 0.06 & 875 & pe + & & & & & & + adams et al . ( 1983 ) & 3.0 - 6.0 & @xmath26 & cohen & kuhi ( 1979 ) & [ 0.06 ] & [ 800 ] & pg + & & & & & & + sagar & joshi ( 1983 ) & 5 & @xmath27 & cohen & kuhi ( 1979 ) & @xmath280.12 & 794 & pe + & & & & & & + prez et al . ( 1987 ) & ... & @xmath29 & ... & 0.06 & 950 & pe + & & & & & & + feldbrugge & van genderen ( 1991 ) & @xmath30 & @xmath31 & ... & 0.04 & 700 & pe + & & & & & & + neri et al . ( 1993 ) & ... & @xmath32 & ... & 0.05 & 910 & pe + & & & & & & + sung et al . ( 1997 ) & 0.88.0 & @xmath33 & s94 , bm96 & 0.071 & 760 & ccd + & & & & & & + flaccomio et al . ( 1999 ) & 0.110.0 & @xmath34 & dm97 & [ 0.06 ] & [ 760 - 950 ] & ccd + & & & & & & + park et al . ( 2000 ) & 0.94.3 & @xmath35 & dm94 , s94 , b98 & 0.066 & 760 & ccd + & & & & & & + rebull et al . ( 2002 ) & 0.16.0 & @xmath36 & dm94 , sdf00 & 0.15 & [ 760 ] & ccd + & & & & & & + sung et al . ( 2004 ) & 3.1 & @xmath37 & sdf00 & 0.070.15 & ... & ccd + + + + + + the age of ngc2264 has long been inferred to be young given the large ob stellar population of the cluster and the short main sequence lifetimes of these massive stars . walker ( 1956 ) derived an estimate of 3 myr , based upon the main sequence contraction time of an a0-type star ( the latest spectral type believed to be on the zams ) from the theoretical work of henyey et al . iben & talbot ( 1966 ) , however , directly compared theoretical time - constant loci or isochrones with the color - magnitude diagram of ngc2264 , concluding that star formation began within the cluster more than 65 myr ago . they further suggested that the star formation rate has been increasing exponentially with time , and that the average mass of each subsequent generation of stars has also increased exponentially . strom et al . ( 1971 ) and strom et al . ( 1972 ) , however , in their insightful investigations of balmer line emission and infrared excesses among a and f - type members of ngc2264 , concluded that dust and gas shells ( spherical or possibly disk - like ) were common among pre - main sequence stars , making strict interpretation of age dispersions from color - magnitude diagrams difficult if not impossible . strom et al . ( 1972 ) conclude that an intrinsic age dispersion of 1 to 3 myr is supported by the a and f - type members of ngc2264 . adams et al . ( 1983 ) revisited the question of age dispersion within ngc2264 in their deep photometric survey of the cluster . from the theoretical hr diagram of probable cluster members , a significant age spread of more than 10 myr is inferred . adams et al . ( 1983 ) further suggest that sequential star formation has occurred within the cluster , beginning with the low - mass stars and continuing with the formation of more massive cluster members . from the theoretical models of cohen & kuhi ( 1979 ) , they derive a mean age of the low - mass stellar population of 45 myr . modern ccd investigations of ngc2264 have yielded similar ages , but they are strongly dependent upon the pre - main sequence models adopted for use . sung et al . ( 1997 ) use the pre - main sequence models of bernasconi & maeder ( 1996 ) and swenson et al . ( 1994 ) to find that the ages of most suspected pre - main sequence members of ngc2264 are from 0.8 to 8 myr , while the main sequence stars range from 1.4 to 16 myr . park et al . ( 2000 ) compare ages and age dispersions of ngc2264 from four sets of pre - main sequence models : those of swenson et al . ( 1994 ) , dantona & mazzitelli ( 1994 ) , baraffe et al . ( 1998 ) , and the revised models of baraffe et al . ( 1998 ) , which incorporate a different ratio of mixing length to pressure scale height . from the distribution of pre - main sequence candidates , the median ages and age dispersions ( respectively ) from the models of swenson et al . ( 1994 ) are 2.1 and 8.0 myr ; dantona & mazzitelli ( 1994 ) 0.9 and 5.5 myr ; baraffe et al . ( 1998 ) 4.3 and 15.3 myr , and for the revised models of baraffe et al . ( 1998 ) 2.7 and 10 myr . figure 4 is a reproduction of the hr diagrams from park et al . ( 2000 ) , with the evolutionary tracks for the various models superposed . the cluster ages and age spreads are represented by the solid and dashed lines , respectively . these surveys used photometry alone to place stars on the hr diagram . ] rebull et al . ( 2002 ) compare the derived ages and masses from dantona & mazzitelli ( 1994 ) and siess et al . ( 2000 ) for a spectroscopically classified sample of stars in ngc2264 , finding systematic differences between the models of up to a factor of two in mass and half an order of magnitude in age . dahm & simon ( 2005 ) use the evolutionary models of dantona & mazzitelli ( 1997 ) to determine a median age of 1.1 myr and to infer an age dispersion of @xmath05 myr for the nearly 500 h@xmath4 emission stars identified within the immediate cluster vicinity . star formation , however , continues within the cluster as evidenced by _ observations of the star forming cores near irs1 and irs 2 . young et al . ( 2006 ) suggest that a group of embedded , low - mass protostars coincident with irs2 exhibits a velocity dispersion consistent with a dynamical age of several @xmath7 yr . the presence of this young cluster as well as other deeply embedded protostars ( allen 1972 ; castelaz & grasdalen 1988 ; margulis et al . 1989 ; thompson et al . 1998 ; young et al . 2006 ) among the substantial number of late b - type dwarfs in ngc2264 implies that an intrinsic age dispersion of at least 35 myr exists among the cluster population . soderblom et al . ( 1999 ) obtained high resolution spectra for 35 members of ngc2264 in order to determine li abundances , radial velocities , rotation rates , and chromospheric activity levels . their radial velocities indicate that the eight stars in their sample lying below the 5 myr isochrone of the cluster are non - members , implying that the age spread within the cluster is only @xmath04 myr . the hierarchical structure of the cluster would indicate that star formation has occurred in different regions of the molecular cloud over the last several myr . we can speculate that from the large quantities of molecular gas remaining within the various cloud cores , star formation will continue in the region for several additional myr . the cluster ages adopted or derived by selected previous investigations of ngc2264 are also presented in table 1 . s mon ( 15 mon ) dominates the northern half of the cluster and is believed to be the ionizing source for the cone nebula ( schwartz et al . 1985 ) as well as many of the observed bright rims in the region including sharpless 273 . in addition to exhibiting slight variability ( hundredths of a magnitude ) , gies et al . ( 1993 , 1997 ) determined s mon to be a visual and spectroscopic binary from speckle interferometry , _ hst _ imaging , and radial velocity data . with a semi - major axis of over 27 au ( assuming a distance of 800 pc ) , the 24-year orbit of the binary is illustrated in figure 5 , taken from gies et al . ( 1997 ) and based upon an orbital inclination of 35@xmath8 to the plane of the sky . the mass estimates for the primary component of s mon ( o7 v ) , 35 m@xmath3 , and the secondary ( o9.5 v ) , 24 m@xmath3 , assume a distance of 950 pc ( prez et al . 1987 ) , which is significantly greater than the currently accepted cluster distance . using their derived orbital elements and with the adopted cluster distance ( 800 pc ) , a mass ratio of q@xmath380.75 is derived , leading to primary and secondary masses 18.1 and 13.5 m@xmath3 , respectively . these mass estimates , however , are inconsistent with the spectral type of s mon and its companion . over the last 13 years , observations of the star using the fine guidance sensor onboard hst have revised earlier estimates of the orbital elements , and the period of the binary now appears to be significantly longer than previously determined ( gies , private communication ) . s mon also exhibits uv resonance line profile variation as well as fluctuation in soft x - ray flux ( snow et al . 1981 ; grady et al . both of these phenomenon are interpreted as being induced by variations in mass loss rate . , width=288 ] the ob - stellar population of ngc2264 consists of at least two dozen stars , several of which were spectroscopically classified by walker ( 1956 ) . morgan et al . ( 1965 ) list 17 early - type stars within the cluster , ranging in spectral type from o7 to b9 . most of these stars are concentrated around s mon or the small rosette - shaped nebulosity lying to its southwest , however , several do lie in the southern half of the cluster . table 2 lists the suspected early - type members of ngc2264 with their positions , available photometry , spectral types , and any relevant notes of interest . several members of the greater mon ob 1 association are also included in table 1 for completeness , some obtained from the compilation of turner ( 1976 ) . the hipparcos survey of de zeeuw et al . ( 1999 ) was unable to provide any rigorous kinematic member selection for mon ob1 members given the distance of the association . many of the early - type stars are known binaries including s mon , hd 47732 , and hd 47755 ( gies et al . 1993 , 1997 ; morgan et al . 1965 ; dahm et al . 2007 ) . the b3 v star hd 47732 is identified by morgan et al ( 1965 ) as a double - line spectroscopic binary and is examined in detail by beardsley & jacobsen ( 1978 ) and koch et al . another early - type cluster member , hd 47755 , was classified by trumpler ( 1930 ) as a b5-type spectroscopic binary , but assigned a low probability of membership by the proper motion survey of vasilevskis et al . koch et al . ( 1986 ) derive a 1.85 day period for the shallow 0.1 mag eclipse depth . dahm et al . ( 2007 ) list several early - type adaptive optics ( ao ) binaries and give their separations and position angles ( see their table 2 ) . other massive cluster members include several mid - to - late b - type stars clustered near s mon including bd@xmath3910@xmath81222 ( b5v ) , hd 261938 ( b6v ) , w 181 ( b9v ) , hd 261969 ( b9iv ) , and @xmath07 to the east , hd 47961 ( b2v ) . although on the cloud periphery , hd 47961 is surrounded by several x - ray and h@xmath4 emission stars and its derived distance modulus , m@xmath40m@xmath21@xmath389.74 , matches that of the cluster . the proper motion data of vasilevskis et al . ( 1965 ) also support its cluster membership status . within the rosette - shaped emission / reflection nebula southwest of s mon is a second clustering of massive stars including w 67 ( b2v ) , hd 47732 ( b3v ) , hd 47755 ( b5v ) , w 90 ( b8e ) , and hd 261810 ( b8v ) . between this grouping and hd 47887 ( w 178 ) , the b2 iii binary just north of the cone nebula , there are just a handful of candidate early - type members : hd 47777 ( b3v ) , hd 262013 ( b5v ) , and hd 261940 ( b8v ) . hd 47887 was assigned a low probability of membership by vasilevskis et al . ( 1965 ) , but if assumed to lie on the zams , the distance modulus of this b2 binary equals that of the cluster . the other component of hd 47887 is the b9 dwarf , hd 47887b , several arcseconds southwest of the primary . hd 262042 , the b2 dwarf directly south of the cone nebula , is a probable background star given its distance modulus ( @xmath41 ) . far to the west is the b5 star hd 47469 , a probable non - member lying @xmath0200 pc in front of the cluster . most of the ob stellar population of the cluster lies concentrated within the boundaries of the molecular cloud , matching the distribution of suspected low - mass members ( i.e. x - ray detected sources and h@xmath4 emitters ) . walker ( 1956 ) notes the presence of five yellow giants within the cluster region and suggests they might lie in the foreground . he assigns to these potential interlopers spectral types and luminosity classes of g5 iii to w73 , k2 ii - iii to w237 , k3 ii - iii to w229 , k3 ii - iii to w69 , and k5 iii to w37 . underhill ( 1958 ) presents radial velocities of these stars , finding that only one ( w73 ) is consistent with the mean radial velocities of the early - type stars . ccccccc + + identifier & ra ( j2000 ) & @xmath42 ( j2000 ) & @xmath43 & @xmath23 & sp t & notes + hd 44498 & 06 22 22.5 & + 08 19 36 & 8.82 & @xmath44 & b2.5v & mon ob1 + hd 45789a & 06 29 55.9 & + 07 06 43 & 7.10 & @xmath45 & b2.5iv & + hd 45827 & 06 30 05.5 & + 09 01 46 & 6.57 & 0.11 & a0iii & + hd 46300 & 06 32 54.2 & + 07 19 58 & 4.50 & 0.00 & a0ib & 13 mon + hd 46388 & 06 33 20.3 & + 04 38 58 & 9.20 & 0.15 & b6v & mon ob1 + hd 261490 & 06 39 34.3 & + 08 21 01 & 8.91 & 0.18 & b2iii & mon ob1 + hd 261657 & 06 40 04.8 & + 09 34 46 & 10.88 & 0.03 & b9v & + hd 47732 & 06 40 28.5 & + 09 49 04 & 8.10 & @xmath46 & b3v & bin + bd+09 1331b & 06 40 37.2 & + 09 47 30 & 10.79 & 0.62 & b2v & + hd 47755 & 06 40 38.3 & + 09 47 15 & 8.43 & @xmath45 & b5v & eb + hd 47777 & 06 40 42.2 & + 09 39 21 & 7.95 & @xmath47 & b3v & + hd 261810 & 06 40 43.2 & + 09 46 01 & 9.15 & @xmath45 & b8v & var + v v590 mon & 06 40 44.6 & + 09 48 02 & 12.88 & 0.15 & b8pe & w90 + bd+10 1220b & 06 40 58.4 & + 09 53 42 & 7.6 & ... & ob & s mon b + hd 261902 & 06 40 58.5 & + 09 33 31 & 10.20 & 0.06 & b8v & + hd 47839 & 06 40 58.6 & + 09 53 44 & 4.66 & @xmath48 & o7ve & s mon a + bd+10 1222 & 06 41 00.2 & + 09 52 15 & 9.88 & @xmath49 & b5v & var + hd 261938 & 06 41 01.8 & + 09 52 48 & 8.97 & @xmath49 & b6v & var + hd 261903 & 06 41 02.8 & + 09 27 23 & 9.16 & @xmath50 & b9v & + hd 261940 & 06 41 04.1 & + 09 33 01 & 10.0 & 0.04 & b8v & + hd 261937 & 06 41 04.5 & + 09 54 43 & 10.36 & 0.35 & b8v & bin + hd 47887 & 06 41 09.6 & + 09 27 57 & 7.17 & @xmath51 & b2iii & bin + hd 47887b & 06 41 09.8 & + 09 27 44 & 9.6 & ... & b9v & + irs 1 & 06 41 10.1 & + 09 29 33 & ... & ... & b2b5 & + hd 261969 & 06 41 10.3 & + 09 53 01 & 9.95 & 0.00 & b9iv & var + w 181 & 06 41 11.2 & + 09 52 55 & 10.03 & @xmath52 & b9vn & bin + hd 262013 & 06 41 12.9 & + 09 35 49 & 9.34 & @xmath46 & b5vn & + hd 262042 & 06 41 18.7 & + 09 12 49 & 9.02 & 0.03 & b2v & + hd 47934 & 06 41 22.0 & + 09 43 51 & 8.88 & @xmath53 & b9v & + hd 47961 & 06 41 27.3 & + 09 51 14 & 7.51 & @xmath47 & b2v & bin + hd 48055 & 06 41 49.7 & + 09 30 29 & 9.00 & @xmath53 & b9v & var + + the full extent of the intermediate and low - mass ( @xmath54 m@xmath3 ) stellar population of ngc 2264 and its associated molecular cloud complex has not been assessed , but several hundred suspected members exhibiting h@xmath4 emission , x - ray emission , or photometric variability have been cataloged principally in the vicinity of the known molecular cloud cores . the traditional means of identifying young , low - mass stars has been the detection of h@xmath4 emission either using wide - field , low - resolution , objective prism imaging or slitless grating spectroscopy . in pre - main sequence stars , strong h@xmath4 emission is generally attributed to accretion processes as gas is channeled along magnetic field lines from the inner edge of the circumstellar disk to the stellar photosphere . weak h@xmath4 emission is thought to arise from enhanced chromospheric activity , similar to that observed in field dme stars . herbig ( 1954 ) conducted the first census of ngc2264 , discovering 84 h@xmath4 emission stars concentrated in two groups in a @xmath55 rectangular area centered roughly between s mon and the cone nebula . the larger of the two groups lies north of hd 47887 , the bright b2 star near the tip of the cone nebula . the second group is concentrated southwest of s mon , near w67 and the prominent rosette of emission / reflection nebulosity . marcy ( 1980 ) re - examined ngc2264 using the identical instrumentation of herbig ( 1954 ) and discovered 11 additional h@xmath4 emitters , but was unable to detect emission from a dozen of the original lh@xmath4 stars . the implication was that h@xmath4 emission varied temporally among these low - mass stars , and that a reservoir of undetected emitters was present within the cluster and other young star forming regions . narrow - band filter photometric techniques have also been used to identify strong h@xmath4 emitters in ngc2264 . the first such survey , by adams et al . ( 1983 ) , was photographic in nature and capable of detecting strong emitters , i.e. classical t tauri stars ( ctts ) . their deep @xmath56 photographic survey of the cluster found @xmath0300 low - luminosity ( 7.5@xmath5712.5 ) pre - main sequence candidates . ogura ( 1984 ) conducted an objective prism survey of the mon ob1 and r1 associations , focusing upon regions away from the core of ngc2264 that had been extensively covered by herbig ( 1954 ) and marcy ( 1980 ) . this kiso h@xmath4 survey discovered 135 new emission - line stars , whose distribution follows the contours of the dark molecular cloud complex . more than three dozen of these h@xmath4 emitters lie within or near ngc2245 to the west . in the molecular cloud region north of ngc2264 , another two dozen emitters were identified by ogura ( 1984 ) . reipurth et al . ( 2004b ) , however , noticed a significant discrepancy with ogura s ( 1984 ) emitters , many of which were not detected in their objective prism survey . it is possible that many of ogura s ( 1984 ) sources are m - type stars which exhibit depressed photospheric continua near h@xmath4 caused by strong tio band absorption . the resulting `` peak '' can be confused for h@xmath4 emission . sung et al . ( 1997 ) conducted a ccd narrow - band filter survey of ngc2264 , selecting pre - main sequence stars based upon differences in @xmath58 color with that of normal main sequence stars . using this technique , sung et al . ( 1997 ) identified 83 pre - main sequence stars and 30 pre - main sequence candidates in the cluster . of these 83 pre - main sequence stars , dahm & simon ( 2005 ) later established that 61 were cttss , 12 were weak - line t tauri stars ( wtts ) and several could not be positively identified . while capable of detecting strong h@xmath4 emission , the narrow - band filter photometric techniques are not able to distinguish the vast majority of weak - line emitters in the cluster . lamm et al . ( 2004 ) also obtained narrow - band h@xmath4 photometry for several hundred stars in their periodic variability survey of ngc2264 . more recently , reipurth et al . ( 2004b ) carried out a 3@xmath8@xmath13@xmath8 objective prism survey of the ngc2264 region using the eso 100/152 cm schmidt telescope at la silla , which yielded 357 h@xmath4 emission stars , 244 of which were newly detected . reipurth et al . ( 2004b ) ranked emission strengths on an ordinal scale from 1 to 5 , with 1 indicating faint emission and strong continuum and 5 the inverse . comparing with the slitless grism survey of dahm & simon ( 2005 ) , the technique adopted by reipurth et al . ( 2004b ) is capable of detecting weak emission to a limit of w(h@xmath4)@xmath59 . shown in figure 6 ( from reipurth et al . 2004b ) is the distribution of their 357 detected h@xmath4 emitters , superposed upon a @xmath60co ( 10 ) map of the region obtained with the at&t bell laboratories 7 m offset cassegrain antenna . the main concentrations of h@xmath4 emitters were located within or near the strongest peaks in co emission strength . a halo of h@xmath4 emitters in the outlying regions does not appear to be associated with the co cloud cores and may represent an older population of stars that have been scattered away from the central molecular cloud . reipurth et al . ( 2004b ) speculate that this may be evidence for prolonged star formation activity within the molecular cloud hosting ngc2264 . the slitless wide - field grism spectrograph ( wfgs ) survey of the cluster by dahm & simon ( 2005 ) increased the number of h@xmath4 emitters in the central @xmath61 region between s mon and the cone nebula to nearly 500 stars . capable of detecting weak emission with _ w_(h@xmath4 ) @xmath62 , the wfgs observations added more than 200 previously unknown h@xmath4 emitters to the growing number of suspected cluster members . combining the number of known emitters outside of the survey of dahm & simon ( 2005 ) identified by ogura ( 1984 ) and reipurth et al . ( 2004b ) , we derive a lower limit for the low - mass ( 0.22 m@xmath3 ) stellar population of the cluster region of more than 600 stars . from a compilation of 616 candidate cluster members with @xmath63 , ( from two deep _ chandra _ observations ) , flaccomio ( private communication ) finds that 202 do not exhibit detectable h@xmath4 emission . accounting for lower masses below the detection threshold of the various h@xmath4 surveys , x - ray selected stars that lack detectable h@xmath4 emission , and embedded clusters of young stars still emerging from their parent molecular cloud cores , the stellar population of the cluster could exceed 1000 members . emitters in ngc2264 from a 3@xmath8@xmath13@xmath8 objective prism survey of the region by reipurth et al . ( 2004b ) , superposed upon a grayscale map of @xmath60co ( 10 ) by john bally . the emitters are concentrated along the broad ridgeline of the molecular cloud and near s mon . north and west of the cluster , h@xmath4 emitters are scattered around and within the molecular cloud perimeter . the stars that appear unassociated with molecular emission may have been scattered from the central cluster region , implying that star formation has been occurring with the molecular cloud complex for several myr . [ f6],scaledwidth=95.0% ] x - ray emission is another well - established discriminant of stellar youth , with pre - main sequence stars often exhibiting x - ray luminosities ( @xmath64 ) 1.5 dex greater than their main sequence counterparts . the importance of x - ray emission as a youth indicator was immediately recognized in young clusters and associations where field star contamination is significant . the ability to select pre - main sequence stars from field interlopers , most notably after strong h@xmath4 emission has subsided , led to the discovery of a large population of weak - line t tauri stars associated with many nearby star forming regions . among field stars , however , the e - folding time for x - ray emission persists well into the main sequence phase of evolution . consequently , some detected x - ray sources are potentially foreground field stars not associated with a given star forming region . for ngc2264 , flaccomio ( private communication ) estimates the fraction of _ chandra _ detected x - ray sources that are foreground interlopers to be @xmath07% . detailed discussion of x - ray observations made to date with rosat , _ xmm - newton _ , or _ chandra _ is given in section 9 , but here we summarize these results in the context of the low - mass stellar population of ngc2264 . patten et al . ( 1994 ) used the high resolution imager ( hri ) onboard rosat to image ngc2264 and identified 74 x - ray sources in the cluster , ranging from s mon to late - k type pre - main sequence stars . this shallow survey was later incorporated by flaccomio et al . ( 2000 ) who combined six pointed rosat hri observations of the cluster for a total integration time of 25 ks in the southern half and 56 ks in the northern half of the cluster . in their sample of 169 x - ray detections ( 133 with optical counterparts ) , @xmath65 were gkm spectral types , including many known cttss and wttss . ramirez et al . ( 2004 ) cataloged 263 x - ray sources in a single 48.1 ks observation of the northern half of the cluster obtained with the advanced ccd imaging spectrometer ( acis ) onboard _ chandra_. flaccomio et al . ( 2006 ) added considerably to the x - ray emitting population of the cluster in their analysis of a single 97 ks _ acis - i observation of the southern half of the cluster , finding an additional 420 sources . of the nearly 700 x - ray sources detected by _ chandra _ in the cluster region , 509 are associated with optical or nir counterparts . of these , the vast majority are still descending their convective tracks . dahm et al . ( 2007 ) identify 316 x - ray sources in ngc2264 in two 42 ks integrations of the northern and southern cluster regions made with the european photon imaging camera ( epic ) onboard _ xmm - newton_. the vast majority of these sources have masses @xmath66 2 m@xmath3 . the positions of these x - ray selected stars as well as several embedded sources lacking optical counterparts are shown in figure 7 , superposed upon the red dss image of the cluster . square red image of ngc2264 obtained from the digitized sky survey ( dss ) showing the field of view for each of the _ xmm _ epic exposures from the survey of dahm et al . the black circles mark 300 detected x - ray sources having optical counterparts , presumably cluster members , while the white circles denote embedded sources or background objects observed through the molecular cloud . the distribution of x - ray sources closely matches the distribution of h@xmath4 emitters seen in figure 6 . [ f7],width=480 ] photometric variability is among the list of criteria established by joy ( 1945 ) for membership in the class of t tauri stars . young , low - mass stars are believed to experience enhanced , solar - like magnetic activity in which large spots or plage regions cause photometric variability at amplitudes typically @xmath670.2 mag . if active accretion is still taking place from a circumstellar gas disk , hot spots from the impact point on the stellar surface are believed to induce large amplitude , irregular variations . wolf ( 1924 ) first examined ngc2264 for photometric variables , identifying 24 from nine archived plates of the cluster taken between 1903 and 1924 . herbig ( 1954 ) noted that 75% ( 18 of 24 ) of these variables also exhibited h@xmath4 emission . herbig ( 1954 ) further suggested that given the variable nature of h@xmath4 emission , continued observation of the cluster would likely remove some of the variables from the non - emission group of stars . nandy ( 1971 ) found that among the h@xmath4 emitters in ngc2264 that are also photometric variables , a positive correlation exists between infrared ( out to @xmath68band ) and ultraviolet excesses . nandy & pratt ( 1972 ) show that the range of variability among the t tauri population in ngc2264 is less in @xmath68band than in other filters ( @xmath69 ) and also demonstrate a technique of studying the extinction properties of dust grains enveloping t tauri stars from their color variations . breger ( 1972 ) undertook an extensive photometric variability study of ngc2264 , finding that only 25% of the pre - main sequence a and f - type stars exhibit short - period variations . breger ( 1972 ) also found a strong correlation between `` shell indicators '' and variability , particularly among the t tauri stars . w90 ( lh@xmath4 25 ) was also found to have brightened by @xmath70 mag in @xmath43 since 1953 , an interesting trend for this peculiar herbig aebe star . koch & perry ( 1974 ) undertook an extensive photographic study of the cluster , identifying over 50 new variables , 16 of which were suspected eclipsing binaries , but no periods could be established . koch & perry ( 1974 ) conclude that many more low - amplitude ( @xmath670.2 mag ) variables must remain unidentified within the cluster , a prescient statement that would remain unproven until the coming of modern ccd photometric surveys . variability studies of individual cluster members include those of rucinski ( 1983 ) for w92 , koch et al . ( 1978 ) for hd 47732 , and walker ( 1977 ) also for w92 . ccd photometric monitoring programs have proven highly effective at identifying pre - main sequence stars from variability analysis . kearns et al . ( 1997 ) reported the discovery of nine periodic variables in ngc2264 , all g and k - type stars . an additional 22 periodic variables were identified by kearns & herbst ( 1998 ) including the deeply eclipsing ( 3.5 mag . in @xmath71 ) , late k - type star , kh-15d . two large - scale photometric variability campaigns have added several hundred more periodic variables to those initially identified by kearns et al . ( 1997 ) and kearns & herbst ( 1998 ) . the surveys of makidon et al . ( 2004 ) and lamm et al . ( 2004 ) were completed in the 2000 - 2001 and 1996 - 1997 observing seasons , respectively . their derived periods for the commonly identified variables show extraordinary agreement , confirming the robustness of the technique . the makidon et al . ( 2004 ) survey used the 0.76 m telescope at mcdonald observatory imaging upon a single 2048@xmath12048 ccd . the short focal length of the set - up resulted in a total field of view of approximately 46square . their observations were made over a 102-day baseline with an average of 5 images per night on 23 separate nights . makidon et al . ( 2004 ) identified 201 periodic variables in a period distribution that was indistinguishable from that of the orion nebula cluster ( herbst et al . this surprising result suggests that spin - up does not occur over the age range from @xmath671 to 5 myr and radii of 1.2 to 4 r@xmath3 ( rebull et al . makidon et al . ( 2004 ) also found no conclusive evidence for correlations between rotation period and the presence of disk indicators , specifically @xmath72 , @xmath73 , and @xmath74 colors as well as strong h@xmath4 emission . the lamm et al . ( 2004 ) survey was made using the wide - field imager ( wfi ) on the mpg / eso 2.2 m telescope at la silla . the wfi is composed of eight 2048@xmath14096 ccds mosaiced together , resulting in a field of view of @xmath033 square . this @xmath75 survey was undertaken over 44 nights with between one and 18 images obtained per night . the resulting photometric database from the lamm et al . ( 2004 ) survey is one of the most extensive available for ngc2264 and includes spectral types from rebull et al . ( 2002 ) as well as other sources . over 10,600 stars were monitored by the program , of which 543 were found to be periodic variables and 484 irregular variables with 11.4@xmath67@xmath76@xmath6719.7 . of these variables , 405 periodic and 184 irregulars possessed other criteria that are indicative of pre - main sequence stars . lamm et al . ( 2004 ) estimate that 70% of all pre - main sequence stars in the cluster ( @xmath76@xmath6718.0 , @xmath77@xmath671.8 ) were identified by their monitoring program ( implying a cluster population of @xmath0850 stars ) . lamm et al . ( 2005 ) use this extensive study to conclude that the period distribution in ngc2264 is similar to that of the orion nebula cluster , but shifted toward shorter periods ( recall that no difference was noted by makidon et al . 2004 ) . for stellar masses less than 0.25 m@xmath3 , the distribution is unimodal , but for more massive stars , the period distribution is bimodal . lamm et al . ( 2005 ) suggest that a large fraction of stars in ngc2264 are spun up relative to their counterparts in the younger orion nebula cluster . no significant spin up , however , was noted between older and younger stars within ngc2264 . lamm et al . ( 2005 ) also find evidence for disk locking among 30% of the higher mass stars , while among lower mass stars , disk locking is less significant . the locking period for the more massive stars is found to be @xmath08 days . for less massive stars a peak in the period distribution at 23 days suggests that these stars while not locked have undergone moderate angular momentum loss from star - disk interaction . other photometric surveys of the low - mass stellar population of ngc2264 include those of prez et al . ( 1987 , 1989 ) and fallscheer & herbst ( 2006 ) , who examined @xmath78 photometry of 0.41.2 m@xmath3 members and found a significant correlation between @xmath72 excess and rotation such that slower rotators are more likely to exhibit ultraviolet excess . numerous spectroscopic surveys of the intermediate and low - mass stellar populations of ngc2264 have been completed including those of walker ( 1954 , 1956 , 1972 ) , young ( 1978 ) , barry et al . ( 1979 ) , simon et al . ( 1985 ) , king ( 1998 ) , soderblom et al . ( 1999 ) , king et al . ( 2000 ) , rebull et al . ( 2002 ) , makidon et al . ( 2004 ) , dahm & simon ( 2005 ) , and furesz et al . walker ( 1972 ) examined the spectra of 25 uv excess stars in the orion nebula cluster and in ngc2264 having masses between 0.20.5 m@xmath3 and bolometric magnitudes ranging from @xmath79 to @xmath80 . from the radial velocities of the h i and ca ii lines , walker ( 1972 ) concluded that nine of these stars were experiencing accretion either from an enveloping shell of gas or from a surrounding disk . the rate of infall was estimated to be @xmath28 10@xmath81 m@xmath3 yr@xmath82 . young ( 1978 ) presented spectral types for 69 suspected cluster members ranging from b3v to k5v . combining the spectra with broad - band photometry , he concluded that differential reddening is present within the cluster , likely induced by intracluster dust clouds . king ( 1998 ) examined li features among six f and g - type stars within the cluster , finding the mean non - lte abundance to be log n(li)@xmath83 . this is identical to meteoritic values within our solar system , implying that no galactic li enrichment has occurred over the past 4.6 gyr . from rotation rates of 35 candidate members of ngc2264 , soderblom et al . ( 1999 ) conclude that the stars in ngc2264 are spun - up relative to members of the orion nebula cluster . furesz et al . ( 2006 ) present spectra for nearly 1000 stars in the ngc2264 region from the hectochelle multiobject spectrograph on the mmt . of these , 471 are confirmed as cluster members based upon radial velocities or h@xmath4 emission . furesz et al . ( 2006 ) also find that spatially coherent structure exists in the radial velocity distribution of suspected members , similar to that observed in the molecular gas associated with the cluster . the substellar population of ngc2264 remains relatively unexplored given the distance of the cluster . several deep imaging surveys have been conducted of the region , including that of sung et al . ( 2004 ) and kendall et al . sung et al . ( 2004 ) observed ngc2264 with the cfht 12k array in the broadband @xmath84 filters and a narrowband h@xmath4 filter . their deep survey extends to the substellar limit . the optical photometry was combined with _ chandra _ x - ray observations of the cluster to derive the cluster imf . using the ( @xmath85 ) color - color diagram to remove background giants , sung et al . ( 2004 ) conclude that the overall shape of the cluster imf is similar to that of the pleiades or the orion nebula cluster . kendall et al . ( 2005 ) identified 236 low - mass candidates lying redward of the 2 myr isochrones of the dusty evolutionary models generated by baraffe et al . ( 1998 ) . of these substellar candidates , 79 could be cross - correlated with the 2mass database , thereby permitting dereddening from nir excesses . most of these candidates range in mass between that of the sun and the substellar limit . the reddest objects with the lowest a@xmath21 values are possible brown dwarfs , but deeper optical and nir ( @xmath86 ) surveys are needed to further probe the substellar population of ngc2264 . for the brown dwarf candidates identified thus far within the cluster , spectroscopic follow - up is needed for confirmation . shown in figure 8 is the ( @xmath87 ) color - magnitude diagram from kendall et al . ( 2005 ) , which shows all 236 low - mass candidates identified in the cluster . , @xmath71 ) color - magnitude diagram of ngc2264 from the survey of kendall et al . ( superposed are the solid isochrones of the dusty models of baraffe et al . for ages of 2 and 5 myr . the dashed isochrones are the nextgen stellar models for the same ages . the 236 substellar candidates ( squares ) were selected to be redward of the sloping straight line . distance assumed for the isochrones is 760 pc . the mass scale on the left is for the 2 myr models . [ f8],scaledwidth=90.0% ] among the earliest radio surveys of ngc2264 is that of menon ( 1956 ) who observed the region at 21 cm and found a decrease in neutral hydrogen intensity toward the molecular cloud , leading to speculation that the formation of molecular hydrogen depleted the region of neutral atoms . minn & greenberg ( 1975 ) surveyed the dark clouds associated with the cluster at both the 6 cm line of h@xmath88co and the 21 cm line of h i. minn & greenberg ( 1975 ) discovered that the h@xmath88co line intensity decreased dramatically outside of the boundaries of the dark cloud , implying that the molecule was confined within the visual boundaries of the cloud . depressions within the 21 cm h i profiles observed along the line of sight were also found to be well - correlated with the h@xmath88co line velocities , but no further quantitative estimates were made . molecular gas dominates the mon ob1 association , with stars accounting for less than a few percent of the total mass of the cluster - cloud complex . zuckerman et al . ( 1972 ) searched for hcn and cs molecular line emission in the cone nebula , but later discovered their pointing was actually 4 north of the cone . emission , however , was serendipitously discovered from the background molecular cloud , presumably the region associated with irs1 . riegel & crutcher ( 1972 ) detected oh emission from several pointings within ngc2264 including a position near that observed by zuckerman et al . ( 1972 ) . from the relative agreement among line radial velocities , riegel & crutcher ( 1972 ) concluded that the oh emission arises from the same general region as that of the other molecular species . mayer et al . ( 1973 ) also detected nh@xmath89 emission at the position reported by zuckerman et al . ( 1972 ) , finding comparable velocity widths among the nh@xmath89 , hcn and cs lines . the measured velocity widths , however , were larger than the thermal doppler broadening inferred from the nh@xmath89 kinetic temperatures . from this , mayer et al . ( 1973 ) suggested that a systematic radial velocity gradient exists over the region where irs1 is located . rickard et al . ( 1977 ) mapped 6 cm continuum emission and h@xmath88co absorption over an extended region of ngc2264 , finding that h@xmath88co absorption toward the cluster was complex and possibly arises from multiple cloud components . blitz ( 1979 ) completed a co ( 10 ) 2.6 mm survey of ngc2264 at a resolution of 8and identified two large molecular clouds in the region with a narrow bridge of gas between them . one of the clouds is centered upon ngc2264 and the other lies 2@xmath8 northwest of the cluster and contains several reflection nebulae ( ngc2245 and ngc2247 ) . crutcher et al . ( 1978 ) mapped the region in the j@xmath3810 lines of @xmath90co and @xmath60co at somewhat better resolution ( half - power beam diameter of @xmath916 ) . shown in figure 9 are their resulting maps for @xmath90co and @xmath60co . the primary cloud core lying approximately 8 north of the cone nebula is elongated with a position angle roughly parallel to the galactic plane . the peak antenna temperature of 22 k occurs over an extended region , which includes irs1 ( allen s source ) but is not centered upon this luminous infrared source . two less prominent peaks were identified by crutcher et al . ( 1978 ) about 4 south and 8 west of s mon . co and @xmath60co of the ngc2264 region from crutcher et al . the coordinate scale is for equinox and epoch b1950 . the contour intervals are 4 and 2 k for the @xmath90co and @xmath60co maps , respectively , with the lowest contours representing the 8 k isotherm . the cross , plus symbol , and square denote the positions of s mon , irs1 , and the cone nebula . [ f9],width=288 ] the kinematic structure of the molecular clouds is somewhat complex with a 2 km s@xmath82 velocity gradient across the primary cloud and three distinct velocity components near s mon . individual clouds identified by crutcher et al . ( 1978 ) are listed in table 3 , reproduced from their table 1 . included with their identifiers are the positions ( b1950 ) , radii in pc , lsr velocity for @xmath90co , antenna temperatures for @xmath90co , and mass . summing these individual fragments , a total cloud mass of @xmath07@xmath110@xmath92 m@xmath3 is estimated . this , however , is a lower limit based upon co column densities . crutcher et al . ( 1978 ) also determined a virial mass of 3@xmath110@xmath2 m@xmath3 for the molecular cloud complex from the velocities of the individual cloud fragments . assuming this to be an upper limit , crutcher et al . ( 1978 ) adopt a middle value of 2@xmath110@xmath2 m@xmath3 for the total cloud mass associated with the cluster , half of which is contained within the primary cloud core north of the cone nebula . considering the energetics of the cloud , they further suggest that the luminous stars in ngc2264 are not capable of heating the core of this massive cloud . this conclusion is challenged by sargent et al . ( 1984 ) on the basis of a revised cooling rate . shown in figure 10 is an optical image of the cloud core c region from crutcher et al . ( 1978 ) , which lies just west of s mon . 15 , with north up and east left . image obtained at the cfht . courtesy j .- c . cuillandre and g. anselmi . [ f10],scaledwidth=90.0% ] cccccccc + + identifier & @xmath4 & @xmath42 & r & v & t@xmath93 & n(co ) & mass + & ( b1950 ) & ( b1950 ) & ( pc ) & ( km / s ) & ( k ) & 10@xmath94 @xmath95 & ( m@xmath3 ) + a & 6 37.6 & + 9 37.0 & 1.7 & 4.9 & 12 & 8 & 1200 + b & 6 37.2 & + 9 56.0 & 0.9 & 8.8 & 19 & 8 & 400 + c & 6 37.7 & + 9 56.0 & 0.9 & 10.4 & 22 & 6 & 300 + d & 6 37.9 & + 9 49.0 & 0.4 & 11.4 & 14 & 2 & 20 + e & 6 38.2 & + 9 53.0 & 0.8 & 8.9 & 23 & 10 & 400 + f & 6 38.2 & + 9 40.0 & 0.8 & 5.5 & 22 & 53 & 1800 + g & 6 38.4 & + 9 32.0 & 0.9 & 7.5 & 22 & 54 & 2300 + h & 6 38.9 & + 9 22.0 & 0.9 & 4.5 & 11 & 5 & 300 + schwartz et al . ( 1985 ) completed co ( 10 ) , c@xmath96o , and cs ( 32 ) cs observations of the irs1 and irs2 regions with the 11 meter nrao antenna on kitt peak as well as nh@xmath89 observations using the nrao 43 meter antenna at green bank . the nh@xmath89 and cs observations were used to examine high density gas while co and c@xmath96o were employed as tracers of h@xmath88 column density . schwartz et al . ( 1985 ) combined these molecular gas observations with far - infrared ( 35250 @xmath5 m ) data obtained with the multibeam photometer system onboard the kuiper airborne observatory . the general morphology of the co and cs maps of the regions were found to be similar , but two key differences were noted : first , the irs2 core appears as two unresolved knots in the cs map and second , the high density gas near irs1 ( allen s source ) lies east of the far - infrared peak and appears extended along the northeast to southwest axis . schwartz et al . ( 1985 ) conclude that allen s source is an early - type star ( @xmath0 b3v ) , which is embedded within a dense molecular cloud . they further speculate that the molecular cloud associated with irs1 is ring - shaped , lying nearly edge - on with respect to the observer ( see their figure 7 ) . this conclusion was drawn from the gas - density and velocity structure maps of the region . krugel et al . ( 1987 ) completed a more spatially resolved survey of the irs1 region in both nh@xmath89 and co , mapping a 10@xmath110 area . two distinct cloud cores were identified , each of roughly 500 m@xmath3 . kinetic gas temperatures were found by the authors to be 18 k in the north and 25 k in the south with a peak of 60 k around irs1 . complex structure was noted in the southern cloud , which exhibits multiple subclouds with differing temperatures and velocities . krugel et al . ( 1987 ) find that the smooth velocity gradient across the region observed at lower resolution disappears completely in their higher resolution maps . they also failed to detect high velocity wings toward irs1 that might be associated with a molecular outflow . tauber et al . ( 1993 ) observed one of the bright rims in ngc2264 west of s mon in the @xmath90co and @xmath60co ( 32 ) transitions with the caltech submillimeter observatory . in their high spatial resolution maps ( 20 ) , they find a morphology which suggests interaction with the ionizing radiation from s mon . the @xmath60co maps reveal a hollow shell of gas broken into three components : two eastern clumps with their long axes pointing toward s mon and a southern clump , which exhibits kinematic structure indicative of an embedded , rotating torus of dense gas . oliver et al . ( 1996 ) completed a sensitive , unbiased co ( 10 ) line emission survey of the mon ob1 region using the 1.2 meter millimeter - wave radio telescope at the harvard smithsonian center for astrophysics . the survey consisted of over 13,400 individual spectra and extended from @xmath97 , with individual pointings uniformly separated by 3.75 . oliver et al . ( 1996 ) find that the molecular gas along the line of sight of the mon ob1 association possesses radial velocities consistent with the local spiral arm and the outer perseus arm . within the local arm , they identify 20 individual molecular clouds ranging in mass from @xmath0100 to 5.2@xmath110@xmath2 m@xmath3 . their table 5 ( reproduced here as table 4 ) summarizes the properties of these molecular clouds including position , v@xmath98 , distance , radius , mass , and associated clusters , molecular clouds , or bright nebulae . the most massive of these molecular clouds hosts ngc2264 , but their derived mass estimates assume the distance of prez et al . ( 1987 ) , 950 pc , which probably overestimates the actual cluster distance by a factor of 1.2 . the revised cloud mass assuming a distance of 800 pc is 3.7 @xmath1 10@xmath2 m@xmath3 , which is in somewhat better agreement with the value derived by crutcher et al . ( 1978 ) . oliver et al . ( 1996 ) also identify six arc - like molecular structures in the mon ob1 region , which may be associated with supernova remnants or wind - blown shells of gas . these structures , if associated with the local spiral arm and at an appropriate distance , may imply that star formation within the region was triggered by an energetic supernova event . c@c@c@c@c@c@c@c@c + + cloud & _ l _ & _ b _ & v@xmath99 & @xmath42v & d & r & mass ( co ) & associations + & @xmath8 & @xmath8 & km / s & km / s & kpc & kpc & m@xmath3 & + 1 & 196.25 & @xmath400.13 & + 4.7 & 4.6 & 0.9 & 9.4 & 3.7@xmath110@xmath92 & + 2 & 196.75 & + 1.50 & + 5.3 & 2.3 & 0.9 & 9.7 & 9@xmath110@xmath92 & + 3 & 196.88 & @xmath401.13 & + 5.3 & 3.3 & 0.9 & 9.4 & 1.1@xmath110@xmath92 & + 4 & 198.88 & + 0.00 & @xmath406.9 & 3.4 & 0.8 & 9.3 & 1.0@xmath110@xmath92 & + 5 & 199.31 & @xmath400.50 & + 3.3 & 2.6 & 0.9 & 9.4 & 6.7@xmath110@xmath100 & + 6 & 199.56 & @xmath400.44 & + 6.4 & 4.9 & 0.9 & 9.4 & 2.0@xmath110@xmath92 & + 7 & 199.81 & + 0.94 & + 6.9 & 4.6 & 0.9 & 9.4 & 2.4@xmath110@xmath2 & l1600 , l1601 + & & & & & & & & l1604 , lbn886 + & & & & & & & & lbn889 + 8 & 200.19 & + 3.44 & @xmath406.8 & 3.4 & 0.8 & 9.3 & 1.9@xmath110@xmath100 & + 9 & 200.81 & + 0.13 & @xmath4011.0 & 5.2 & 0.8 & 9.3 & 1.5@xmath110@xmath92 & vy mon , lbn895 + 10 & 201.38 & + 0.31 & @xmath401.0 & 3.2 & 0.9 & 9.3 & 4.3@xmath110@xmath92 & mon r1 , l1605 + & & & & & & & & vdb76,77,78 + & & & & & & & & lbn898,903 + 11 & 201.44 & + 0.69 & + 5.1 & 4.4 & 0.9 & 9.3 & 2.1@xmath110@xmath2 & mon r1 , vy mon + & & & & & & & & ngc2245 , ngc2247 + & & & & & & & & l1605 , lbn901/904 + 12 & 201.44 & + 2.56 & + 0.7 & 3.1 & 0.9 & 9.4 & 2.0@xmath110@xmath92 & ngc2259 + 13 & 201.50 & + 2.38 & + 4.6 & 6.9 & 0.95 & 9.4 & 6.3@xmath110@xmath92 & ngc2259 , lbn899 + 14 & 202.06 & + 1.44 & + 1.1 & 2.9 & 0.95 & 9.4 & 1.5@xmath110@xmath92 & l1609 + 15 & 202.25 & + 1.69 & + 4.7 & 2.5 & 0.95 & 9.4 & 9@xmath110@xmath100 & l1610 + 16 & 203.25 & + 2.06 & + 6.9 & 5.0 & 0.95 & 9.4 & 5.2@xmath110@xmath2 & ngc2264 , irs1 , l1613 + & & & & & & & & s mon , lbn911/912/922 + 17 & 203.75 & + 1.25 & + 8.7 & 2.8 & 0.80 & 9.2 & 1.0@xmath110@xmath92 & ngc2261 + & & & & & & & & hh 39(a f ) + & & & & & & & & r mon , lbn920 + 18 & 204.13 & + 0.44 & + 5.2 & 2.8 & 0.80 & 9.2 & 1.9@xmath110@xmath92 & ngc2254 , lbn929 + 19 & 204.44 & @xmath400.13 & @xmath403.1 & 2.6 & 0.80 & 9.2 & 1.1@xmath110@xmath100 & ngc2254 + 20 & 204.81 & + 0.44 & + 9.4 & 2.6 & 0.80 & 9.2 & 8.2@xmath110@xmath100 & ngc2254 + wolf - chase & gregersen ( 1997 ) analyzed observations of numerous transitions of cs and co in the irs1 region . taken as a whole , the observations suggest that gravitational infall is taking place . schreyer et al . ( 1997 ) mapped the region around irs1 in various molecular transitions of cs , co , methanol , and c@xmath96o . complementing the millimeter survey , schreyer et al . ( 1997 ) also imaged the region in the nir ( @xmath101 ) , revealing the presence of several suspected low - mass stars surrounding irs1 . to the southeast , a small , deeply embedded cluster was noted in the @xmath102band mosaic of the field , which coincided with a second cloud core identified by the millimeter survey . extending northwest from irs1 , schreyer et al . ( 1997 ) noted a jet - like feature in all nir passbands that connects to the fan - shaped nebula evident on optical images of the field . two bipolar outflows were also detected in their cs mapping of the irs1 region , one originating from irs1 itself and another from the millimeter source lying to the southeast . thompson et al . ( 1998 ) examined irs1 with nicmos onboard hst and discovered six point sources at projected separations of 26 to 49 from allen s source ( irs1 ) . from the nir colors and magnitudes of these sources , thompson et al . ( 1998 ) suggest that these faint stars are pre - main sequence stars whose formation was possibly triggered by the collapse of irs1 . ward - thompson et al . ( 2000 ) observed the irs1 region in the millimeter and submillimeter continuum ( 1.3 mm 350 @xmath5 m ) , using the 7-channel bolometer array on the iram 30 meter telescope and the ukt14 detector on the james clerk maxwell telescope ( jcmt ) on mauna kea . the resulting maps revealed a ridge of bright millimeter emission as well as a clustering of five sources with masses ranging from 1050 m@xmath3 . table 4 of ward - thompson et al . ( 2000 ) summarizes the properties of these five massive condensations , which are assumed to be the progenitor cores of intermediate - mass stars . the third millimeter source ( mms3 ) of ward - thompson et al . ( 2000 ) was identified as the source of one of the bipolar outflows discovered by schreyer et al . ( 1997 ) in their millimeter survey of the region . williams & garland ( 2002 ) completed 870 @xmath5 m continuum emission maps and ( 32 ) line surveys of hco@xmath103 and h@xmath60co@xmath103 of the irs1 and irs2 regions of ngc2264 . the submillimeter continuum emission was used to trace dust around the young clusters of protostars , while line emission was used as a diagnostic of gas flow within the region . shown in figure 11 is the 870 @xmath5 m continuum emission map of williams & garland ( 2002 ) , their figure 1 , which clearly demonstrates the fragmented nature of irs2 . although irs1 possesses a much higher peak flux density , the masses of both regions are found to be comparable , @xmath010@xmath92 m@xmath3 . williams & garland ( 2002 ) find evidence for large - scale collapse for both irs1 and irs2 with infall velocities of @xmath00.3 km s@xmath82 . from their derived virial mass of the irs2 protostellar cluster , they conclude that the system is very likely gravitationally unbound . m continuum emission map of the irs1 and irs2 regions from williams & garland ( 2002 ) . the intensity scale and beamsize are annotated in the lower right and left corners of the figure , respectively . the elongated shape of irs1 exhibits signs of substructure , while irs2 is more fragmented and therefore suggestive of a more evolved cluster of protostars . [ f11],width=336 ] nakano et al . ( 2003 ) completed high resolution h@xmath60co@xmath103 ( 10 ) and 93 ghz continuum observations of the irs1 region using the nobeyama millimeter array and the nobeyama 45-meter telescope . four sources were identified in the resulting map , three of which were coincident with sources identified by ward - thompson et al . nakano et al . ( 2003 ) conclude that a dense shell of gas @xmath104 pc in diameter envelops irs1 , the interior of which has been evacuated on timescales of @xmath105 myr . schreyer et al . ( 2003 ) examined irs1 at high resolution using the iram plateau de bure interferometer at 3 mm and in the cs ( 21 ) transition . they complement this data with 2.2 , 4.6 , and 11.9 @xmath5 m imaging to interpret the immediate environment around irs1 . no circumstellar disk was found around irs1 . this young b - type star and several low - mass companions appear to be within a low - density cavity of the remnant cloud core . the source of the large bipolar outflow is also identified as a deeply embedded sources lying 20 north of irs1 . shown in figure 12 is the hst nicmos image of the irs1 field with several embedded sources identified , including source 8 , a binary which exhibits a centrosymmetric polarization pattern consistent with circumstellar dust emission . m image ( blue ) , the mean of 1.6 and 2.2 @xmath5 m images ( green ) , and the 2.2 @xmath5 m image ( red ) . numerous embedded sources are evident around irs1 including source 8 , a binary which exhibits a centrosymmetric polarization pattern consistent with dust emission . [ f12],width=288 ] hedden et al . ( 2006 ) used the heinrich hertz telescope and the aro 12 meter millimeter - wave telescope to map several outflows in the northern cloud complex of the mon ob1 association in @xmath90co ( 32 ) , @xmath60co ( 32 ) , @xmath90co ( 10 ) , and 870 @xmath5 m continuum . several continuum emission cores were identified and the seds of these sources were constructed to derive their column densities , masses , luminosities , and temperatures . hedden et al . ( 2006 ) conclude that the molecular cloud complexes are maintaining their integrity except along the axes of outflows . the outflows are found to deposit most of their energy outside of the cloud , leading to a weak correlation between outflow kinetic energy and turbulent energy within the clouds . peretto et al . ( 2006 ) surveyed the two massive cloud cores associated with irs1 and irs2 in dust continuum and molecular line emission , finding 12 and 15 compact millimeter continuum sources within each core , respectively . the millimeter sources have typical diameters of @xmath00.04 pc and range in mass from @xmath0241 m@xmath3 . although similar in size to cores within the @xmath106 oph star forming region , the millimeter sources in ngc2264 exhibit velocity dispersions two to five times greater than those of the @xmath106 oph main cloud and the isolated cores in the taurus - auriga complex . as many as 70% of the sources within the irs1 cloud core host class 0 protostars that are associated with jets of shocked h@xmath88 . in the irs2 cloud , only 25% of the millimeter sources are associated with 2mass or mid - infrared point sources . peretto et al . ( 2006 ) also find evidence for widespread infall within both cloud cores and suggest that the irs1 core is collapsing along its long axis in a free - fall timescale of @xmath01.7@xmath107 years . this is consistent with the findings of williams & garland ( 2002 ) . within this core , peretto et al . ( 2006 ) conclude that a high - mass ( 1020 m@xmath3 ) protostar is currently forming . reipurth et al . ( 2004c ) completed a 3.6 cm radio continuum survey of young outflow sources including ngc2264 irs1 using the very large array in its a configuration . their map reveals 8 sources in the general region of irs1 , including a source that is coincident with irs1 itself . the sources vla2 ( mms4 ) and vla7 appear extended and show significant collimation . three arcminutes southeast of irs1 , reipurth et al . ( 2004c ) find a bipolar radio jet with a 3.6 cm flux density of @xmath013 mjy . the eastern portion of this jet is comprised of at least 8 well - resolved knots that appear flattened perpendicular to the axis of the flow . the extent of this well - collimated jet is estimated to be 28 . the western lobe of the outflow exhibits only one large clump . deep optical and @xmath102band imaging of the central region of the jet reveal no evidence for a source , suggesting that it may be extragalactic in origin . trejo & rodrguez ( 2008 ) compare 3.6 cm observations of this non - thermal radio jet obtained in 2006 with archived data from 1995 and find no evidence for proper motion or polarization changes . flux density variations were found in one knot , which is tentatively identified as the core of a quasar or radio galaxy . teixeira et al . ( 2007 ) present high angular resolution ( 1 ) 1.3 mm continuum observations of the core d - mm1 in the spokes cluster obtained using the submillimeter array ( sma ) . they find a dense cluster of 7 class 0 objects within a 20@xmath120 region with masses ranging from 0.4 to 1.2 m@xmath3 . teixeira et al . ( 2007 ) conclude that the 1.3 mm continuum emission arises from the envelopes of the class 0 sources , which are found to be @xmath0600 au in diameter . the sources within the d - mm1 cluster have projected separations that are consistent with hierarchical fragmentation . one of the earliest infrared surveys of the cluster by allen ( 1972 ) identified several sources that were correlated with known stars , but a single bright source was identified near the tip of the cone nebula that lacked an optical counterpart . this source , now referred to as allen s infrared source or ngc2264 irs1 , is recognized to be an embedded early - type ( b spectral class ) star . allen ( 1972 ) suggested that this source was the most massive and luminous star within the cluster , a claim that has since been refuted by subsequent infrared observations . harvey et al . ( 1977 ) surveyed ngc2264 and ngc2244 in the mid- and far - infrared ( 53175 @xmath5 m ) using the kuiper airborne observatory . irs1 remained unresolved in the far infrared , but possessed relatively cool ( 53175 @xmath5 m ) colors . from their luminosity estimate for irs1 of 10@xmath92 @xmath108 , harvey et al . ( 1977 ) concluded that relative to compact hii regions believed to be the progenitors of massive o - type stars , irs1 was significantly less luminous , implying that it was an embedded intermediate - mass star of 510 m@xmath3 . sargent et al . ( 1984 ) completed a balloon - borne , large - scale mapping of ngc2264 at 70 and 130 @xmath5 m , identifying a number of far infrared sources within the cluster . the luminosity of irs1 was found to be 3.8@xmath110@xmath92 @xmath108 , which was consistent with the earlier higher resolution observations of harvey et al . ( 1977 ) . warner et al . ( 1977 ) obtained @xmath109 band observations of 66 members of ngc2264 using an indium antimonide detector on the mount lemmon 1.5 meter telescope . they confirmed the existence of infrared excesses for a significant fraction ( 30% ) of stars with spectral types later than a0 . margulis et al . ( 1989 ) identified 30 discrete iras sources in the mon ob1 molecular cloud complex , 18 of which were found to have class i spectral energy distributions . from the large population of class i sources , margulis et al . ( 1989 ) concluded that active star formation is still taking place within the molecular cloud complex hosting the cluster . neri et al . ( 1993 ) presented @xmath110 and @xmath111 band photometry for @xmath112 peculiar stars within the cluster , from which they determined a revised distance estimate and mean extinction value . the authors also examined optical and infrared variability among the sample stars and derive effective temperatures and log g values for each . they found no evidence for a or b - type stars with infrared fluxes lower than expected for their observed optical magnitudes . such stars had been reported previously in the ori i ob association . the first extensive nir imaging surveys of ngc2264 were completed by pich ( 1992 , 1993 ) and lada et al . ( 1993 ) , who mapped most of the cluster region in the @xmath113bands . lada et al . ( 1993 ) detected over 1650 @xmath102band sources in their survey and concluded that 360@xmath114130 were probable cluster members . of these , 50@xmath11420% possessed infrared excess emission , possibly implying the presence of circumstellar disks . rebull et al . ( 2002 ) undertook an optical and nir survey of the cluster , presenting photometry for over 5600 stars and spectral types for another 400 . three criteria were used to identify circumstellar disk candidates within the cluster : excess ( @xmath72 ) emission , excess nir emission ( @xmath115 and @xmath74 ) , and large h@xmath4 equivalent widths , if spectra were available . rebull et al . ( 2002 ) established an inner disk fraction ranging from 21% to 56% . no statistically significant variation was found in the disk fraction as a function of age , mass , @xmath68band mag , or ( @xmath116 ) color . mass accretion rates were derived from @xmath117band excesses with typical values on the order of 10@xmath118 m@xmath3 yr@xmath82 . wang et al . ( 2002 ) completed narrowband h@xmath88 , @xmath119 s(1 ) imaging of the irs1 region and identified at least four highly collimated jets of emission as well as several isolated knots of h@xmath88 emission . some of the jets are associated with millimeter and submillimeter sources identified by ward - thompson et al . aspin & reipurth ( 2003 ) imaged the irs2 region in the nir ( @xmath101 ) and thermal @xmath120 and @xmath121bands , finding two stars with spectra similar to those characteristic of fu ori type stars . the stars form a close ( 28 ) binary and exhibit arcuate reflection nebulae . aspin & reipurth ( 2003 ) compile the ( @xmath122 ) magnitudes for @xmath032 stars in the irs2 region . _ spitzer _ space telescope is revolutionizing our understanding of the star formation process and circumstellar disk evolution . three - color infrared array camera ( irac ) and multiband imaging photometer for _ spitzer _ ( mips ) images of ngc2264 have been released that reveal significant structure within the molecular cloud cores as well as embedded clusters of class i sources . teixeira et al . ( 2006 ) identify primordial filamentary substructure within one cluster such that the protostars are uniformly spaced along cores of molecular gas in a semi - linear fashion and at intervals consistent with the jeans length . shown in figure 13 is the three - color composite image of the embedded `` spokes '' cluster from teixeira et al . ( 2006 ) constructed from irac 3.6 @xmath5 m ( blue ) , irac 8.0 @xmath5 m ( green ) , and mips 24.0 @xmath5 m ( red ) images of the region . several quasi - linear structures appear to be coincident with dust emission from dense cores of molecular gas traced at 850 @xmath5 m with scuba by wolf - chase et al . figure 14 , obtained from teixeira et al . ( 2006 ) , compares the spatial distribution of the dust emission cores with 24 @xmath5 m point sources . the sizes of the stars representing 24 @xmath5 m sources are proportional to their magnitudes . m data as well as mips 24 @xmath5 m band imaging were used to create the image . [ f13 ] ] when complete , the _ spitzer _ irac and mips surveys of ngc2264 will unambiguously identify the disk - bearing population of the cluster and provide tentative characterizations of disk structure based upon the stellar optical to mid - infrared seds . when merged with extant rotational period data , the _ spitzer _ results may resolve long - standing questions regarding the impact of circumstellar disks upon stellar rotation and angular momentum evolution . population statistics for the embedded clusters will also add to the stellar census of ngc2264 and the greater mon ob1 association . m dust emission from wolf - chase et al . ( 2003 ) with 24 @xmath5 m point sources superposed as five - pointed stars . the squares mark the positions of the two brightest 24 @xmath5 m sources . the size of the stars and squares are proportional to the magnitudes of the sources . the beam size for the 850 @xmath5 m data is indicated in the lower right . [ f14],width=384 ] the earliest x - ray survey of ngc2264 was undertaken by simon et al . ( 1985 ) with the imaging proportional counter ( ipc ) aboard the _ einstein _ observatory . these early x - ray observations had 1 spatial resolution and an energy bandwidth of 0.44.0 kev . all three images obtained were centered upon s mon and had short integration times of 471 , 1660 , and 1772 s. in addition to s mon , seven x - ray sources were identified by the _ einstein _ program with x - ray luminosities ranging from 2.45.2@xmath110@xmath123 ergs s@xmath82 . these sources were among the most x - ray luminous of all young cluster stars observed with _ einstein_. rosat observed the cluster in 1991 march and 1992 september using the high resolution imager , hri . exposure times were significantly longer , 19.4 and 10.9 ks , respectively , resulting in the detection of 74 x - ray sources with s / n @xmath124 3.0 ( patten et al . 1994 ) . with the exception of a through early f - type stars , the hri observations detected cluster members over a range of spectral types from o7v to late - k . patten et al . ( 1994 ) also compared x - ray surface fluxes of solar analogs in the pleiades , ic 2391 , and ngc2264 , finding all three to be comparable . flaccomio et al . ( 2000 ) combined three archived rosat images of ngc2264 with three new observations made approximately 15 southeast of the earlier epochs of data . from these images , 169 distinct x - ray sources were identified : 133 possessed single optical counterparts , 30 had multiple counterparts and six had no optical counterparts . flaccomio et al . ( 2000 ) used optical ( @xmath125 ) photometry from flaccomio et al . ( 1999 ) and sung et al . ( 1997 ) to construct an hr diagram of the x - ray emission population of the cluster . ages and masses were also derived from the models of dantona and mazzitelli ( 1997 ) using the 1998 updates . comparing the ages and masses of the x - ray population with those derived from their earlier optical survey of the southern half of the cluster , flaccomio et al . ( 2000 ) concluded that the x - ray sample was representative of the entire pre - main sequence population of the cluster . flaccomio et al . ( 2000 ) also found x - ray luminosities of known cttss and wttss to be comparable , implying that accretion was not a significant source of x - ray emission . using the same data set , but with an improved determination of x - ray luminosities and a better reference optical sample , flaccomio et al . ( 2003 ) did find that cttss have on average lower x - ray luminosities with respect to wttss . nakano et al . ( 2000 ) used the advanced satellite for cosmology and astrophysics ( asca ) to observe ngc2264 in 1998 october with the gas - imaging spectrometer ( gis ) and the solid state imaging spectrometer ( sis ) . the field center of the observation was near w157 , several arcmin northwest of the cone nebula . given the moderate resolution of sis ( 30 ) , establishing optical or infrared counterparts of the x - ray emission sources was difficult . a dozen x - ray sources within the cluster were identified including two class i sources , irs1 and irs2 , as well as several known h@xmath4 emitters . nakano et al . ( 2000 ) suggest that most of the detected hard x - ray flux originates from intermediate mass class i sources , similar to allen s source . the improved spatial resolution ( @xmath01 ) and sensitivity of _ xmm - newton _ and _ chandra _ have revolutionized x - ray studies of young clusters and star forming regions . optical and nir counterparts of most x - ray sources can now be unambiguously identified , even in clustered regions . simon & dahm ( 2005 ) used deep ( 42 ks ) _ xmm - newton _ epic observations of the northern and southern halves of ngc2264 to probe sites of active star formation . the resulting integrations revealed strong x - ray emission from three deeply embedded ysos near irs1 and irs2 . the brightest x - ray source was located 11 southwest of allen s source and had a x - ray luminosity of 10@xmath126 ergs s@xmath82 and a temperature of 100 mk . follow - up 14 @xmath5 m , moderate - resolution spectra of the sources revealed deep water ice absorption bands at 3.1 @xmath5 m as well as many emission and absorption features of hi , co , and various metals . the nir images of the irs1 and irs2 regions with the _ xmm _ contours superposed are shown in figure 15 . within the _ xmm _ epic frames , over 316 confirmed x - ray sources were identified , 300 of which have optical or nir counterparts . dahm et al . ( 2007 ) find that most of these x - ray sources lie on or above the 3 myr isochrone of siess et al . ( 2000 ) . given the estimated low - mass population of ngc2264 from variability studies and h@xmath4 emission surveys , the _ xmm _ sample represents only the most x - ray luminous members of the cluster . ) false color images of the embedded x - ray sources near irs1 . contours for the brightest x - ray emission levels are overlaid . the strongest x - ray source in the cluster is exs-1 , which is identifiable with the infrared source 2mass j06410954 + 0929250 , which lies 11 southwest of allen s source . ( _ b _ ) in the irs2 field , the x - ray sources are from left to right : w164 , w159 , rno - e ( exs-26 ) , and 2mass j06405767 + 0936082 ( exs-10 ) . x - ray contours are superposed in green and white . [ f15],title="fig:",scaledwidth=80.0% ] ) false color images of the embedded x - ray sources near irs1 . contours for the brightest x - ray emission levels are overlaid . the strongest x - ray source in the cluster is exs-1 , which is identifiable with the infrared source 2mass j06410954 + 0929250 , which lies 11 southwest of allen s source . ( _ b _ ) in the irs2 field , the x - ray sources are from left to right : w164 , w159 , rno - e ( exs-26 ) , and 2mass j06405767 + 0936082 ( exs-10 ) . x - ray contours are superposed in green and white . [ f15],title="fig : " ] the advanced ccd imaging spectrometer ( acis ) onboard the _ chandra _ x - ray observatory was used by ramirez et al . ( 2004 ) to observe the northern half of ngc 2264 in 2002 february . the field of view of the imaging array ( acis - i ) is approximately 17@xmath117 and the total integration time of the observation was 48.1 ks . the pipeline reduction package detected 313 sources , 50 of which were rejected as cosmic ray artifacts or duplicate detections , leaving 263 probable cluster members . of these sources , 41 exhibited flux variability and 14 were consistent with flaring sources ( rapid rise followed by slow decay of x - ray flux ) . of the confirmed sources , 213 were identified with optical or nir counterparts . the deepest x - ray survey of ngc2264 to date is that of flaccomio et al . ( 2006 ) , who obtained a 97 ks _ acis - i integration of the southern half of the cluster in 2002 october . within the field of view of acis , 420 x - ray sources were detected , 85% of which have optical and nir counterparts . flaccomio et al . ( 2006 ) found that the median fractional x - ray luminosity , @xmath127/@xmath128 , of the sample is slightly less than 10@xmath129 . cttss were found to exhibit higher levels of x - ray variability relative to wttss , which was attributed to the stochastic nature of accretion processes . flaccomio et al . ( 2006 ) also found that cttss for a given stellar mass exhibit lower activity levels than wttss , possibly because accretion modifies magnetic field geometry resulting in mass loading of field lines and thus damping the heating of plasma to x - ray temperatures ( preibisch et al . 2005 ; flaccomio et al . flaccomio et al . ( 2006 ) also find , however , that the plasma temperatures of cttss are on average higher than their wtts counterparts . rebull et al . ( 2006 ) combine both of these _ chandra _ observations of ngc2264 in order to examine the x - ray properties of the full cluster population . they find that the level of x - ray emission is strongly correlated with internal stellar structure , as evidenced by an order of magnitude drop in x - ray flux among 12 m@xmath3 stars as they turn onto their radiative tracks . among the sample of x - ray detected stars with established rotation periods , rebull et al . ( 2006 ) find no correlation between @xmath127 and rotation rate . they also find no statistically significant correlation between the level of x - ray flux and the presence or absence of circumstellar accretion disks or accretion rates as determined by ultraviolet excess . herbig - haro ( hh ) objects and outflows are regarded as indicators of recent star formation activity . early surveys of the ngc2264 region by herbig ( 1974 ) identified several candidate hh objects . adams et al . ( 1979 ) conducted a narrow - band h@xmath4 emission survey of the cluster and identified 5 additional hh condensations that they conclude are associated with the molecular cloud complex behind the stellar cluster . walsh et al . ( 1992 ) discovered two additional hh objects in ngc2264 ( hh 124 and hh 125 ) using narrow - band imaging and low - dispersion spectra . hh 124 lies north of the cluster region and emanates from the cometary cloud core brc 25 , which contains iras 06382 + 1017 . hh 124 is composed of at least 6 knots of emission with the western condensations exhibiting negative high velocity wings and the eastern components positive ( up to + 100 km s@xmath82 ) . from a large - scale co ( 32 ) map of brc 25 , reipurth et al . ( 2004a ) report a significant molecular outflow along the axis of hh 124 with no source identified . ogura ( 1995 ) identified a giant ( 1 pc ) bow - shock structure associated with hh 124 using narrowband ( [ s ii ] @xmath130@xmath1306717 , 6731 ) ccd imaging and slit spectroscopy . the large projected distance of the shock from its proposed source implies a dynamical age in excess of 10,000 yrs . image of the hh 576 and 577 region with the co ( 32 ) emission contours superposed from reipurth et al . ( 2004a ) . the extent of the region mapped in co is shown by the light dashed white line . solid black contours represent blueshifted ( -3 to + 3 km s@xmath82 ) co emission , and dashed black contours redshifted ( + 13 to + 19 km s@xmath82 ) emission . [ f16],width=377 ] walsh et al . ( 1992 ) found that hh 125 is composed of at least 16 knots of emission and lies near other known hh objects identifed previously by adams et al . the angular extent of the hh object implies a projected linear dimension of @xmath00.78 pc . walsh et al . ( 1992 ) suggested that iras 06382 + 0945 is the exciting source for hh 125 . wang et al . ( 2003 ) completed an extensive 2@xmath8 wide - field [ s ii ] narrowband imaging survey of the mon ob1 region . in the northern part of the molecular cloud , two new hh objects were discovered ( hh 572 and hh 575 ) . reipurth et al . ( 2004a ) identify 15 additional hh objects in the region , some of which have parsec - scale dimensions . one these is the giant ( @xmath05.2 pc ) bipolar flow hh 571/572 which also originates from a source within brc 25 , possibly iras 06382 + 1017 . the co map of reipurth et al . ( 2004a ) revealed two large molecular outflows with position angles similar to those of hh 576 and 577 , suggesting a physical association . figure 16 shows the co contours overlaying an h@xmath4 image of hh 576 and 577 . the southwestern quadrant of the brc 25 cloud core also shows optical features that suggest significant outflow activity , but no co counterpart was identified . reipurth et al . ( 2004a ) suggested that hh 125 , 225 , and 226 form a single giant outflow , possibly originating in the irs2 region . table 5 summarizes known hh objects in the ngc2264 region . ccccc + + identifier & ra ( j2000 ) & @xmath42 ( j2000 ) & notes & ref + hh 39 & 06 39 07 & + 08 51.9 & ngc2261 & h74 + hh 575a & 06 40 31.6 & + 10 07 56 & brightest knot & r04 + hh 576 & 06 40 35.9 & + 10 39 48 & bow shock ( west ) & r04 + hh 577 & 06 40 36.6 & + 10 34 02 & brightest knot & r04 + hh 571 & 06 40 46.5 & + 10 05 15 & brightest point & r04 + hh 580 & 06 40 56.7 & + 09 32 52 & tip of bow & r04 + hh 582 & 06 40 56.9 & + 09 31 20 & western knot & r04 + hh 581 & 06 41 00.5 & + 09 32 56 & southern knot & r04 + hh 573a & 06 41 01.9 & + 10 14 51 & diffuse knot in brc 25 & r04 + hh 226 & 06 41 02.2 & + 09 39 49 & & w03 + hh 124 & 06 41 02.7 & + 10 15 03 & bow shock & o95 + hh 225 & 06 41 02.7 & + 09 44 16 & & w03 + hh 125 & 06 41 02.8 & + 09 46 07 & & w92 + hh 583 & 06 41 06.5 & + 09 33 16 & middle knot & r04 + hh 574 & 06 41 07.9 & + 10 16 19 & brightest knot & r04 + hh 578 & 06 41 10.6 & + 10 20 50 & star at end of jet & r04 + hh 579 & 06 41 14.4 & + 09 31 10 & tip of bow & r04 + hh 585 & 06 41 25.3 & + 09 24 01 & middle of bow & r04 + hh 572 & 06 41 28.8 & + 10 23 45 & eastern knot in bow & r04 + hh 584 & 06 41 38.8 & + 09 28 28 & central knot & r04 + + + * irs1 ( allen s infrared source ) : * first discovered by allen ( 1972 ) in his near infrared survey of ngc2264 , irs1 ( iras 06384 + 0932 ) is now recognized as a deeply embedded , early - type ( b2b5 ) star lying within a massive molecular cloud core . critical investigations into the nature of irs1 include those of allen ( 1972 ) , thompson & tokunaga ( 1978 ) , schwartz et al . ( 1985 ) , schreyer et al . ( 1997 , 2003 ) , and thompson et al . at least one molecular outflow is associated with irs1 ( schreyer et al . 1997 ; wolf - chase & gregersen 1997 ) as well as a jet - like structure detected in the near infrared . hst nicmos imaging has revealed several point sources surrounding irs1 assessed as solar - mass , pre - main sequence stars ( thompson et al . the millimeter continuum and molecular line observations and mid - infrared imaging of schreyer et al . ( 2003 ) suggest that irs1 is not associated with a circumstellar disk of primordial gas and dust . + * w90 ( lh@xmath425 ) : * no discussion of ngc2264 would be complete without addressing the remarkable herbig aebe star , w90 ( lh@xmath4 25 ) . herbig ( 1954 ) lists the star as an early - a spectral type with bright h@xmath4 emission and possible evidence for photometric variation ( @xmath00.1 mag ) relative to earlier brightness estimates by trumpler ( 1930 ) . he further noted that for a normal a2a3 type star , lh@xmath4 25 is three mag fainter than expected for the adopted distance of the cluster . walker ( 1956 ) found the star to lie well below the zams of ngc2264 . he further states that the star appears unreddened , but that structure within the balmer lines and the presence of several [ fe ii ] lines in its spectrum argue for a weak shell enveloping the star . herbig ( 1960 ) revised the spectral type of w90 to b8pe + shell , but noted that the balmer line wings are weak relative to other late b - type stars . poveda ( 1965 ) suggested that stars below the zams similar to w90 are surrounded by optically thick dust and gas shells , which induce neutral extinction . strom et al . ( 1971 ) estimated the surface gravity of w90 using its balmer line profiles , finding log g @xmath131 , consistent with giant atmospheres . they further speculated that w90 is surrounded by a dust shell that absorbs 95% of the radiated visible light . strom et al . ( 1972 ) confirmed the presence of dust around w90 , finding an extraordinary infrared ( @xmath132 ) excess of 3 mag . rydgren & vrba ( 1987 ) presented an sed for the star spanning from 0.3520 @xmath5 m , and concluded that w90 is observed through an edge - on dust disk . dahm & simon ( 2005 ) presented a nir spectrum of the star ( 0.852.4 @xmath5 m ) , which reveals strong brackett and paschen series line emission as well as he i and fe ii emission . w90 remains something of an enigma , but is a key representative of the herbig aebe population . + * kh-15d ( v582 mon ) : * this deeply eclipsing ( @xmath03.5 mag in @xmath71 ) k7 tts lies just north of the cone nebula and the b2 star hd 47887 . the eclipse profile of kh-15d ( kearns & herbst 1998 ) suggests that an inclined knife - edge screen is periodically occulting the star ( herbst et al . 2002 ) . during eclipse , herbst et al . ( 2002 ) find that the color of the star is bluer than when outside of eclipse . these observations and the polarization measurements of agol et al . ( 2004 ) support the conclusion that the flux received during eclipse is scattered by large dust grains within the obscuring screen ( hamilton et al . 2005 ) . by combining archived observations with recent photometry , hamilton et al . ( 2005 ) were able to analyze a 9-year baseline of eclipse data . their finding suggest that the eclipse is evolving rapidly , with its duration lengthening at a rate of 2 days per year . johnson et al . ( 2004 ) present results of an high resolution spectroscopic monitoring program for kh-15d and find it to be a single - line spectroscopic binary with a period of 48.38 days , identical to the photometric period . they conclude that the periodic dimming of kh-15d is caused by the binary motion that moves the visible stellar component above and behind the edge of an obscuring cloud . estimates for the eccentricity and mass function are given . johnson et al . ( 2005 ) present historical @xmath133 photometric observations of kh-15d obtained between 1954 and 1997 from multiple observatories . they find that the system has been variable at the level of 1 mag since at least 1965 . no evidence for color variation is found , and johnson et al . ( 2005 ) conclude that kh-15d is being occulted by an inclined , precessing , circumbinary ring . winn et al . ( 2006 ) use radial velocity measurements , ccd and photographic photometry obtained over the past 50 years to examine whether a model of kh-15d that incorporates a circumbinary disk can successfully account for its observed flux variations . after making some refinements such as the inclusion of disk scattering , they find that the model is successful in reproducing the observed eclipses . + , width=453 ] * ngc2261 ( hubble s variable nebula ) : * over a degree south of s mon lies the small reflection nebula ngc2261 , first noted by friedrich wilhelm herschel in his catalog of nebulae and stellar clusters . shown in figure 17 is a composite image of this object that reveals some of the extraordinary detail associated with the nebulosity . hubble ( 1916 ) describes ngc2261 as a cometary nebula in the form of an equilateral triangle with a sharp stellar nucleus at the extreme southern point . " what drew hubble s attention to the object , however , were indisputable changes in the outline and structure of the nebula that occurred between march 1908 and march 1916 . within the nucleus of ngc2261 is the irregular variable and herbig aebe star , r mon , which ranges in brightness from @xmath1349.5 to 13 mag . the changes within the nebula , however , did not appear to coincide with the brightness variations of r mon . slipher ( 1939 ) obtained a spectrum of ngc2261 that revealed bright hydrogen emission lines superposed upon a faint continuum . the nova - like spectrum of the nebula was identical to that of r mon , even in its outlying regions . lampland ( 1948 ) examined several hundred photographic plates of ngc2261 obtained over more than two decades and concluded that changes in the nebula were caused by varying degrees of veiling or obscuration , not physical motion . several polarization studies of the nebula have been made including those of hall ( 1964 ) , kemp et al . ( 1972 ) , aspin et al . ( 1985 ) , menard et al . ( 1988 ) , and close et al . ( 1997 ) . spectroscopic studies of ngc2261 include those of slipher ( 1939 ) , herbig ( 1968 ) , and stockton et al . herbig ( 1968 ) discovered the presence of hh 39 lying 8 north of the apex of ngc2261 . co ( 10 ) observations of ngc2261 by cant et al . ( 1981 ) identified an elongated molecular cloud centered upon r mon that is interpreted as being disk - shaped in structure . stellar winds from r mon are proposed to have created a bipolar cavity within the cloud , the northern lobe of which is the visible nebulosity of ngc2261 . brugel et al . ( 1984 ) identify both components of the highly collimated bipolar outflow associated with r mon . the measured velocities for the two flows are @xmath135 and @xmath136 km / s , respectively . brugel et al . ( 1984 ) argue that the flow collimation occurs within 2000 au of the star . walsh & malin ( 1985 ) obtained deep @xmath137 and @xmath138band ccd images of hh 39 , revealing a knot of nebulosity ( hh 39 g ) that has varied in brightness and has a measured proper motion . a filament is also observed between hh 39 and r mon that may be related to a stellar wind - driven flow . movsessian et al . ( 2002 ) determined radial velocities for several knots in the hh 39 group and find that the kinematics of the system as a whole suggest the precession of the outflow . polarization maps of ngc2261 by aspin et al . ( 1985 ) identified small lobes close to r mon that support the bipolar model proposed by cant et al . ( 1981 ) . among recent investigations of r mon and ngc2261 is the adaptive optics @xmath139band imaging polarimetry survey of close et al . ( 1997 ) who find that r mon is a close binary ( 0.69 separation ) . the companion is believed to be a 1.5 m@xmath3 star that dereddens to the classical t tauri star locus . r mon itself appears to be an unresolved point source , but exhibits a complex of twisted filaments that extend from 1000 - 100,000 au from the star and possibly trace the magnetic field in the region . weigelt et al . ( 2002 ) use near infrared speckle interferometry to examine structure within the immediate vicinity of r mon at 55 mas ( in @xmath140band ) scales . the primary ( r mon ) appears marginally extended in @xmath102band and significantly extended in @xmath140band . weigelt et al . ( 2002 ) also identify a bright arc - shape feature pointing away from r mon in the northwesterly direction , which is interpreted as the surface of a dense structure near the circumstellar disk surrounding r mon . their images confirm the presence of the twisted filaments reported by close et al . ( 1997 ) . m ( blue ) , 5.8 @xmath5 m ( cyan ) , 8 @xmath5 m ( green ) , and 24 @xmath5 m ( red ) emission . [ f18],width=458 ] ngc2264 has remained a favored target for star formation studies for more than half a century , but significant work remains unfinished . analysis of extensive _ spitzer _ irac and mips surveys of ngc2264 is nearing completion and will soon be available . the small sampling of _ spitzer _ data presented by teixeira et al . ( 2006 ) of the star forming core near irs2 provides some insight into the details of the star formation process that will be revealed . from these datasets the disk - bearing population of the cluster will be unambiguously identified , and tentative classifications of disk structure will be possible by comparing observed seds with models . a preview of the final _ spitzer _ image is shown in figure 18 , which combines irac and mips data to create a composite 5-color image . the spokes cluster is readily apparent near image center . ngc2264 is perhaps best described not as a single cluster , but rather as multiple sub - clusters in various stages of evolution spread across several parsecs . other future observations of the cluster that are needed include : high resolution sub / millimeter maps of all molecular cloud cores within the region ; deep optical and near infrared photometry for a complete substellar census of the cluster ; and a modern proper motion survey for membership determinations . also of interest will be high fidelity ( hst wfpc2/wfc3 ) photometric studies that may be capable of reducing the inferred age dispersion in the color - magnitude diagram of the cluster . because of its relative proximity , significant and well - defined stellar population , and low foreground extinction , ngc2264 will undoubtedly remain a principal target for star formation and circumstellar disk evolution studies throughout the foreseeable future . + * acknowledgments . * i wish to thank the referee , ettore flaccomio , and the editor , bo reipurth , for many helpful comments and suggestions that significantly improved this work . i am also grateful to t. hallas for permission to use figure 2 , t.a . rector and b.a . wolpa for figure 3 , and to j .- c . cuillandre and g. anselmi for figure 10 , and to carole westphal and adam block for figure 17 . sed is supported by an nsf astronomy and astrophysics postdoctoral fellowship under award ast-0502381 . adams , m. t. , strom , k. m. , & strom , s. e. 1979 , apj , 230 , l183 adams , m. t. , strom , k. m. , & strom , s. e. 1983 , apjs , 53 , 893 agol , e. , barth , aaron j. , wolf , s. , & charbonneau , d. 2004 , apj , 600 , 781 aspin , c. , mclean , i. s. , & coyne , g. v. 1985 , a&a , 149 , 158 aspin , c. & reipurth , b. 2003 , aj , 126 , 2936 allen , d. a. 1972 , , 172 , l55 baraffe , i. , chabrier , g. , allard , f. , & hauschildt , p. h. 1998 , a&a , 337 , 403 barry , d. c. , cromwell , r. h. , & schoolman , s. a. 1979 , apjs , 41 , 119 beardsley , w. r. & jacobsen , t. s. 1978 , apj , 222 , 570 bernasconi , p. a. & maeder , a. 1996 , a&a , 307 , 829 blitz , l. 1979 , ph.d . thesis columbia univ . , new york , ny breger , m. 1972 , apj , 171 , 267 brugel , e. w. , mundt , r. , & buehrke , t. 1984 , apj , 287 , 73 cant , j. , rodriguez , l. f. , barral , j. f. , & carral , p. 1981 , apj , 244 , 102 castelaz , m. w. , & grasdalen , g. 1988 , apj , 335 , 150 close , l. m. , roddier , f. , hora , j. l. , graves , j. e. , northcott , m. , et al . 1997 , apj , 489 , 210 cohen , m. & kuhi , l. v. 1979 , apjs , 41 , 743 crutcher , r. m. , hartkopf , w. i. , & giguere , p. t. 1978 , apj , 226 , 839 dahm , s. e. , & simon , t. 2005 , aj , 129 , 829 dahm , s. e. , simon , t. , proszkow , e. m. , & patten b. m. 2007 , aj , 134 , 999 dantona , f. , & mazzitelli , i. 1994 , apjs , 90 , 467 dantona , f. , & mazzitelli , i. 1997 , in _ cool stars in clusters and associations _ , ed . g. micela & r. pallavicini ( firenze : soc . astron . italiana ) , 807 de zeeuw , p. t. , hoogerwerf , r. , & de bruijne , j. h. j. 1999 , 117 , 354 fallscheer , c. & herbst , w. 2006 , apj , l155 feigelson , e. d. & montmerle , t. 1999 , ara&a , 37 , 363 feldbrugge , p. t. m. & van genderen , a. m. 1991 , a&as , 91 , 209 flaccomio , e. , micela , g. , sciortino , s. , favata , f. , corbally , c. , & tomaney , a. 1999 , a&a , 345 , 521 flaccomio , e. , micela , g. , sciortino , s. , damiani , f. , favata , f. , harnden , f. r. , jr . , & schachter , j. 2000 , a&a , 355 , 651 flaccomio , e. , micela , g. , & sciortino , s. 2003 , a&a , 402 , 277 flaccomio , e. , micela , g. , & sciortino , s. 2006 , a&a , 455 , 903 fukui , y. 1989 , in _ proceedings of the eso workshop on low mass star formation and pre - main sequence objects _ , ed . b. reipurth ( garching : european southern observatory ) , 95 furesz , g. , hartmann , l. w. , szentgyorgyi , a. h. , ridge , n. a. , rebull , l. , et al . 2006 , apj , 648 , 109 gies , d. r. , mason , b. d. , hartkopf , w. i. , mcalister , h. a. , frazin , r. a. et al . 1993 , aj , 106 , 207 gies , d. r. , mason , b. d. , bagnuolo , w. g. , jr . , hahula , m. e. , hartkopf , w. i. et al . 1996 , apj , 475 , l49 grady , c. a. , snow , t. p. , & cash , w. c. 1984 , apj , 283 , 218 hall , r. c. 1968 , apj , 139 , 759 hamilton , c. m. , herbst , w. , vrba , f. j. , ibrahimov , m. a. , mundt , r. , et al . 2005 , aj , 130 , 1896 harvey , p. m. , campbell , m. f. , & hoffmann , w. f. 1977 , apj , 215 , 151 hedden , a. s. , walker , c. k. , groppi , c. e. , & butner , h. m. 2006 , apj , 645 , 345 henyey , l. g. , lelevier , r. , & leve , r. d. 1955 , pasp , 67 , 154 herbig , g. h. 1954 , , 119 , 483 herbig , g. h. 1968 , apj , 152 , 439 herbig , g. h. 1974 , lick obs . bull . 658 herbst , w. , hamilton , c. m. , vrba , f. j. , ibrahimov , m. a. , bailer - jones , c. a. l. et al . 2002 , pasp , 114 , 1167 hodapp , k - w . 1994 , apjs , 94 , 615 hubble , e. p. 1916 , apj , 44 , 190 iben , i. & talbot , r. j. 1966 , apj , 144 , 968 johnson , h. l. & morgan , w. w. 1953 , apj , 117 , 313 johnson , h. l. & hiltner , w. a. 1956 , apj , 123 , 267 johnson , j. a. , marcy , g. w. , hamilton , c. m. , herbst , w. , & johns - krull , c. m. 2004 , aj , 128 , 1265 johnson , j. a. , winn , j. n. , rampazzi , f. , barbieri , c. , mito , h. et al . 2005 , aj , 129 , 1978 joy , a. h. 1945 , apj , 102 , 168 kearns , k. e. , eaton , n. l. , herbst , w. , & mazzurco , c. j. 1997 , aj , 114 , 1098 kearns , k. e. & herbst , w. 1998 , aj , 116 , 261 kemp , j. c. , wolstencroft , r. d. , & swedlund , j. b. 1972 , apj , 177 , 177 kendall , t. r. , bouvier , j. , moraux , e. , james , d. j. , & menard , f. 2005 , a&a , 434 , 939 king , j. r. 1998 , aj , 116 , 254 king , j. r. , soderblom , d. r. , fischer , d. , jones , b. f. 2000 , apj , 533 , 944 koch , r. h. & perry , p. m. 1974 , aj , 79 , 379 koch , r. h. , sutton , c. s. , choi , k. h. , kjer , d. e. , & arquilla , r. 1978 , apj , 222 , 574 koch , r. h. , bradstreet , d. h. , hrivnak , b. j. , pfeiffer , r. j. , & perry , p. m. 1986 , aj , 91 , 590 kruegel , e. , guesten , r. , schulz , a. , & thum , c. 1987 , a&a , 185 , 283 lada , c. j. , young , e. t. , & greene , t. p. 1993 , apj , 408 , 471 lamm , m. h. , bailer - jones , c. a. l. , mundt , r. , herbst , w. , & scholz , a. 2004 , a&a , 417 , 557 lamm , m. h. , mundt , r. , bailer - jones , c. a. l. , & herbst , w. 2005 , a&a , 430 , 1005 lampland , c. o. 1948 , aj , 54 , 42 makidon , r. b. , rebull , l. m. , strom , s. e. , adams , m. t. , & patten , b. m. 2004 , aj , 127 , 2228 marcy , g. w. 1980 , aj , 85 , 230 margulis , m. & lada , c. j. 1986 , apj , 309 , 87 margulis , m. , lada , c. j. , & young , e. t. 1989 , apj , 345 , 906 mayer , c. h. , waak , j. a. , cheung , a. c. , & chui , m. f. 1973 , apj , 182 , 65 menard , f. , bastien , p. , & robert , c. 1988 , apj , 335 , 290 mendoza , e. e. 1966 , apj , 143 , 1010 mendoza , e. e. & gmez , t. 1980 , mnras , 190 , 623 menon , t. k. 1956 , aj , 61 , 9 minn , y. k. & greenberg , j. m. 1975 , a&a , 38 , 81 minn , y. k. & greenberg , j. m. 1979 , a&a , 77 , 100 morgan , w. w. , hiltner , w. a. , neff , j. s. , garrison , r. , & osterbrock , d. e. 1965 , apj , 142 , 974 movsessian , t.a . , magakian , t. yu . , & afanasiev , v.l . 2002 , a&a , 390 , l5 nakano , m. , yamauchi , s. , sugitani , k. , & ogura , k. 2000 , pasj , 52 , 437 nakano , m. , sugitani , k. , & morita , k. 2003 , pasj , 55,1 nandy , k. 1971 , mnras , 153 , 521 nandy , k. & pratt n. 1972 , ap&ss , 19 , 219 neri , l. j. , chavarria - k . , c. , & de lara , e. 1993 , a&as , 102 , 201 ogura , k. 1984 , , 36 , 139 ogura , k. 1995 , apj , 450 , 230 oliver , r. j. , masheder , m. r. w. , & thaddeus , p. 1996 , a&a , 315 , 578 park , b. , sung , h. , bessell , m. s. , & kang , y. h. 2000 , aj , 120 , 894 patten , b. m. , simon , t. , strom , s. e. , strom , k. m. 1994 , in _ cool stars , stellar systems , and the sun _ jean - pierre caillault ( san francisco : astronomical society of the pacific ) , p.125 peres , g. , orlando , s. , reale , f. , rosner , r. , & hudson , h. 2000 , apj , 528 , 537 peretto , n. , andre , p. , & belloche , a. 2006 , a&a , 445 , 979 prez , m. r. 1991 , rmxaa , 22 , 99 prez , m. r. , th , p. s. , & westerlund , b. e. 1987 , pasp , 99 , 1050 prez , m. r. , joner , m. d. , th , p. s. , & westerlund , b. e. 1989 , pasp , 101 , 195 pich , f. 1992 , ph.d . thesis , university of washington pich , f. 1993 , pasp , 105 , 324 poveda , a. 1965 , bol . tonantzintla tacubaya , 4 , 15 ramirez , s. v. , rebull , l. , stauffer , j. , hearty , t. , hillenbrand , l. a. et al . 2004 , aj , 127 , 2659 rebull , l. m. , makidon , r. b. , strom , s. e. , hillenbrand , l. a. , birmingham , a. , et al . 2002 , aj , 123 , 1528 rebull , l. m. , wolff , s. c. , & strom , s. e. 2004 , aj , 127 , 1029 rebull , l. m. , stauffer , j. r. , ramirez , s. v. , flaccomio , e. , sciortino , s. et al . 2006 , aj , 131 , 2934 reipurth , b. , yu , k. c. , moriarty - schieven , g. , bally , j. , aspin , c. , & heathcote , s. 2004a , aj , 127 , 1069 reipurth , b. , pettersson , b. , armond , t. , bally , j. , & vaz , l. p. r. 2004b , aj , 127 , 1117 reipurth , b. , rodrguez , l. f. , anglada , g. , & bally , j. 2004c , aj , 127 , 1736 rickard , l. j. , palmer , p. , buhl , d. , & zuckerman , b. 1977 , apj , 213 , 654 riegel , k. w. & crutcher , r. m. 1972 , apj , 172 , l107 rucinski , s. m. 1983 , ap&ss , 89 , 395 rydgren , a. e. 1977 , pasp , 89 , 823 rydgren , a. e. & vrba , f. j. 1987 , pasp , 99 , 482 sagar , r. & joshi , u. c. 1983 , mnras , 205 , 747 sargent , a. i. , van duinen , r. j. , nordh , h. l. , fridlund , c. v. m. , aalders , j. w. , & beintema , d. 1984 , a&a , 135 , 377 schreyer , k. , helmich , f. p. , van dishoeck , e. f. , & henning , t. 1997 , a&a , 326 , 347 schreyer , k. , stecklum , b. , linz , h. , & henning , t. 2003 , apj , 599 , 335 schwartz , p. r. , thronson , h. a. , odenwald , s. f. , glaccum , w. , loewenstein , r. f. , & wolf , g. 1985 , apj , 292 , 231 schwartz , p. r. 1987 , apj , 320 , 258 siess , l. , dufour , e. , & forestini , m. 2000 , , 358 , 593 simon , t. , cash , w. , & snow , t. p. 1985 , apj , 293 , 542 simon , t. & dahm , s. e. 2005 , apj , 618 , 795 slipher , v. m. 1939 , pasp , 51 , 115 snow , t. p. , jr . , cash , w. , & grady , c. a. 1981 , apj , 244 , l19 soderblom , d. r. , king , j. r. , siess , l. , jones , b. f. , & fischer , d. 1999 , aj , 118 , 130 stockton , a. , chesley , d. , & chesley , s. 1975 , apj , 199 , 406 strom , s. e. , strom , k. m. , & yost , j. 1971 , apj , 165 , 479 strom , s. e. , strom , k. m. , brooke , a. l. , bregman , j. , & yost , j. 1972 , apj , 171 , 267 sung , h. , bessell , m. s. , & lee , s .- w . 1997 , aj , 114 , 2644 sung , h. , bessell , m. s. , & chun , m. y. 2004 , , 128 , 1684 swenson , f. j. , faulkner , . , rogers , f. j. , & iglesias , c. a. 1994 , apj , 425 , 286 tauber , j. a. , lis , d. c. , & goldsmith , p. f. 1993 , apj , 403 , 202 teixeira , p. s. , lada , c. j. , young , e. t. , marengo , m. , muench , a. et al . 2006 , apj , 636 , 45 teixeira , p. s. , zapata , l. a. , & lada , c. j. 2007 , apj , 667 , l179 thompson , r. i. & tokunaga , a. t. 1978 , apj , 226 , 119 thompson , r. i. , corbin , m. r. , young , e. , & schneider , g. 1998 , apj , 492 , 177 trejo , a. & rodrguez , l.f . 2008 , aj , 135 , 575 trumpler , r. j. 1930 , lick observatory bulletin , 420 , 188 turner , d. g. 1976 , apj , 210 , 65 underhill , a. b. 1958 , pasp , 70 , 607 vasilevskis , s. , sanders , w. l. , & balz , jr . , a. g. a. 1965 , , 70 , 797 walker , m. f. 1954 , aj , 59 , 333 walker , m. f. 1956 , apjs , 2 , 365 walker , m. f. 1972 , apj , 175 , 89 walker , m. f. 1977 , pasp , 89 , 874 walsh , j. r. & malin , d. f. 1985 , mnras , 217 , 31 walsh , j. r. , ogura , k. , & reipurth , b. 1992 , mnras , 257 , 110 wang , h. , yang , j. , wang , m. , & yan , j. 2002 , a&a , 389 , 1015 wang , h. , yang , j. , wang , m. , & yan , j. 2003 , aj , 125 , 842 ward - thompson , d. , zylka , r. , mezger , p. g. , & sievers , a. w. 2000 , a&a , 355 , 1122 warner , j. w. , strom , s. e. , & strom , k. m. 1977 , apj , 213 , 427 weigelt , g. , balega , y. y. , hofmann , k .- h . , & preibisch , t. 2002 , a&a , 392 , 937 williams , j. p. & garland , c. a. 2002 , apj , 568 , 259 winn , j. n. , hamilton , c. m. , herbst , w. j. , hoffman , j. l. , holman , m. j. , et al . 2006 , apj , 644 , 510 wolf , m. 1924 , astronomische nachrichten , 221 , 379 wolf - chase , g. & gregersen , e. 1997 , apj , 479 , 67 wolf - chase , g. , moriarty - schieven , g. , fich , m. , & barsony , m. 2003 , mnras , 344 , 809 young , a. 1978 , pasp , 90 , 144 young , e. t. , teixeira , p. s. , lada , c. j. , muzerolle , j. , persson , s. e. et al . 2006 , apj , 642 , 972 zuckerman , b. , morris , m. , palmer , p. , & turner , b. e. 1972 , apj , 173 , 125

ngc2264 is a young galactic cluster and the dominant component of the mon ob1 association lying approximately 760 pc distant within the local spiral arm . the cluster is hierarchically structured , with subclusters of suspected members spread across several parsecs . associated with the cluster is an extensive molecular cloud complex spanning more than two degrees on the sky . the combined mass of the remaining molecular cloud cores upon which the cluster is superposed is estimated to be at least @xmath03.7@xmath110@xmath2 m@xmath3 . star formation is ongoing within the region as evidenced by the presence of numerous embedded clusters of protostars , molecular outflows , and herbig - haro objects . the stellar population of ngc2264 is dominated by the o7 v multiple star , s mon , and several dozen b - type zero - age main sequence stars . x - ray imaging surveys , h@xmath4 emission surveys , and photometric variability studies have identified more than 600 intermediate and low - mass members distributed throughout the molecular cloud complex , but concentrated within two densely populated areas between s mon and the cone nebula . estimates for the total stellar population of the cluster range as high as 1000 members and limited deep photometric surveys have identified @xmath0230 substellar mass candidates . the median age of ngc2264 is estimated to be @xmath03 myr by fitting various pre - main sequence isochrones to the low - mass stellar population , but an apparent age dispersion of at least @xmath05 myr can be inferred from the broadened sequence of suspected members . infrared and millimeter observations of the cluster have identified two prominent sites of star formation activity centered near ngc2264 irs1 , a deeply embedded early - type ( b2b5 ) star , and irs2 , a star forming core and associated protostellar cluster . ngc2264 and its associated molecular clouds have been extensively examined at all wavelengths , from the centimeter regime to x - rays . given its relative proximity , well - defined stellar population , and low foreground extinction , the cluster will remain a prime candidate for star formation studies throughout the foreseeable future .

in physics , formal simplicity is often a reliable guide to the significance of a result . the concept of weak measurement , due to aharonov and his coworkers @xcite , derives some of its appeal from the formal simplicity of its basic formulae . one can extend the basic concept to a sequence of weak measurements carried out at a succession of points during the evolution of a system @xcite , but then the formula relating pointer positions to weak values turns out to be not quite so simple , particularly if one allows arbitrary initial conditions for the measuring system . i show here that the complications largely disappear if one takes the cumulants of expected values of pointer positions ; these are related in a formally satisfying way to weak values , and this form is preserved under all measurement conditions . the goal of weak measurement is to obtain information about a quantum system given both an initial state @xmath0 and a final , post - selected state @xmath1 . since weak measurement causes only a small disturbance to the system , the measurement result can reflect both the initial and final states . it can therefore give richer information than a conventional ( strong ) measurement , including in particular the results of all possible strong measurements @xcite . to carry out the measurement , a measuring device is coupled to the system in such a way that the system is only slightly perturbed ; this can be achieved by having a small coupling constant @xmath2 . after the interaction , the pointer s position @xmath3 is measured ( or possibly some other pointer observable ; e.g. its momentum @xmath4 ) . suppose that , following the standard von neumann paradigm , @xcite , the interaction between measuring device and system is taken to be @xmath5 , where @xmath4 is the momentum of a pointer and the delta function indicates an impulsive interaction at time @xmath6 . it can be shown @xcite that the expectation of the pointer position , ignoring terms of order @xmath7 or higher , is @xmath8 where @xmath9 is the _ weak value _ of the observable @xmath10 given by @xmath11 as can be seen , ( [ qclassic ] ) has an appealing simplicity , relating the pointer shift directly to the weak value . however , this formula only holds under the rather special assumption that the initial pointer wavefunction @xmath12 is a gaussian , or , more generally , is real and has zero mean . when @xmath12 is a completely general wavefunction , i.e. is allowed to take complex values and have any mean value @xcite , equation ( [ qclassic ] ) is replaced by @xmath13 where , for any pointer variable @xmath14 , @xmath15 denotes the initial expected value @xmath16 of @xmath14 ; so for instance @xmath17 and @xmath18 are the means of the initial pointer position and momentum , respectively . ( again , this formula ignores terms of order @xmath7 or higher . ) equation ( [ complex - version ] ) seems to have lost the simplicity of ( [ qclassic ] ) , but we can rewrite it as @xmath19 where @xmath20 and equation ( [ firstxi ] ) is then closer to the form of ( [ qclassic ] ) . as will become clear , this is part of a general pattern . one can also weakly measure several observables , @xmath21 , in succession @xcite . here one couples pointers at several locations and times during the evolution of the system , taking the coupling constant @xmath22 at site @xmath23 to be small . one then measures each pointer , and takes the product of the positions @xmath24 of the pointers . for two observables , and in the special case where the initial pointer distributions are real and have zero mean , e.g. a gaussian , one finds @xcite @xmath25,\end{aligned}\ ] ] ignoring terms in higher powers of @xmath26 and @xmath27 . here @xmath28 is the _ sequential weak value _ defined by @xmath29 where @xmath30 is a unitary taking the system from the initial state @xmath0 to the first weak measurement , @xmath31 describes the evolution between the two measurements , and @xmath32 takes the system to the final state . ( note the reverse order of operators in @xmath33 , which reflects the order in which they are applied . ) if we drop the assumption about the special initial form of the pointer distribution and allow an arbitrary @xmath12 , then the counterpart of ( [ abmean ] ) becomes extremely complicated : see appendix , equation [ horrible ] . even the comparatively simple formula ( [ abmean ] ) is not quite ideal . by analogy with ( [ qclassic ] ) we would hope for a formula of the form @xmath34 , but there is an extra term @xmath35 . what we seek , therefore , is a relationship that has some of the formal simplicity of ( [ qclassic ] ) and furthermore preserves its form for all measurement conditions . it turns out that this is possible if we take the _ cumulant _ of the expectations of pointer positions . as we shall see in the next section , this is a certain sum of products of joint expectations of subsets of the @xmath36 , which we denote by @xmath37 . for a set of observables , we can define a formally equivalent expression using sequential weak values , which we denote by @xmath38 . then the claim is that , up to order @xmath39 in the coupling constants @xmath22 ( assumed to be all of the same approximate order of magnitude ) : @xmath40 where @xmath41 is a factor dependent on the initial wavefunctions for each pointer . equation ( [ cumulant - equation ] ) holds for any initial pointer wavefunction , though different wavefunctions produce different values of @xmath41 . the remarkable thing is that all the complexity is packed into this one number , rather than exploding into a multiplicity of terms , as in ( [ horrible ] ) . note also that ( [ firstxi ] ) has essentially the same form as ( [ cumulant - equation ] ) since , in the case @xmath42 , @xmath43 . however , there is an extra term @xmath44 in ( [ firstxi ] ) ; this arises because the cumulant for @xmath42 is anomalous in that its terms do not sum to zero . given a collection of random variables , such as the pointer positions @xmath36 , the cumulant @xmath45 is a polynomial in the expectations of subsets of these variables @xcite ; it has the property that it vanishes whenever the set of variables @xmath36 can be divided into two independent subsets . one can say that the cumulant , in a certain sense , picks out the maximal correlation involving all of the variables . we introduce some notation to define the cumulant . let @xmath14 be a subset of the integers @xmath46 . we write @xmath47 for @xmath48 , where @xmath49 is the size of @xmath14 and the indices of the @xmath3 s in the product run over all the integers @xmath50 in @xmath14 . then the cumulant is given by @xmath51 where @xmath52 runs over all partitions of the integers @xmath46 and the coefficient @xmath53 is given by @xmath54 for @xmath42 we have @xmath55 , and for @xmath56 @xmath57 there is an inverse operation for the cumulant @xcite : [ anti ] @xmath58 to see that this equation holds , we must show that the term @xmath59 obtained by expanding the right - hand side is zero unless @xmath60 is the partition consisting of the single set @xmath46 . replacing each subset @xmath61 by the integer @xmath62 , this is equivalent to @xmath63 , where the sum is over all partitions of @xmath64 by subsets of sizes @xmath65 and the @xmath66 s are given by ( [ coefficients ] ) . in this sum we distinguish partitions with distinct integers ; e.g. @xmath67 and @xmath68 . there are @xmath69 such distinct partitions with subset sizes @xmath70 , where @xmath71 is the number of @xmath23 s equal to @xmath72 , so our sum may be rewritten as @xmath73 , where the sum is now over partitions in the standard sense @xcite . this is @xmath74 times the coefficient of @xmath75 in @xmath76 thus the sum is zero except for @xmath77 , which corresponds to the single - set partition @xmath60 . if @xmath46 can be written as the disjoint union of two subsets @xmath78 and @xmath79 , we say the variables corresponding to these subsets are independent if @xmath80 for any subsets @xmath81 . we now prove the characteristic property of cumulants : [ indep - lemma ] the cumulant vanishes if its arguments can be divided into two independent subsets . for @xmath56 this follows at once from ( [ q2 ] ) and ( [ indep ] ) , and we continue by induction . from ( [ anticumulant ] ) and the inductive assumption for @xmath82 , we have @xmath83 this holds because any term on the right - hand side of ( [ anticumulant ] ) vanishes when any subset of the partition @xmath60 includes elements of both @xmath78 and @xmath79 . using ( [ anticumulant ] ) again , this implies @xmath84 and by independence , @xmath85 . thus the inductive assumption holds for @xmath39 . in fact , the coefficients @xmath53 in ( [ cumulant ] ) are uniquely determined to have the form ( [ coefficients ] ) by the requirement that the cumulant vanishes when the variables form two independent subsets @xcite . for @xmath56 , the cumulant ( [ q2 ] ) is just the covariance , @xmath86 , and the same is true for @xmath87 , namely @xmath88 . for @xmath89 , however , there is a surprise . the covariance is given by @xmath90 where the sums include all distinct combinations of indices , but the cumulant is @xmath91 which includes terms like @xmath92 that do not occur in the covariance . note that , if the subsets @xmath93 and @xmath94 are independent , the covariance does not vanish , since independence implies we can write the first term in ( [ 4covariance ] ) as @xmath95 and there is no cancelling term . however , as we have seen , the cumulant does contain such a term , and it is a pleasant exercise to check that the whole cumulant vanishes . to carry out a sequential weak measurement , one starts a system in an initial state @xmath0 , then weakly couples pointers at several times @xmath96 during the evolution of the system , and finally post - selects the system state @xmath1 . one then measures the pointers and finally takes the product of the values obtained from these pointer measurements . it is assumed that one can repeat the whole process many times to obtain the expectation of the product of pointer values . if one measures pointer positions @xmath24 , for instance , one can estimate @xmath97 , but one could also measure the momenta of the pointers to estimate @xmath98 . if the coupling for the @xmath23th pointer is given by @xmath99 , and if the individual initial pointer wavefunctions are gaussian , or , more generally , are real with zero mean , then it turns out @xcite that these expectations can be expressed in terms of sequential weak values of order @xmath39 or less . here the sequential weak value of order @xmath39 , @xmath100 , is defined by @xmath101 where @xmath102 defines the evolution of the system between the measurements of @xmath103 and @xmath104 . when the @xmath105 are projectors , @xmath106 , we can write the sequential weak value as @xcite @xmath107 which shows that , in this case , the weak values has a natural interpretation as the amplitude for following the path defined by the @xmath108 . figure [ cumulant ] shows an example taken from @xcite where the path ( labelled by 1 and 2 successively ) is a route taken by a photon through a pair of interferometers , starting by injecting the photon at the top left ( with state @xmath109 ) and ending with post - selection by detection at the bottom right ( with final state @xmath110 ) . in the last section , the cumulant was defined for expectations of products of variables . one can define the cumulant for other entities by formal analogy ; for instance for density matrices @xcite , or hypergraphs @xcite . we can do the same for sequential weak values , defining the cumulant by ( [ cumulant ] ) with @xmath111 replaced by @xmath112 , where the arrow indicates that the indices , which run over the subset @xmath61 , are arranged in ascending order from right to left . for example , for @xmath42 , @xmath113 , and for @xmath89 @xmath114 there is a notion of independence that parallels ( [ indep ] ) : given a disjoint partition @xmath115 such that @xmath116 for any subsets @xmath81 , then we say the observables labelled by the two subsets are _ weakly independent_. there is then an analogue of lemma [ indep - lemma ] : the cumulant @xmath117 vanishes if the @xmath105 are weakly independent for some subsets @xmath78 , @xmath79 . as an example of this , if one is given a bipartite system @xmath118 , and initial and final states that factorise as @xmath119 and @xmath120 , then observables on the @xmath10- and @xmath121-parts of the system are clearly weakly independent . another class of examples comes from what one might describe as a `` bottleneck '' construction , where , at some point the evolution of the system is divided into two parts by a one - dimensional projector ( the bottleneck ) and its complement , and the post - selection excludes the complementary part . then , if all the measurements before the projector belong to @xmath78 and all those after the projector belong to @xmath79 , the two sets are weakly independent . this follows because we can write @xmath122 where @xmath123 is the part of @xmath124 lying in the post - selected subspace . as an illustration of this , suppose we add a connecting link ( figure [ bottleneckfig ] , `` @xmath125 '' ) between the two interferometers in figure [ cumulantfig ] , so @xmath126 , the bottleneck , is the projection onto @xmath125 , and post - selection discards the part of the wavefunction corresponding to the path @xmath127 . then measurements at ` 1 ' and ` 2 ' are weakly independent ; in fact @xmath128 , @xmath129 and @xmath130 . note that the same measurements are _ not _ independent in the double interferometer of figure [ cumulantfig ] , where @xmath131 , @xmath132 , and yet , surprisingly , @xmath133 , @xcite . consider @xmath39 system observables @xmath21 . suppose @xmath134 , for @xmath135 , are observables of the @xmath23th pointer , namely hermitian functions @xmath136 of pointer position @xmath24 and momentum @xmath137 , and the interaction hamiltonian for the weak measurement of system observable @xmath105 is @xmath138 , where @xmath22 is a small coupling constant ( all @xmath22 being assumed of the same order of magnitude @xmath2 ) . suppose further that the pointer observables @xmath139 are measured after the coupling . let @xmath140 be the @xmath23-th pointer s initial wave - function . for any variable @xmath108 associated to the @xmath23-th pointer , write @xmath141 for @xmath142 . we are now almost ready to state the main theorem , but first need to clarify the measurement procedure . when we evaluate expectations of products of the @xmath139 for different sets of pointers , for instance when we evaluate @xmath143 , we have a choice . we could either couple the entire set of @xmath39 pointers and then select the data for pointers 1 and 2 to get @xmath143 . or we could carry out an experiment in which we couple just pointers 1 and 2 to give @xmath143 . these procedures give different answers . for instance , if we couple three pointers and measure pointers 1 and 2 to get @xmath143 , in addition to the terms in @xmath26 , @xmath27 and @xmath144 we also get terms in @xmath145 and @xmath146 involving the observable @xmath147 . this means we get a different cumulant @xmath148 , depending on the procedure used . in what follows , we regard each expectation as being evaluated in a separate experiment , with only the relevant pointers coupled . it will be shown elsewhere that , with the alternative definition , the theorem still holds but with a different value of the constant @xmath41 . [ main - theorem ] for @xmath149 , for any pointer observables @xmath139 and @xmath134 , and for any initial pointer wavefunctions @xmath140 , up to total order @xmath39 in the @xmath22 , @xmath150 where @xmath41 ( sometimes written more explicitly as @xmath151 ) is given by @xmath152 for @xmath42 the same result holds , but with the extra term @xmath153 : @xmath154 we use the methods of @xcite to calculate the expectations of products of pointer variables for sequential weak measurements . let the initial and final states of the system be @xmath0 and @xmath1 , respectively . consider some subset @xmath155 of @xmath46 , with @xmath156 . the state of the system and the pointers @xmath157 after the coupling of those pointers is @xmath158 and following post - selection by the system state @xmath1 , the state of the pointers is @xmath159 expanding each exponential , we have @xmath160 where @xmath161 are integers , @xmath162 means that @xmath163 for @xmath164 , and @xmath165 let us write ( [ sumratio ] ) as @xmath166 where @xmath167 and @xmath168 denotes the index set @xmath169 , etc .. define @xmath170 then @xmath171 set @xmath172 , where @xmath60 in the product ranges over all distinct subsets of the integers @xmath173 . then @xmath174 is an ( infinite ) weighted sum of terms @xmath175 where @xmath176 denotes the set of all the index sets that occur in @xmath177 . the strategy is to show that , when the size of the index set @xmath178 is less than @xmath39 , the coefficient of @xmath177 vanishes ; by ( [ alpha ] ) this implies that all coefficients of order less than @xmath39 in @xmath2 vanish . we then look at the index sets of size @xmath39 , corresponding to terms of order @xmath179 , and show that the relevant terms sum up to the right - hand side of ( [ main - result ] ) . but if @xmath180 for some x , then we also have @xmath181 , since @xmath182 . let @xmath183 be a partition of @xmath46 . we say that @xmath60 is a _ valid _ partition for @xmath178 if a. for each @xmath184 with @xmath185 , @xmath186 , for some @xmath187 , and we can associate a distinct @xmath187 to each @xmath184 . ( here @xmath188 means the index set @xmath189 . ) b. for each @xmath184 with @xmath190 , @xmath191 , for some subset @xmath192 that is not in the partition @xmath60 , i.e. for which @xmath193 for any @xmath194 , and we can associate a distinct @xmath195 to each @xmath184 . let @xmath196 be the number of ways of associating a subset @xmath195 to each @xmath184 . [ vanishing ] the coefficient of @xmath177 in @xmath174 is zero if all the index sets in @xmath178 have a zero at some position @xmath184 . if we expand @xmath174 using ( [ cxy2 ] ) , each term in this expansion is associated with a partition @xmath60 of @xmath197 . let @xmath60 be a valid partition for @xmath178 , and let @xmath198 denote the partition derived from @xmath60 by removing @xmath184 from the subset @xmath187 that contains it , and deleting that subset if it contains only @xmath184 . then the following partitions include @xmath60 and are all valid : @xmath199 each partition @xmath200 , for @xmath201 contributes @xmath196 to the coefficient of @xmath177 in @xmath202 , and since this term has coefficient @xmath203 in ( [ cxy2 ] ) for partitions @xmath204 , and @xmath205 for @xmath206 , the sum of all contributions is zero . from equations ( [ alpha ] ) and ( [ index ] ) , the power of @xmath2 in the term @xmath177 is @xmath207 . this , together with the preceding lemma , implies that the lowest order non - vanishing terms in @xmath174 are @xmath177 s that have a 1 occurring once and once only in each position ; we call these _ complete lowest - degree _ terms . [ one - index - set ] the coefficient of a complete lowest - degree term @xmath177 in @xmath174 is zero unless only one of the four classes of indices in @xmath178 , viz . @xmath208 , @xmath209 , @xmath210 or @xmath211 , has non - zero terms . consider first the case where the indices in @xmath209 and @xmath211 are zero , and where both @xmath208 and @xmath210 have some non - zero indices . let @xmath212 be the partition whose subsets consists of the non - zero positions in index sets @xmath213 in @xmath208 , and let @xmath214 be some partition of the remaining integers in @xmath46 . suppose @xmath215 . then we can construct a set of partitions by mixing @xmath60 and @xmath216 ; these have the form @xmath217 where each @xmath218 is either empty or consists of some @xmath219 , and all the subsets @xmath219 are present once only in the partition . if any @xmath220 is eligible , all the other mixtures will also be eligible . furthermore , the set of all eligible partitions can be decomposed into non - overlapping subsets of mixtures obtained in this way . any mixture @xmath220 gives the same value of @xmath221 , which we denote simply by @xmath222 ; so to show that all the contributions to the coefficient of @xmath177 cancel , we have only to sum over all the mixtures , weighting a partition with @xmath6 subsets by @xmath223 . this gives @xmath224 the above argument applies equally well to the situation where @xmath208 and @xmath211 both have some non - zero indices and indices in @xmath209 and @xmath210 are zero . if the non - zero indices are present in @xmath208 and @xmath209 , we can take any eligible partition @xmath225 and divide each subset @xmath53 into two subsets @xmath226 and @xmath227 with the indices from @xmath208 in @xmath226 and those from @xmath209 in @xmath227 . all the mixtures of type ( [ mix ] ) are eligible , and they include the original partition @xmath228 . by the above argument , the coefficients of @xmath177 arising from them sum to zero . other combinations of indices are dealt with similarly . note that , for @xmath89 and for the index sets @xmath229 and @xmath230 , the `` mixture '' argument shows that coefficient of @xmath177 coming from @xmath231 cancels that coming from @xmath232 to give zero . this cancellation occurs with the cumulant ( [ 4cumulant ] ) , but not with the covariance ( [ 4covariance ] ) , where the term @xmath232 is absent . the only terms that need to be considered , therefore , are complete lowest - degree terms with non - zero indices only in one of the sets @xmath208 , @xmath209 , @xmath210 and @xmath211 . it is easy to calculate the coefficients one gets for such terms . consider the case of @xmath208 . we only need to consider the single partition @xmath60 whose subsets are the index sets of @xmath208 . for this partition , by ( [ z ] ) , ( [ x ] ) and ( [ y ] ) , @xmath233 from ( [ cxy2 ] ) , @xmath177 appears in @xmath234 with a coefficient @xmath223 . so , summing over all @xmath177 with indices in @xmath208 , one obtains @xmath235 . similarly , from ( [ alpha ] ) , ( [ u ] ) and ( [ v ] ) , summing over the @xmath177 with indices in @xmath209 gives the complex conjugate of @xmath236 . thus @xmath208 and @xmath209 together give @xmath237 . this corresponds to ( [ main - result ] ) , but with only the first half of @xmath41 as defined by ( [ xi ] ) . the rest of @xmath41 comes from the index sets @xmath210 and @xmath211 . however , the sum of the coefficients of @xmath177 for the same index set in @xmath208 and @xmath210 is zero . this is true because , for any complete lowest degree index set , the sum of coefficients for all @xmath177 with the indices divided in any manner between @xmath208 and @xmath210 is zero , being the number ways of obtaining that index set from @xmath238 times @xmath239 . but by lemma [ one - index - set ] , the coefficient of @xmath177 is zero unless the index set comes wholly from @xmath208 or @xmath210 . now ( [ z ] ) , ( [ x ] ) and ( [ y ] ) tell us that , for an index set in @xmath210 , @xmath240 and from the above argument , this appears appears in @xmath241 with coefficient @xmath242 . again , the index sets in @xmath211 give the complex conjugate of those in @xmath210 . thus we obtain the remaining half of @xmath41 , which proves ( [ main - result ] ) for @xmath149 . for @xmath42 the constant terms ( of order zero in @xmath2 ) in @xmath243 do not vanish , but the proof goes through if we consider @xmath244 instead . consider first the simplest case , where @xmath42 and @xmath245 . we take @xmath246 throughout this section , so @xmath247 . then ( [ main - result1 ] ) and ( [ xi ] ) give @xmath248 which we have already seen as equations ( [ firstxi ] ) and ( [ xi1 ] ) . if we measure the pointer momentum , so @xmath249 , we find @xmath250 which is equivalent to the result obtained in @xcite . for two variables , our theorem for @xmath251 , is @xmath252 with @xmath253 the calculations in the appendix allow one to check ( [ qq ] ) and ( [ xiqq ] ) by explicit evaluation ; see ( [ explicit ] ) . note in passing that , if one writes @xmath254 , the cauchy - schwarz inequality @xmath255 implies a heisenberg - type inequality @xmath256 relating the pointer noise distributions of two weak measurements carried out at different times during the evolution of the system . when one or both of the @xmath24 in ( [ qq ] ) is replaced by the pointer momentum @xmath137 , we get @xmath257 with @xmath258 consider now the special case where @xmath12 is real with zero mean . then the very complicated expression for @xmath259 in ( [ horrible ] ) reduces to @xmath260,\end{aligned}\ ] ] as shown in @xcite . two further examples from @xcite are @xmath261,\\ \label{4q } \langle q_1q_2q_3q_4 \rangle&=\frac{g_1g_2g_3g_4}{8}\ re \left [ ( a_4,a_3,a_2,a_1)_w+(a_4,a_3,a_2)_w({\bar a}_1)_w+\ldots + ( a_4,a_3)_w(\overline{a_2,a_1})_w+\ldots \right].\end{aligned}\ ] ] we can use these formulae to calculate the cumulant @xmath262 , and thus check theorem [ main - theorem]for this special class of wavefunctions @xmath12 . each formula contains on the right - hand side a leading sequential weak value , but there are also extra terms , such as @xmath263 in ( [ 2q ] ) and @xmath264 in ( [ 3q ] ) . all these extra terms are eliminated when the cumulant is calculated , and we are left with ( [ main - result ] ) with @xmath265 . this gratifying simplification depends on the fact that the cumulant is a sum over all partitions . for instance , it does not occur if one uses the covariance instead of the cumulant . to see this , look at the case @xmath89 : the term @xmath266 in @xmath267 , the covariance of pointer positions , gives rise via ( [ 4q ] ) to weak value terms like @xmath268 . however , ( [ 4covariance ] ) together with ( [ 2q ] ) , ( [ 3q ] ) and ( [ 4q ] ) show that @xmath267 has no other terms that generate any multiple of @xmath268 , and consequently this weak value expression can not be cancelled and must be present in @xmath267 . this means that there can not be any equation relating @xmath267 and @xmath269 . this negative conclusion does not apply to the cumulant @xmath270 , as this includes terms such as @xmath271 ; see ( [ 4cumulant ] ) . we have treated the interactions between each pointer and the system individually , the hamiltonian for the @xmath23th pointer and system being @xmath272 , but of course we can equivalently describe the interaction between all the pointers and the system by @xmath273 . for sequential measurements we implicitly assume that all the times @xmath96 are distinct . however , the limiting case where there is no evolution between coupling of the pointers and all the @xmath96 s are equal is of interest , and is the _ simultaneous _ weak measurement considered in @xcite . in this case , the state of the pointers after post - selection is given by @xmath274 the exponential @xmath275 here differs from the sequential expression @xmath276 in ( [ bigstate ] ) in that each term in the expansion of the latter appears with the operators in a specific order , viz . the arrow order @xmath277 as in ( [ 4weak ] ) , whereas in the expansion of the former the same term is replaced by a symmetrised sum over all orderings of operators . for instance , for arbitrary operators @xmath278 , @xmath279 and @xmath280 , the third degree terms in @xmath281 include @xmath282 , @xmath283 and @xmath284 , whose counterparts in @xmath285 are , respectively , @xmath282 , @xmath286 and @xmath287 . apart from this symmetrisation , the calculations in section [ theorem - section ] can be carried through unchanged for simultaneous measurement . thus if we replace the sequential weak value by the _ simultaneous weak value _ @xcite @xmath288 where the sum on the right - hand side includes all possible orders of applying the operators , we obtain a version of theorem [ main - theorem ] for simultaneous weak measurement : @xmath289 likewise , relations such ( [ 2q ] ) , ( [ 3q ] ) , etc . , hold with simultaneous weak values in place of the sequential weak values ; indeed , these relations were first proved for simultaneous measurement @xcite . from ( [ swv ] ) we see that , when the operators @xmath105 all commute , the sequential and simultaneous weak values coincide . one important instance of this arises when the operators @xmath105 are applied to distinct subsystems , as in the case of the simultaneous weak measurements of the electron and positron in hardy s paradox @xcite . when the operators do not commute , the meaning of simultaneous weak measurement is not so obvious . one possible physical interpretation follows from the well - known formula @xmath290 and its analogues for more operators . suppose two pointers , one for @xmath291 and one for @xmath292 , are coupled alternately in a sequence of @xmath293 short intervals ( figure [ alternate ] , top diagram ) with coupling strength @xmath294 for each interval . this is an enlarged sense of sequential weak measurement @xcite in which the same pointer is used repeatedly , coherently preserving its state between couplings . the state after post - selection is @xmath295 from ( [ formula ] ) we deduce that @xmath296 this picture readily extends to more operators @xmath105 . one can also simulate a simultaneous measurement by averaging the results of a set of sequential measurements with the operators in all orders ; in effect , one carries out a set of experiments that implement the averaging in ( [ swv ] ) . there is then no single act that counts as simultaneous measurement , but weak measurement in any case relies on averaging many repeats of experiments in order to extract the signal from the noise . in a certain sense , therefore , sequential measurement includes and extends the concept of simultaneous measurement . however , if we wish to accomplish simultaneous measurement in a single act , then we need a broader concept of weak measurement where pointers can be re - used ; indeed , we can go further , and consider generalised weak coupling between one time - evolving system and another , followed by measurement of the second system . however , even in this case , the measurement results can be expressed algebraically in terms of the sequential weak values of the first system @xcite . lundeen and resch @xcite showed that , for a gaussian initial pointer wavefunction , if one defines an operator @xmath228 by @xmath297 then the relationship @xmath298 holds . they argued that @xmath299 can be interpreted physically as a lowering operator , carrying the pointer from its first excited state @xmath300 , in number state notation , to the gaussian state @xmath301 ( despite the fact that the pointer is not actually in a harmonic potential ) . although @xmath299 is not an observable , @xmath302 can be regarded as a prescription for combining expecations of pointer position and momentum to get the weak value . if instead of @xmath299 one takes @xmath303 then the even simpler relationship @xmath304 holds . we refer to @xmath228 as a generalised lowering operator . lundeen and resch also extended their lowering operator concept to simultaneous weak measurement of several observables @xmath105 . rephrased in terms of our generalised lowering operators @xmath53 defined by ( [ gaussian ] ) , their finding @xcite can be stated as @xmath305 this is of interest for two reasons . first , the entire simultaneous weak value appears on the right - hand side , not just its real part ; and second , the `` extra terms '' in the simultaneous analogues of ( [ 2q ] ) , ( [ 3q ] ) and ( [ 4q ] ) have disappeared . the lowering operator seems to relate directly to weak values . we can generalise these ideas in two ways . first , we extend them from simultaneous to sequential weak measurements . secondly , instead of assuming the initial pointer wavefunction is a gaussian , we allow it be arbitrary ; we do this by defining a generalised lowering operator @xmath306 for a gaussian @xmath12 , @xmath307 , so the above definition reduces to ( [ gaussian ] ) in this case . in general , however , @xmath12 will not be annihilated by @xmath228 and is therefore not the number state @xmath301 ( this state is a gaussian with complex variance @xmath308 ) . nonetheless , there is an analogue of theorem [ main - theorem ] in which the whole sequential weak value , rather than its real part , appears : [ lowering - theorem ] for @xmath309 @xmath310 where @xmath311 is given by @xmath312 for @xmath42 the same result holds , but with the extra term @xmath313 : @xmath314 put @xmath315 , @xmath316 . then @xmath317,\end{aligned}\ ] ] where we used theorem [ main - theorem ] to get the last line , and where @xmath311 is given by ( [ constant ] ) and @xmath318 by @xmath319 ( note the bar over @xmath320 that is absent in the definition of @xmath311 by ( [ constant ] ) ) . we want to prove @xmath321 , and to do this it suffices to prove that the complex conjugate of the numerator is zero , i.e. @xmath322 let @xmath323 , @xmath324 , @xmath325 , @xmath326 . using the definition of @xmath41 in ( [ xi ] ) , the above equation can be written @xmath327 suppose the interaction hamiltonian has the standard von neumann form @xmath328 , so @xmath247 in the definition of @xmath41 by equation ( [ xi ] ) . then for @xmath42 , since @xmath329 and @xmath330 , @xmath331 , so we get the even simpler result @xmath332 this is valid for all initial pointer wavefunctions , and therefore extends lundeen and resch s equation ( [ lr1 ] ) . it seems almost too simple : there is no factor corresponding to @xmath41 in equation ( [ qmean ] ) . however , a dependency on the initial pointer wavefunction is of course built into the definition of @xmath228 through @xmath333 . for @xmath309 it is no longer true that @xmath334 , even with the standard interaction hamiltonian . however , if in addition @xmath335 , then @xmath336 thus @xmath337 for all @xmath39 . applying the inverse operation for the cumulant , given by propostion [ anti ] , we deduce : if @xmath338 , e.g. if the initial pointer wavefunction @xmath12 is real , then for @xmath309 @xmath339 this is the sequential weak value version of the result for simultaneous measurements , ( [ simul ] ) , but is more general than the gaussian case treated in @xcite . we might be tempted to try to repeat the above argument for pointer positions @xmath24 instead of the lowering operators @xmath53 by applying the anti - cumulant to both sides of ( [ main - result ] ) . this fails , however , because of the need to take the real part of the weak values ; in fact , this is one way of seeing where the extra terms come from in ( [ 2q ] ) , ( [ 3q ] ) and ( [ 4q ] ) and their higher analogues . note also that ( [ nice ] ) does not hold for general @xmath12 , since then different subsets of indices may have different values of @xmath311 . the procedure for sequential weak measurement involves coupling pointers at several stages during the evolution of the system , measuring the position ( or some other observable ) of each pointer , and then multiplying the measured values together . in @xcite it was argued that we would really like to measure the product of the values of the operators @xmath340 , and that this corresponds to the sequential weak value @xmath341 . multiplication of the values of pointer observables is the best we can do to achieve this goal . however , this brings along extra terms , such as @xmath342 in ( [ 2q ] ) , which are an artefact of this method of extracting information . from this perspective , the cumulant extracts the information we really want . in @xcite , a somewhat idealised measuring device was being considered , where the pointer position distribution is real and has zero mean . when the pointer distribution is allowed to be arbitrary , the expressions for @xmath97 become wildly complicated ( see for instance ( [ horrible ] ) ) . yet the cumulant of these terms condenses into the succinct equation ( [ main - result ] ) with all the complexity hidden away in the one number @xmath41 . why does the cumulant have this property ? recall that the cumulant vanishes when its variables belong to two independent sets . the product of the pointer positions @xmath343 will include terms that come from products of disjoint subsets of these pointer positions , and the cumulant of these terms will be sent to zero , by lemma [ indep - lemma ] . for instance , with @xmath56 , the pointers are deflected in proportion to their individual weak values , according to ( [ firstxi ] ) , and the cumulant subtracts this component leaving only the component that arises from the @xmath344-influence of the weak measurement of @xmath291 on that of @xmath292 . the subtraction of this component corresponds to the subtraction of the term @xmath345 from ( [ 2q ] ) . in general , the cumulant of pointer positions singles out the maximal correlation involving all the @xmath36 , and the theorem tells us that this is directly related to the corresponding `` maximal correlation '' of sequential weak values , @xmath346 , which involves all the operators . in fact , the theorem tells us something stronger : that it does not matter what pointer observable @xmath347 we measure , e.g. position , momentum , or some hermitian combination of them , and that likewise the coupling of the pointer with the system can be via a hamiltonian @xmath348 with any hermitian @xmath349 . different choices of @xmath184 and @xmath350 lead only to a different multiplicative constant @xmath41 in front of @xmath117 in ( [ main - result ] ) . we always extract the same function of sequential weak values , @xmath351 , from the system . this argues both for the fundamental character of sequential weak values and also for the key role played by their cumulants . i am indebted to j. berg for many discussions and for comments on drafts of this paper ; i thank him particularly for putting me on the track of cumulants . i also thank a. botero , p. davies , r. jozsa , r. koenig and s. popescu for helpful comments . a preliminary version of this work was presented at a workshop on `` weak values and weak measurement '' at arizona state university in june 2007 , under the aegis of the center for fundamental concepts in science , directed by p. davies . to calculate @xmath352 for arbitrary pointer wavefunctions @xmath353 and @xmath354 , we use ( [ bigstate ] ) to determine the state of the two pointers after the weak interaction , and then evaluate the expectation using ( [ expectation ] ) , keeping only terms up to order @xmath7 . we define @xmath355

a weak measurement on a system is made by coupling a pointer weakly to the system and then measuring the position of the pointer . if the initial wavefunction for the pointer is real , the mean displacement of the pointer is proportional to the so - called weak value of the observable being measured . this gives an intuitively direct way of understanding weak measurement . however , if the initial pointer wavefunction takes complex values , the relationship between pointer displacement and weak value is not quite so simple , as pointed out recently by r. jozsa @xcite . this is even more striking in the case of sequential weak measurements @xcite . these are carried out by coupling several pointers at different stages of evolution of the system , and the relationship between the products of the measured pointer positions and the sequential weak values can become extremely complicated for an arbitrary initial pointer wavefunction . surprisingly , all this complication vanishes when one calculates the cumulants of pointer positions . these are directly proportional to the cumulants of sequential weak values . this suggests that cumulants have a fundamental physical significance for weak measurement .

recently , we proposed @xcite and generalized @xcite the stochastic point process models generating a variety of monofractal and multifractal time series exhibiting power laws of the spectrum @xmath0 and of the distribution @xmath1 of the signal intensity and applied them for the analysis of the financial systems @xcite . these models can generate @xmath2 noise with a very large hooge parameter . they may be used as the theoretical framework for understanding huge fluctuations ( see , e.g. , @xcite ) and as well for description of a large variety of observable statistics , i.e. , jointly power spectral density ( psd ) , @xmath3 , signal probability distribution function ( pdf ) , @xmath4 , with different slopes , different distributions , @xmath5 , of the interevent time @xmath6 and different multifractality . here we will present the extensions and generalizations of the point process models for the poissonian - like processes with slowly diffusing mean interevent time @xcite . we will adjust the parameters of the generalized model to the empirical data of the trading activity in the financial markets @xcite and to the frequencies of the word occurrences in the language , reproducing the pdf and psd . we investigate stochastic time series as a sequence of events which occur at discrete times @xmath7 and can be considered as identical point events . such point process equivalently is defined by the set of stochastic interevent times @xmath8 . let us consider the flow of events as the poissonian - like process driven by the multiplicative stochastic equation . we define the stochastic rate @xmath9 of event flow by continuous stochastic differential equation @xmath10\tau^{2\mu-2}\mathrm{d}t+\sigma\tau^{\mu-1/2}\mathrm{d}w , \label{eq : taustoch2}\ ] ] where @xmath11 is a standard wiener process , @xmath12 denotes the standard deviation of the white noise , @xmath13 is a coefficient of the nonlinear damping and @xmath14 defines the power of noise multiplicativity . the diffusion of @xmath6 is restricted from the side of high values by an additional term @xmath15 , which produces the exponential diffusion reversion . @xmath16 and @xmath17 are the power and value of the diffusion reversion , respectively . the associated fokker - plank equation with the zero flow gives the simple stationary pdf @xmath18\label{eq : taudistrib}\ ] ] with @xmath19 and @xmath20 . ( [ eq : taustoch2 ] ) describes continuous stochastic variable @xmath6 , defines rate @xmath9 with stationary distribution and psd @xmath0 @xcite , @xmath21 @xmath22 here we define the fractal point process driven by the stochastic differential equation ( [ eq : taustoch2 ] ) , i.e. , we assume @xmath23 as slowly diffusing mean interevent time of the poissonian - like process with the stochastic rate @xmath24 . within this assumption the conditional probability of interevent time @xmath25 in the poissonian - like process with the stochastic rate @xmath26 is @xmath27.\label{eq : taupoisson}\ ] ] then the long time distribution @xmath28 of interevent time @xmath25 in @xmath29-space @xcite has the integral form @xmath30\tau^{\alpha-1}\exp\left[-\left(\frac{\tau}{\tau_{0}}\right)^m\right]\mathrm{d } \tau,\label{eq : taupdistrib}\ ] ] with @xmath31 defined from the normalization , @xmath32 . the distributions of interevent time @xmath25 have their explicit forms for the integer values of power @xmath16 . for @xmath33 and for @xmath34 they are expressed by the modified bessel function @xcite and in terms of the hypergeometric functions , respectively . of the point process for word `` eye '' occurrences in the novels of jack london . the straight line approximates the power - law with the exponent @xmath35 . ( b ) the interevent interval @xmath6 distribution of the word `` eye '' occurrences calculated from the histogram in the same novels . the smooth line represents the integral formula ( [ eq : taupdistrib ] ) with @xmath36 , @xmath37 and @xmath38.,title="fig : " ] of the point process for word `` eye '' occurrences in the novels of jack london . the straight line approximates the power - law with the exponent @xmath35 . ( b ) the interevent interval @xmath6 distribution of the word `` eye '' occurrences calculated from the histogram in the same novels . the smooth line represents the integral formula ( [ eq : taupdistrib ] ) with @xmath36 , @xmath37 and @xmath38.,title="fig : " ] we consider two applications of the proposed model . the frequencies of the word occurrences in the language depend on the content and diffuse in the text . we consider the flow of words in the text as the steps of discrete events , i.e. , one word is the unit of the interval . then the number of other words in between of the two successive occurrences of the same noun measures the interevent interval @xmath25 of the point process defined for the sequence of selected noun . one can easily calculate the sequence of all selected word occurrence intervals @xmath25 and so define the realization of the point process . here we demonstrate the statistics of the word `` eye '' in the selected novels of jack london , over 1.2 mln . words totally . first of all , we demonstrate that the point process , defined in such a way , has long memory as the exponent of psd @xmath35 [ fig . 1 ( a ) ] . with the assumption of pure multiplicative process with @xmath39 from the relation @xmath40 one defines the parameter @xmath38 related with the probability distribution functions . the histogram of @xmath25 distribution coincides with theoretical pdf defined by its integral form ( [ eq : taupdistrib ] ) when @xmath36 [ fig . 1 ( b ) ] . the exponent @xmath41 of the power - law distribution for number of the word `` eye '' occurrences in the 1000 words length pieces of text is @xmath42 . as we will see later , the presented example of the word statistics resembles the statistical properties of trading activity in the financial markets . of the stock cvx trade sequence traded on the nyse . ( b ) power spectral density of the poissonian - like process driven by eq . ( [ eq : taucontinuous ] ) with the parameters @xmath43 , @xmath44 , @xmath45 , and @xmath46 . straight lines approximate power - law spectrum with exponents @xmath47 and @xmath48 . ( c ) the empirical distribution of @xmath49 calculated from the histogram of cvx trades on nyse . the smooth line represents the integral formula ( [ eq : taupdistrib ] ) for @xmath36 , @xmath43 and @xmath38 . ( d ) the model distribution of @xmath49 calculated with same parameters as in ( b ) , the smooth line is the same as in ( c).,title="fig : " ] of the stock cvx trade sequence traded on the nyse . ( b ) power spectral density of the poissonian - like process driven by eq . ( [ eq : taucontinuous ] ) with the parameters @xmath43 , @xmath44 , @xmath45 , and @xmath46 . straight lines approximate power - law spectrum with exponents @xmath47 and @xmath48 . ( c ) the empirical distribution of @xmath49 calculated from the histogram of cvx trades on nyse . the smooth line represents the integral formula ( [ eq : taupdistrib ] ) for @xmath36 , @xmath43 and @xmath38 . ( d ) the model distribution of @xmath49 calculated with same parameters as in ( b ) , the smooth line is the same as in ( c).,title="fig : " ] of the stock cvx trade sequence traded on the nyse . ( b ) power spectral density of the poissonian - like process driven by eq . ( [ eq : taucontinuous ] ) with the parameters @xmath43 , @xmath44 , @xmath45 , and @xmath46 . straight lines approximate power - law spectrum with exponents @xmath47 and @xmath48 . ( c ) the empirical distribution of @xmath49 calculated from the histogram of cvx trades on nyse . the smooth line represents the integral formula ( [ eq : taupdistrib ] ) for @xmath36 , @xmath43 and @xmath38 . ( d ) the model distribution of @xmath49 calculated with same parameters as in ( b ) , the smooth line is the same as in ( c).,title="fig : " ] of the stock cvx trade sequence traded on the nyse . ( b ) power spectral density of the poissonian - like process driven by eq . ( [ eq : taucontinuous ] ) with the parameters @xmath43 , @xmath44 , @xmath45 , and @xmath46 . straight lines approximate power - law spectrum with exponents @xmath47 and @xmath48 . ( c ) the empirical distribution of @xmath49 calculated from the histogram of cvx trades on nyse . the smooth line represents the integral formula ( [ eq : taupdistrib ] ) for @xmath36 , @xmath43 and @xmath38 . ( d ) the model distribution of @xmath49 calculated with same parameters as in ( b ) , the smooth line is the same as in ( c).,title="fig : " ] in the case of the financial market we consider every trade of the selected stock as a point event , i.e. the sequence of all trades for the stock composes the stochastic point process , described by the set of time intervals between the successive trades . the power spectral density of the trade sequence serves as a measure of the long range memory property of trading activity . an example of the spectrum for the stock cvx trade sequence traded in the period of two years on nyse , fig.2 ( a ) , reveals the structure of the power spectral density in a wide range of frequencies and shows that the real markets exhibit two power laws with the exponents @xmath47 and @xmath48 . in our recent works @xcite we have proposed the model adjustment , introducing a new form of the modulating stochastic differential equation instead of eq . ( [ eq : taustoch2 ] ) , @xmath50\frac{1}{(\epsilon+\tau)^2}\mathrm{d}t + \sigma\frac{\sqrt{\tau}}{\epsilon+\tau}\mathrm{d}w , \label{eq : taucontinuous}\ ] ] where a new parameter @xmath51 defines the crossover between two areas of @xmath6 diffusion with assumption @xmath52 . the solution of eq . ( [ eq : taucontinuous ] ) has to be scaled by @xmath17 for other values of @xmath17 . the poisonian - like point process modulated by eq . ( [ eq : taucontinuous ] ) reproduces psd , fig.2 ( a ) , of the empirical trade sequence in detail , including two exponents and the crossover point , fig.2 ( b ) . the proposed model with the same parameters reproduces the empirical pdf of @xmath49 , fig.2 ( c ) and ( d ) , very well . moreover , the model describes the distribution of the empirical trading activity , i.e. , the number of transactions per selected time window with the power - law exponent @xmath42 .

we consider stochastic point processes generating time series exhibiting power laws of spectrum and distribution density ( _ phys . rev . e _ * 71 * , 051105 ( 2005 ) ) and apply them for modeling the trading activity in the financial markets and for the frequencies of word occurrences in the language . address = institute of theoretical physics and astronomy of vilnius university , a. gostauto 12 , lt-01108 vilnius , lithuania address = institute of theoretical physics and astronomy of vilnius university , a. gostauto 12 , lt-01108 vilnius , lithuania address = institute of theoretical physics and astronomy of vilnius university , a. gostauto 12 , lt-01108 vilnius , lithuania

the interest in systems undergoing reaction - diffusion processes is experiencing a rapid growth , due to their intrinsic relevance in an extraordinary broad range of fields [ 1 ] . in particular , a great deal of experimental and theoretical work has been devoted to the study of reaction - diffusion processes embedded in _ restricted geometries_. this expression refers to two , possibly concurrent , situations : @xmath1 low dimensionality and _ ii . _ small spatial extent . in the first case , the spectral dimension @xmath2 characterizing the diffusive behaviour of the reactants on the substrate is low @xmath3 , and the substrate underlying the diffusion - reaction lacks spatial homogeneity . this situation is able to model media whose properties are not translationally invariant and where the reactants perform a `` compact exploration '' [ 2 ] . these kinds of structures can lead to a chemical behaviour significantly different from those occurring on substrates displaying a homogeneous spatial arrangement . indeed , while in high dimensions a mean - field approach ( based on classical rate equations ) provides a good description , in low dimension local fluctuations are responsible for significant deviation from mean - field predictions [ 3 ] . there also exists a variety of experimental situations in which reaction - diffusion processes occur on spatial scales too small to allow an infinite volume treatment : in this case finite - size corrections to the asymptotic ( infinite - volume ) behaviour become predominant . here , differently from previous works , we explicitly examine finite size systems , i.e. no thermodynamic limit is taken [ 4 - 6 ] . all the quantities we calculate are hence finite , and we seek their dependence on the finite parameters of the system ( volume of the substrate and concentration of the reactants ) . in particular , we study the dynamics of a system made up of two species particles undergoing irreversible quadratic autocatalytic reactions @xmath0 . all particles move randomly and react upon encounter with probability @xmath4 , i.e. the reaction is strictly local and deterministic . notice that , allowing all the particles to diffuse makes the problem under study a genuine multiparticle - diffusion problem . the latter is generally quite difficult to manage due to the fact that the effects of each single particle do not combine linearly , even in the non - interacting case . for this reason the analytic treatment often relies on simplifying assumptions which , nevertheless , preserve the main generic features of the problem . in the past , autocatalytic reactions have been extensively analyzed on euclidean structures [ 7 ] , within a continuous picture attained by the fisher equation [ 8,9 ] which describes the system in terms of front propagation . evidently , this picture is not suitable for low - density systems , where front propagation can not be defined . in order to describe also the high - dilution regime , here a different approach is introduced which , as we will see , works as well for inhomogeneous structures . this way , we are also able to highlight the role of topology in the temporal evolution of the system . in the following , we shall examine the concentration @xmath5 of @xmath6 particles present in the system at time @xmath7 and its fluctuations ; from @xmath8 it is then possible to derive an estimate for the reaction velocity . furthermore , we consider the average time @xmath9 ( also called `` final time '' ) at which the system achieves its inert state , i.e. @xmath10 . as we will show , @xmath11 depends on the number of particles @xmath12 and on the volume @xmath13 of the underlying structure . more precisely , for small concentrations of the reactants , we find , both numerically and analytically , that the @xmath11 factorizes into two terms depending on @xmath12 and @xmath13 , respectively . one of the most interesting applications of the final time is analytic [ 10,11 ] : as we show , @xmath9 sensitively depends on the initial amount of reactant @xmath12 and , on low dimensional substrates @xmath14 , by reducing the dimension @xmath15 , the sensitivity can be further improved . we consider a system made up of @xmath12 particles of two different chemical species @xmath6 and @xmath16 , diffusing and reacting on a discrete substrate with no excluded volume effects . at time @xmath7 , @xmath17 and @xmath18 represent the number of @xmath6 and @xmath16 particles , respectively , with @xmath19 . being @xmath13 the substrate volume , we define @xmath20 and @xmath21 as the concentrations of the two species at time @xmath7 . different species particles residing at time step @xmath7 , on the same node or on nearest - neighbour nodes react according to the mechanism @xmath0 with reaction probability set equal to one . notice that the previous scheme also includes possible additional products ( other than @xmath22 ) made up of some inert species of no consequences to the overall kinetics . the initial condition at time @xmath23 is @xmath24 ( the source ) , @xmath25 , with all particles distributed randomly throughout the substrate . as a consequence of the chemical reaction defined above , @xmath17 is a monotonic function of @xmath7 and , due to the finiteness of the system , it finally reaches value @xmath12 ; at that stage the system is chemically inert . the average time at which @xmath26 is called `` final time '' and denoted by @xmath11 . the final time @xmath11 is of great experimental importance since it represents the average time when the system is inert and therefore it provides an estimate of the time when reaction - induced effects ( such as side - reactions or photoemission ) vanish [ 12 ] . in this perspective , deviations from the theoretical prediction of @xmath11 are , as well , noteworthy : they could reveal the existence of competitive reactions or explain how the process is affected by external radiation . finally , notice that the autocatalytic reaction can also be used as a model for spreading phenomena : @xmath27 particles may stand for ( irreversibly ) sick ( healthy ) or informed ( unaware ) agents , respectively . for these systems a knowledge of the infection rate or information diffusion is of great importance [ 4,5 ] . as previously said , @xmath11 generally depends on the total number of agents @xmath12 and on the size of the lattice @xmath13 , while its functional form is affected by the topology of the lattice itself . the analytical treatment is carried out in the two limit regimes of high and low density . when @xmath28 , the substrate topology does not qualitatively affects results . we can assume that the set of @xmath6 particles covers a connected region of the substrate whose volume expands with a constant velocity ( depending on the density @xmath29 and dimension @xmath15 ) . in this case ( and exactly in the limit @xmath30 the process can be described as the deterministic propagation of a wave front decoupled from the random motion of the agents . if we suppose the source to be at the center of the lattice at time @xmath23 , at each instant the wave front is the locus of points whose chemical distance from the center is @xmath31 . the connected region spanned by the wave front is entirely occupied by @xmath6 particles , while @xmath16 particles fill the remaining of the lattice . in particular , for a @xmath15-dimensional regular substrate , the region where @xmath6 particles concentrate is a @xmath15-dimensional polyhedron [ 4,5 ] . in general , for a finite system , the average final time is @xmath32 , where @xmath33 is the chemical distance of the most distant point on the lattice , starting from the source . on euclidean geometries this yields @xmath34 for d=1 and @xmath35 for @xmath36 . on the other hand , on inhomogeneous structures , the dependence on @xmath37 is not so simple , since it involves taking the average with respect to all possible starting points for the source . in the case of low density @xmath38 the time an @xmath6 particle walks before meeting a @xmath16 particle becomes very large , so that the process is diffusion - limited . we adopt a mean - field - like approximation by assuming that the time elapsing between a reaction and the successive one is long enough that the spatial distribution of reactants can be considered random . in other words , the particles between each event have the time to redistribute randomly on the lattice and we neglect correlations between their spatial positions . another consequence of the low concentration of reactants , is that we can just focus on two - body interactions since the event of three or more particles interacting together is unlikely . notice that the high - dilution assumption , by itself , generally does not allow to apply the classical rate equations : when diffusion is involved also the substrate topology has to be taken into account . for this reason , in the following we will treat high and low dimensional structures separately . _ high - dimensional structures _ @xmath39 let us consider a given configuration of the system where @xmath40 and @xmath41 particles are present . the probability for a given b particle to encounter and react with any a particle is just the trapping probability @xmath42 for a particle , out of @xmath41 , in the presence of @xmath40 traps , both species diffusing . under the assumptions specified above , for high - dimensional substrates [ 1 ] : @xmath43 where @xmath44 is a constant depending on the given substrate . form the previous equation we can calculate the average trapping time for a b particle as @xmath45 . let us now introduce an early - time @xmath46 approximation for the trapping probability : @xmath47 , where @xmath48 is the probability that , after each reaction , two given particles first encounter at a given time ( in general , this probability depends not only on the volume of the underlying structure , but also on the history of the system ) . this simple form for @xmath49 allows us to go on straightforwardly . in fact , the process can be meant as an absorbing markov chain , with @xmath12 states ( labeled with the total number of @xmath6 particles : @xmath50 ) , and one absorbing state @xmath51 ; the chain starts from state 1 . the transition matrix @xmath52 can be written : the transition probability from a state @xmath53 to a state @xmath54 as a function of n and p is : @xmath55^{m - k}\left [ ( 1 - p)^k \right]^{n - m}\ ] ] for any @xmath12 and @xmath56 . from @xmath52 we can take the submatrix @xmath57 , obtained subtracting the last row and column ( those pertaining to the absorbing state ) , and compute the fundamental matrix @xmath58 . now , by expanding to first order in @xmath56 , a direct calculation shows that @xmath59 is an upper triangular matrix given by @xmath60 the mean time @xmath11 required to reach the absorbing state n , starting from state 1 is given by the sum of the first row of @xmath59 : @xmath61 where @xmath62 is the euler - mascheroni constant . the last result is in perfect agreement with numerical simulations and also emphasizes @xmath11 factorization . _ low - dimensional structures _ @xmath63 . for low dimensional structures the dependence on @xmath12 found above is not correct . the reason is that a non - linear cooperative behaviour among particles emerges . let us define @xmath64 as the average time elapsing between the @xmath65-th first encounter among different particles and the @xmath66-th one . this time just corresponds to the average time during which there are just @xmath67 particles in the system . in our approximation @xmath64 is proportional to the trapping time @xmath68 in the presence of n mobile traps diffusing throughout a volume @xmath13 [ 6 ] . for compact exploration of the space @xmath69 , @xmath70 . this result was derived for infinite lattices , nonetheless , it provides a good approximation also for finite lattices , provided that the time to encounter is not too large . from @xmath71 we obtain @xmath64 as the average trapping time of the first out of @xmath72 particles , that , for rare events , is just @xmath73 with logarithmic corrections in the case @xmath74 . the time @xmath11 can therefore be written as a sum over @xmath66 @xmath75 of @xmath64 . now , by adopting a continuous approximation , we obtain for @xmath76 [ 6 ] : @xmath77,\ ] ] where @xmath78 is the harmonic number . in particular , the leading - order contribution for a one - dimensional system @xmath79 is @xmath80 for a two - dimensional lattice @xmath81 @xmath82 notice that the factorization in eq.([eq3 ] ) is consistent with eq.([eq1 ] ) : in both cases , the factor containing the dependence on @xmath13 represents the average time for two particles to meet . [ fig1 ] with the linear size of the system for a one - dimensional chain ( blue circles ) , a sierpinski gasket ( black triangles ) , a t - fractal ( red squares ) , and a three - dimensional cubic lattice ( green diamonds ) on a double - logarithmic scale . the number of reactants is fixed at @xmath83 for all systems . the spectral dimension for the sierpinski and the t - graph is @xmath841.365 and @xmath85 , respectively . dotted lines highlight the low - concentration regime @xmath86 , corresponding to a power law for all systems . for the one - dimensional chain , the linear high - concentration regime is also pointed up.,width=288,height=268 ] as can be evinced from fig . 2 , for small densities all the data collapse ; moreover , in that region , the fit coefficients introduced are in good agreement with theoretical predictions . [ fig2 ] for the sierpinski gasket ( left ) and the two - dimensional lattice ( right ) . different symbols and colours distinguish different sizes , as explained by the legend . the line provides the best fit in very good agreement with eqs . ( [ eq3 ] ) and ( [ eq4 ] ) , apart from sub - leading corrections in the ( marginal ) case @xmath87.,title="fig:",width=230,height=192 ] for the sierpinski gasket ( left ) and the two - dimensional lattice ( right ) . different symbols and colours distinguish different sizes , as explained by the legend . the line provides the best fit in very good agreement with eqs . ( [ eq3 ] ) and ( [ eq4 ] ) , apart from sub - leading corrections in the ( marginal ) case @xmath87.,title="fig:",width=230,height=196 ] for low densities , the standard deviation @xmath88 displays a dependence on @xmath12 and @xmath37 analogous to @xmath11 ; for high densities , @xmath89 becomes vanishingly small , in fact the process becomes deterministic . as anticipated in section 1 , experimental measures of @xmath9 are useful in monitoring trace reactants [ 6 ] . in the high - dilution regime , our results show that @xmath90 and therefore , once the substrate size is fixed , the initial amount of reactant can be expressed as @xmath91 . a proper estimate of the sensitivity of this method is provided by the derivative @xmath92 : the smaller the derivative and the larger the sensitivity . as can be evinced from fig.3 , which displays numerical results for @xmath12 and @xmath93 , the smaller the concentration and the better the sensitivity of this technique . this makes such technique very suitable for the determination of ultratrace amounts of reactants , which is of great experimental importance [ 13 ] . interestingly , @xmath93 also depends on the substrate topology : when @xmath94 2 and at fixed @xmath13 , the sensitivity can be further improved by lowering the substrates dimension . conversely , when @xmath95 ceases to depend on @xmath2 . [ fig3 ] ( left panel ) and its derivative @xmath93 ( right panel ) vs final time @xmath11 . as shown in the legend , different substrate topologies ( with approximately the same volume ) are compared . lines are guide to the eyes.,title="fig:",width=230,height=192 ] ( left panel ) and its derivative @xmath93 ( right panel ) vs final time @xmath11 . as shown in the legend , different substrate topologies ( with approximately the same volume ) are compared . lines are guide to the eyes.,title="fig:",width=230,height=192 ] in this section we deal with quantities depending explicitly on time @xmath7 . first of all , we consider the concentration @xmath96 of @xmath6 particles present at time t. due to the irreversibility of the reaction taken into account , @xmath96 is a monotonic increasing function ; more precisely it is described by a sigmoidal law , typical of autocatalytic phenomena [ 7 ] . as shown in fig.4 the curves @xmath17 grow faster , and saturate earlier , with increasing @xmath2 ( @xmath12 and @xmath13 being fixed ) . this is consistent with the meaning of the spectral dimension @xmath2 : it describes the long - range connectivity structure of the substrate and the long - time diffusive behaviour of a random walker on the substrate . more precisely , for @xmath97 , the number of different sites visited by each walker grows faster as @xmath2 increases , and analogously the number of meetings between walkers . for @xmath98 ( e.g. , @xmath99 in the figure ) , @xmath17 is independent of @xmath2 and is fitted by a pure sigmoidal function . also notice that deviations between curves relevant to different topologies are especially important at early - times , while at long times they all agree with the pure sigmoidal curve . this result is consistent with the existence of two temporal regimes concerning diffusion on low - dimensional structures [ 1 ] . as a result , the topology of the underlying structure is important only at early times , while , at long times , the system evolves as expected for high - dimensional structures . [ fig1 ] particles @xmath100 vs time @xmath7 for a system made up of @xmath101 particles embedded on different structures , as explained in the legend . the best fit for the cubic lattice a pure sigmoidal function ( see eq . ( 5 ) ) , shown by the green line . the latter also provides the best fit for the long time behaviour of @xmath100 on low dimensional substrates.,title="fig:",width=288,height=268 ] within the analytic framework developed in the last section , it is possible to derive some insights into the temporal behaviour displayed by @xmath17 . being @xmath102 the average time at which the number of @xmath6 particles reaches value @xmath66 , recalling eq . ( [ eq2 ] ) we can write from which @xmath103 , whose numerical solution provides an s - shaped curve consistent with data obtained from simulations . as for transient lattices , the easy form obtained for @xmath104 and the assumption of a uniform distribution for agents positions , allow to write a master equation for the number of @xmath6 particles in the system : @xmath105.\ ] ] to first order in @xmath56 : @xmath106 , being @xmath107 a logistic - like map , with a repelling fixed point in @xmath108 @xmath109 , and an attracting fixed point in @xmath12 @xmath110 . since @xmath111 , the increment of @xmath17 at each time step is very small ( of order @xmath56 ) , and we can take the evolution to be continuous . thus we obtain @xmath112 which is in good agreement with numerical results ( fig.4 ) . [ fig1 ] , fluctuations @xmath113 and concentration @xmath96 versus time for a system of @xmath101 particles diffusing on a sierpinski gasket ; three different generations ( depicted in different colours ) are shown . notice @xmath114.,title="fig:",width=288,height=268 ] from @xmath115 one can derive the rate of reaction @xmath116 which represents the reaction velocity . as you can see from fig.5 , in agreement with the theoretical predictions , @xmath117 is an asymmetrical curve exhibiting a maximum at a time denoted as @xmath118 obviously corresponding to a flex in @xmath17 . interestingly , @xmath119 scales with the volume of the structure according to @xmath120 which is the same dependence shown by @xmath11 . moreover , at @xmath119 the population of the two species are about the same @xmath121 . hence , the efficiency of the autocatalytic reaction is not constant in time but , provided the number @xmath12 of particles is conserved , it is maximum when the number of b particles is about @xmath122 . from eq . we can derive a similar result for the variance @xmath123 of the number of a particles present on the substrate . interestingly , fluctuations @xmath123 peak at a time @xmath124 which , again , depends on the system size with the same law as @xmath11 ; notice that @xmath125 . we introduced an analytic approach to deal with autocatalytic diffusion - reaction processes , also able to take into account the role played by particles discreteness and substrate topology . within such framework , we derived in the low - density regime , for both fractal and euclidean substrates , the exact dependence on system parameters displayed by the average final time , also highlighting how topology affects it . in particular , the case @xmath87 is marginal . exact results are also found for euclidean lattices in the limit of high density . theoretical results concerning the average final time find important applications in analytical fields , where measures of @xmath11 are exploited for detecting trace reactants . our results suggest that the sensitivity of such technique is affected not only by the reactant concentration , but also by the topology of the structure underlying diffusion . [ 1 ] s. havlin , d. ben avraham , diffusion and reactions in fractals and disordered systems , cambridge university press , cambridge , 2000 [ 13 ] a. rose , z. zhu , c.f . madigan , t.m . swager , v. bulovic nature , * 434 * 876 ( 2005 ) ; n.d . priest , j. environ . * 6 * 375 ( 2002 ) ; j.r . mckeachie , w.r . van der veer , l.c . short , r.m . garnica , m.f . appel , t. benter analyst , * 126 * 1221 ( 2001 )

we study the dynamics of a system made up of particles of two different species undergoing irreversible quadratic autocatalytic reactions : @xmath0 . we especially focus on the reaction velocity and on the average time at which the system achieves its inert state . by means of both analytical and numerical methods , we are also able to highlight the role of topology in the temporal evolution of the system .

polarimetry is a powerful diagnostic of specific phenomena at work in cosmic sources in the radio - wave and optical energy bands , but very few results are available at high photon energies : the only significant observation in the x - gamma energy range , to date , is the measurement of a linear polarisation fraction of @xmath0 of the 2.6kev emission of the crab nebula by a bragg polarimeter on board oso-8 @xcite . at higher energies , hard - x - ray and soft - gamma - ray telescopes that have flown to space in the past ( comptel @xcite , batse@xcite ) were not optimized for polarimetry , and their sensitivity to polarisation was poor . presently active missions ( integral ibis@xcite and spi @xcite ) have provided some improvement , with , in particular , mildly significant measurements of @xmath1 ( 130 to 440 kev @xcite ) and @xmath2 ( 200 to 800 kev @xcite ) for the crab nebula . a number of compton polarimeter / telescope projects have been developed , some of which also propose to record photon conversions to @xmath3 pairs . a variety of technologies have been considered , such as scintillator arrays ( pogo @xcite , grape @xcite , polar @xcite ) , si or ge microstrip detectors ( mega @xcite , astrogam @xcite ) or combinations of these ( si @xmath4 labr@xmath5 for grips @xcite , si @xmath4 csi(tl ) for tigre @xcite ) , semiconductor pixel detectors ( cipher @xcite ) and liquid xenon ( lxegrit @xcite ) time projection chambers ( tpc ) . in most compton telescopes the reconstruction of the direction of the incident photon provides an uncertainty area which has the shape of a thin cone arc . the tracking of the recoil electron from the first compton interaction with a measurement of the direction of the recoil momentum , as is within reach with a gas tpc , allows to decrease the length of the arc and therefore to improve dramatically the sensitivity of the detector ( @xcite and references therein ) . some of these telescopes are sensitive to photon energies up to tens of mev in the compton mode , but their sensitivity to polarisation above a few mev is either nonexistent or undocumented . as is well known , the sensitivity to polarisation of compton scattering is excellent at low energies ( thomson scattering ) , as the polarisation asymmetry @xmath6 , also known as the modulation factor and defined by the phase - space dependence of the differential cross section , @xmath7 } \right ] , \label{eq : def : diff : cross : section : compton}\ ] ] reaches @xmath8 at a polar angle @xmath9 of @xmath10 ( fig . 1 of ref . @xcite ) . in this expression , @xmath11 is the azimuthal angle , that is the angle between the scattering plane and the direction of polarisation of the incident photon . unfortunately , @xmath6 is decreasing with energy , and as the precision of the measurement scales as @xmath12 when the background noise is negligible and where @xmath13 is the number of signal event , the sensitivity of compton polarimetry decreases at high energies . with the goal of a quantitative assessment of this sensitivity , in this paper we compute the average polarisation asymmetry @xmath14 from the klein - nishina differential cross section on free electrons at rest @xcite . @xmath14 is defined from the differential cross section in @xmath11 , that is after the full differential cross section ( eq . ( [ eq : def : diff : cross : section : compton ] ) ) has been integrated over the other variables that describe the final state : @xmath15 } \right ] . \label{eq : def : diff : cross : section : compton : int}\ ] ] following heitler @xcite , the doubly differential cross section for linear polarised radiation reads : @xmath16 , \label{eq : diff : cross : section : compton}\ ] ] where @xmath17 , @xmath18 and @xmath19 are the energy of the incident and scattered @xmath20s , respectively ; @xmath9 is the scattering angle , that is the polar angle of the direction of the scattered @xmath20 with respect to the direction of the incident @xmath20 . the differential element @xmath21 is @xmath22 as usual . in the case of partially polarised emission with polarisation fraction @xmath23 , the differential cross section becomes : @xmath24 \sin\theta { \mbox{d}}\theta { \mbox{d}}\phi.\ ] ] the minus sign reflects the fact that photons compton scatter preferentially into the direction perpendicular to the orientation of the electric field of the incoming radiation . the energy of the scattered @xmath20 is related to @xmath9 from energy - momentum conservation : @xmath25 $ ] , @xmath26 $ ] , @xmath27 , @xmath28 and @xmath29 - \left[1/(x k_0 ) - 1/k_0 \right]^2 $ ] . we then obtain @xcite : @xmath30 \left(\cos{(2\phi ) } p + 1 \right ) \right ] { \mbox{d}}x { \mbox{d}}\phi.\ ] ] @xmath19 varies in a range such that @xmath31 , that is @xmath32 . the distributions of these kinematic variables are shown in fig . [ fig : compton : spectra ] . after an elementary integration over @xmath33 , we obtain : @xmath34 { \mbox{d}}\phi , \nonumber\end{aligned}\ ] ] that is a total cross section of @xcite : @xmath35 . \nonumber\end{aligned}\ ] ] equating the constant term and the term proportional to @xmath36 in eqs . ( [ eq : def : diff : cross : section : compton : int ] ) and ( [ eq : phi : diff : cross : section ] ) , we obtain for the average polarisation asymmetry : @xmath37 [ ht ] spectra of the azimuthal and polar angles @xmath11 and @xmath9 , of @xmath38 , of the fraction @xmath33 of the incident photon energy carried away by the scattered photon , and of the 1d and 2d weights @xmath39 and @xmath40 , for incident photon energies @xmath41 , @xmath42 , and @xmath43 , all for a fully polarized beam . , title="fig : " ] [ ht ] absolute value of the average asymmetry in compton scattering on free electrons at rest , as a function of the incident photon energy in electron rest - mass units . thick solid line : full expression ( eq . ( [ eq : compton : asymmetry : full ] ) ) . dashed line : high - energy approximation ( eq . ( [ eq : compton : asymmetry : high ] ) ) . thin solid line : high - energy asymptote . ] we now examine two limiting cases : * at low energies , @xmath44 , eq . ( [ eq : phi : diff : cross : section ] ) reduces to : @xmath45 \frac{k_0}{3 } { \mbox{d}}\phi,\ ] ] which results in a total cross section of @xmath46 , i.e. , the thomson cross section . the low - energy average asymmetry is @xmath47 . * at high energies , @xmath48 { \mbox{d}}\phi,\ ] ] which results in a total cross section of @xmath49 . the high - energy average asymmetry is @xmath50 the average asymmetry decreases at high energies , asymptotically approaching @xmath51 . the variation of the average polarisation asymmetry of photon compton scattering on free electrons at rest ( eq . ( [ eq : compton : asymmetry : full ] ) ) is compared to its high - energy approximation ( eq . ( [ eq : compton : asymmetry : high ] ) ) in fig . [ fig : compton : asymmetry ] . the absence of sensitivity of compton polarimeters at high energies @xcite is due to this strong decrease of @xmath52 . the value of the polarisation fraction @xmath23 is classically obtained by a fit to the @xmath11 distribution . a way to improve the polarisation sensitivity is to make an optimal use of the information contained in the multi - dimensional probability density function ( pdf ) through the use of an optimal variable ( @xcite and references therein ) , that is , of a weight @xmath53 such that the @xmath23 dependence of the expectation value @xmath54 of @xmath39 allows a measurement of @xmath23 , and that the variance of such a measurement is minimal . the solution , up to a multiplicative factor , is ( eg . @xcite ) : @xmath55 in the present case of a polarisation measurement : @xmath56 with @xmath57 and @xmath58 , we obtain : @xmath59 that is , if @xmath60 is small compared to @xmath61 , @xmath62 the @xmath63 moment of @xmath40 is @xmath64 { \mbox{d}}\omega = p \int { \displaystyle\frac{g^2(\omega)}{f(\omega ) } } { \mbox{d}}\omega $ ] , which is proportional to @xmath23 . the expressions for @xmath65 and @xmath66 are obtained from the measured values of @xmath11 and @xmath9 ( and therefore of @xmath33 ) by equating the constant term and the term proportional to @xmath36 in eqs . ( [ eq : p(omega)f+pg ] ) and ( [ eq : def : diff : cross : section : compton ] ) . the spectrum of @xmath40 is shown in fig . [ fig : compton : spectra ] for a fully polarised beam . we can see that @xmath67 is most often much smaller than unity ( beware the vertical log scale ) , so that our neglecting @xmath60 in the expression of @xmath68 was legitimate . the asymmetry , the non - evenness of the @xmath40 distribution makes the non - zero average due to the beam polarisation explicit . moment s methods are equivalent to a likelihood analysis in the case where the pdf is a linear function of the variables that one aims to measure , as is the case here , but they are much simpler to instantiate as one just has to compute @xmath69 , and average it over the whole statistics . although the analysis of experimental data is beyond the scope of this paper , the following considerations apply : * background subtraction reduces to a simple subtraction in computing the average of @xmath40 . their @xmath70-dimensional parametrization is not needed . * likelihood methods need the use of a @xmath70-dimensional parametrization of the acceptance , or efficiency , for correction . this is pretty simple in the case of compton scattering for which the final state is described by only two variables , but for higher - dimensional systems , producing enough monte carlo ( mc ) statistics and determining a parametrization becomes a nightmare : in that case the use of a moments - based efficiency correction becomes mandatory ( for a real - case presentation see eg . , section iv.a , eqs . ( 18)-(24 ) and vi.b eqs . ( 47)-(49 ) of @xcite ) . in the `` reduced '' 1d case of eq . ( [ eq : def : diff : cross : section : compton : int ] ) , @xmath71 becomes @xmath72 and the estimator for @xmath73 is @xmath74 @xcite . the uncertainty then reads : @xmath75 that is , in the case of thomson scattering ( @xmath76 ) , @xmath77 . needless to say , in the case where the direction of the polarisation of the emission of a particular cosmic source `` on the sky '' is unknown , a combined use of @xmath78 and of @xmath79 should be used . [ ht ] ratio @xmath80 of the figures of merit of the 2d to 1d estimators of the linear polarisation fraction , as a function of the incident photon energy in electron rest - mass units . ] the performance of the 2d estimator @xmath81 is compared to that of the 1d @xmath74 by the comparison of the ratios of the rms width normalized to the mean value : @xmath82 in contrast with polarimetry performed with @xmath83 telescopes , for which an improvement in the precision of the measurement of the linear polarisation fraction by a factor of larger than two is at hand ( fig . 21 right of @xcite ) , in the case of compton polarimeters the improvement is found to be marginal , varying from @xmath84 at low energy to @xmath85 at high energy ( fig . [ fig : figure : of : merit : ratio ] ) . these results are in qualitative agreement with those obtained at 100 kev by a likelihood analysis of the doubly differential cross section @xcite . in summary , we have obtained the expression for the average polarisation asymmetry , or modulation factor , of compton scattering on free electrons at rest , eq . ( [ eq : compton : asymmetry : full ] ) , fig . [ fig : compton : asymmetry ] . we have then obtained a simple optimal estimator of the polarisation fraction @xmath23 that makes use of all the information ( azimuthal and polar angles of the scatter ) , avoiding the technicalities of a maximum likelihood analysis but with the same performance . it a pleasure to acknowledge the support by the french national research agency ( anr-13-bs05 - 0002 ) and the scrutiny and the suggestions of referee # 1 of nuclear instruments and methods in physics research a. 99 `` a precision measurement of the x - ray polarization of the crab nebula without pulsar contamination '' m. c. weisskopf _ _ , astrophysical journal * 220 * ( 1978 ) l117 . `` characteristics of comptel as a polarimeter and its data analysis '' f. lei _ et al . _ , astronomy and astrophysics supplement , * 120 * ( 1996 ) 695 . `` evidence of polarisation in the prompt gamma - ray emission from grb 930131 and grb 960924 , '' d. r. willis _ et al . _ , astron . astrophys . * 439 * ( 2005 ) 245 , [ astro - ph/0505097 ] . `` polarization of the crab pulsar and nebula as observed by the integral / ibis telescope , '' m. forot _ et al . _ , astrophys . j. * 688 * ( 2008 ) l29 , [ arxiv:0809.1292 [ astro - ph ] ] . `` status of the integral / ibis telescope modeling and of the response matrices generation '' p. laurent _ et al . _ , a&a * 411 * ( 2003 ) l185 . `` polarimetry in the hard x - ray domain with integral spi , '' m. chauvin _ et al . _ , astrophys . j. * 769 * ( 2013 ) 137 , [ arxiv:1305.0802 [ astro-ph.im ] ] . `` beam test of a prototype detector array for the pogo astronomical hard x - ray / soft gamma - ray polarimeter , '' t. mizuno _ et al . _ , nucl . instrum . a * 540 * , 158 ( 2005 ) [ astro - ph/0411341 ] . `` calibration of the gamma - ray polarimeter experiment ( grape ) at a polarized hard x - ray beam '' , p.f . et al . _ , instrum . a * 600 * ( 2009 ) 424 . `` response of the compton polarimeter polar to polarized hard x - rays '' , s. orsi _ et al . _ , instrum . meth . a * 648 * ( 2011 ) 139 . `` polarization measurements with the mega telescope '' , a. zoglauer _ et al . _ , proceedings of the 5th integral workshop on the integral universe ( esa sp-552 ) . 16 - 20 february 2004 , munich , germany . `` astrogam '' , proposal submitted for the esa m4 mission programme january 15 2015 , http://astrogam.iaps.inaf.it/ grips - gamma - ray imaging , polarimetry and spectroscopy , j. greiner _ et al . _ , exper.astron . * 34 * ( 2012 ) 551 . `` tracking , imaging and polarimeter properties of the tigre instrument '' , t. j. oneill _ et al . _ , astronomy and astrophysics supplement , * 120 * ( 1996 ) 661 . [ cipher ] , `` a cdte position sensitive spectrometer for hard x- and soft -ray polarimetry '' , e. caroli _ _ , nucl . instrum . meth . a * 477 * ( 2002 ) 567 . `` compton imaging of mev gamma - rays with the liquid xenon gamma - ray imaging telescope ( lxegrit ) , '' e. aprile _ et al . _ , instrum . meth . a * 593 * ( 2008 ) 414 , [ arxiv:0805.0290 [ physics.ins-det ] ] . `` an electron - tracking compton telescope for a survey of the deep universe by mev gamma - rays , '' t. tanimori _ et al . _ , accepted for publication in the astrophysical journal , arxiv:1507.03850 [ astro-ph.im ] . `` ber die streuung von strahlung durch freie elektronen nach der neuen relativistischen quantendynamik von dirac '' , o. klein , y. nishina , z. phys . * 52 * ( 1929 ) 853 . `` die polarisation der comptonstreuung nach der diracschen theorie des elektrons '' , y. nishina , z. phys . * 52 * ( 1929 ) 869 . `` the quantum theory of radiation '' , w. heitler , 1954 ( oxford university press , 3rd edition ) . `` quantum electrodynamics '' a. i. akhiezer and v. b. berestetskii , interscience monographs and texts in physics and astronomy , second edition , new york : interscience publishers , 1965 . `` status and prospects for polarimetry in high energy astrophysics '' , m.l . mcconnell & j.m . ryan , new astronomy reviews 48 ( 2004 ) 215 . `` polarimetry of cosmic gamma - ray sources above @xmath83 pair creation threshold , '' d. bernard , nucl . instrum . a * 729 * ( 2013 ) 765 , [ arxiv:1307.3892 [ astro-ph.im ] ] . `` ambiguity - free measurement of @xmath86 : time - integrated and time - dependent angular analyses of @xmath87 , '' b. aubert _ et al . _ [ babar collaboration ] , phys . d * 71 * ( 2005 ) 032005 [ hep - ex/0411016 ] .

we compute the average polarisation asymmetry from the klein - nishina differential cross section on free electrons at rest . as expected from the expression for the asymmetry , the average asymmetry is found to decrease like the inverse of the incident photon energy asymptotically at high energy . we then compute a simple estimator of the polarisation fraction that makes optimal use of all the kinematic information present in an event final state , by the use of `` moments '' method , and we compare its statistical power to that of a simple fit of the azimuthal distribution . in contrast to polarimetry with pair creation , for which we obtained an improvement by a factor of larger than two in a previous work , here for compton scattering the improvement is only of 1020% . hard x - ray , gamma - ray , compton scattering , polarimeter , polarisation asymmetry , optimal variable

beta decays and electron capture reactions play an important role in nuclear physics @xcite and in many astrophysical phenomena like supernovae explosions and nucleosynthesis @xcite . in energetic contexts like supernova explosions neutrino capture reactions are also relevant @xcite . @xmath4 decay has a direct access to the absolute gt transition strengths b(gt ) , allowing the study of half - lives , q@xmath11 -values and branching ratios in the q - window . charge exchange reactions like @xmath12 and ( @xmath6he,@xmath13 are useful tools to study the relative values of b(gt ) strengths up to high excitation energies . recent experimental improvements have made possible to make one - to - one comparisons of gt transitions studied in charge exchange reactions and @xmath4 decays @xcite . employing the isospin symmetry experimental information can be obtained for unstable nuclei . a long series of high quality experiments have provided new experimental information about the gamow - teller strength distribution in medium mass nuclei employing these techniques @xcite . large - scale shell - model calculations , employing a slightly monopole - corrected version of the well - known kb3 interaction , denoted as kb3 g , were able to reproduce the measured gamow - teller strength distributions and spectra of the @xmath10 shell nuclei in the mass range a = 45 - 65 @xcite . the description of electron capture reaction rates , and the strengths and energies of the gamow - teller transitions in @xmath14ni required a new shell - model interaction , gxpf1j @xcite . shell - model calculations in the @xmath10 model space with the kb3 g and gxpf1a interactions qualitatively reproduced experimental gamow - teller strength distributions of 13 stable isotopes with 45@xmath15a@xmath1564 . they were used to estimate electron - capture rates for astrophysical purposes with relatively good accuracy @xcite . shell model diagonalizations have become the appropriate tool to calculate the allowed contributions to neutrino - nucleus cross sections for supernova neutrinos @xcite . recently , f. molina _ et . @xcite , populated the @xmath0ti , @xmath1cr , @xmath2fe and @xmath3ni nuclei by the fragmentation of a @xmath16ni beam at 680 mev / nucleon on a 400 mg/@xmath17 be target and studied the @xmath4-decay . with the help of experimentally observed @xmath4-decay half lives , excitation energies , and @xmath4 branching ratios , they reported the fermi and gamow - teller transition strengths and compared them with the more precise b(gt ) value reported in @xcite with the help of charge - exchange reaction at high excitation energies , finding very good agreement between the both experimental data . the aim of the present study is to present state of the art shell model calculations for the observed transitions in @xmath0ti , @xmath1cr , @xmath2fe and @xmath3ni nuclei , restricted to the @xmath10 model space , employing the kb3 g @xcite and gxpf1a @xcite interactions . the shell model calculations are performed using the code nushellx@msu @xcite . they provide a theoretical description of the experimental results presented in @xcite and @xcite . in order to describe the measured gt strength distribution for @xmath0ti , @xmath1cr , @xmath2fe , and @xmath3ni nuclei we employ the shell - model restricted to the @xmath10 valence space and the effective interactions kb3 g and gxpf1a . the interaction kb3 g @xcite is a monopole - corrected version of the previous kb3 interaction @xcite , whose parameters were fitted using experimental energies of the lower @xmath18 shell nuclei . the gxpf1a is based on the gxpf1 interaction @xcite . initially the two body matrix elements ( tbme ) of the gxpf1 interaction were obtained from the bonn - c bare nucleon - nucleon potential and g - matrix calculations , with a scaling @xmath19 . later on the 195 tbme and 4 spe were determined by fitting 699 experimental energies of 87 nuclei from @xmath20=20 to @xmath20=32 . the modification of five tbme lead to the gxpf1a interaction . the full shell model hilbert space in the @xmath10 shell is employed in the description of @xmath0ti , @xmath1cr and @xmath2fe nuclei . due to the huge matrix dimensions , in the case of @xmath3ni we allowed for a maximum of four nucleon excitation from the @xmath21 shell to the rest of the @xmath10 orbitals . the gamow - teller strength b(gt ) is calculated using following expression , @xmath22 where @xmath23 , @xmath24 , the index @xmath25 runs over the single particle orbitals , @xmath26 and @xmath27 describe the state of the parent and daughter nuclei , respectively . in the present work the b(gt ) values are scaled employing a quenching factor @xmath28 @xcite . in this section the theoretical results are compared with the experimental data reported in @xcite and @xcite . -1.5 cm fig . [ 42ti ] displays a comparison between the shell - model calculations and the experimental gt strength distribution for the transition @xmath0ti @xmath29 @xmath0sc . fig . 1(a ) presents the experimental data observed through the @xmath4-decay @xmath0ti@xmath30sc up to the excitation energy @xmath31sc ) = 1.888 mev @xcite . 1(b ) shows the experimental data obtained through the charge - exchange reaction @xmath0ca(@xmath6he,@xmath7)@xmath0sc up to the excitation energy @xmath31sc ) = 3.688 mev @xcite . 1(c ) depicts the shell - model calculation using the kb3 g interaction , fig . 1(d ) , the shell - model calculation using the gxpf1a interaction , and fig . 1(e ) , the running sums of b(gt ) as a function of the excitation energy . the experimental gt strength is dominated by the transition @xmath0ti@xmath32 @xmath29 @xmath0sc(@xmath33 ) . the reported energy @xmath34 is 611 kev , while the calculated ones are lower . the calculated intensities for this transition are similar to the measured ones . it is noticeably that the interaction kb3 g generated an excitation energy closer to the experimental one than the energy obtained employing the gxpf1a interaction , while the opposite is true for the gt strength . the second excited @xmath35 state at 1888 kev is missed in both calculations , which predict a second , small b(gt ) strength at an excitation energy slightly above 4 mev , which could be the one observed in the ce reaction . both interactions predict a noticeable b(gt ) strength at an excitation energy between 9 and 10 mev , where there is no experimental information . the close similitude in the b(gt ) strength predicted using the gxpf1a interaction and the @xmath36 data is visible in the summed strength plot , -1.5 cm fig . [ 46cr ] shows the experimental and shell - model calculated b(gt ) strength distributions for the transition @xmath1cr @xmath29 @xmath1v . fig . 2(a ) represents the experimental data observed through the @xmath4-decay @xmath1cr(@xmath37 ) @xmath29 @xmath1v(@xmath35 ) up to the excitation energy @xmath38v ) = 3.867 mev @xcite , fig . 2(b ) the experimental data observed through the charge - exchange reaction process @xcite i.e. , @xmath1ti(@xmath6he,@xmath7)@xmath1v up to the excitation energy @xmath38v ) = 5.717 mev , fig . 2(c ) , the shell - model calculation using the kb3 g interaction , fig . 2(d ) , the shell - model calculation using the gxpf1a interaction , and fig . 2(e ) , the running sums of b(gt ) as function of excitation energy . the experimentally observed b(gt ) strength as a function of the excitation energy exhibits two clusters , one between 1 and 1.5 mev , and another between 2.4 and 3.0 mev , plus some small intensities around and above 4 mev . on the theoretical side , the kb3 g and gxpf1a interactions predict a low energy transitions below 1 mev , and the most intense transition close to 3 mev . while the general distribution of b(gt ) strength is similar using both interactions , the kb3 g predicts more fragmentation . the summed b(gt ) intensities obtained from the two calculations are in close agreement , and reproduce well the observed one . -1.5 cm the shell - model calculations and the experimental gt strength distributions for the transition @xmath2fe @xmath29 @xmath2mn are presented in the fig . [ 50fe ] . the experimental data observed through the @xmath4-decay @xmath2fe@xmath39mn up to the excitation energy @xmath40mn ) = 4.315 mev @xcite are shown in fig . 3(a ) , those observed through the charge - exchange reaction process @xmath2cr(@xmath6he,@xmath7)@xmath2mn up to the excitation energy @xmath40mn ) = 5.545 mev @xcite in fig . 3(b ) , the shell - model calculation using the kb3 g interaction in fig . 3(c ) , the shell - model calculation using the gxpf1a interaction in fig . 3(d ) , and the running sums of b(gt ) as function of the excitation energy in fig . there is an intense isolated b(gt ) transition to the first @xmath35 state , observed at 651 kev , which is predicted , but at lower excitation energies , by both interactions . there are a few observed transitions with comparable strength distributed between 2.4 and 4.4 mev , which are described with some detail using the interaction kb3 g . the same strength is concentrated in three transitions when using the interaction gxpf1a . both interactions predict a long tail of small intensity transitions . the calculated summed b(gt ) intensities closely reproduce the experimental ones . -1.5 cm fig . [ 54co ] shows the experimental and shell - model calculated b(gt ) strength distributions for the transition @xmath3ni@xmath41co . 4(a ) displays the experimental data obtained through the @xmath4-decay @xmath3ni@xmath42co up to the excitation energy @xmath43co ) = 5.202 mev @xcite , fig . 4(b ) the experimental data observed through the charge - exchange reactions @xmath3fe(@xmath6he,@xmath7)@xmath3cr up to the excitation energy @xmath43co ) = 5.917 mev @xcite , fig . 4(c ) the shell - model calculation using the kb3 g interaction , fig . 4(d ) , the shell - model calculation using the gxpf1a interaction , and fig . 4(e ) , the running sums of b(gt ) as function of the excitation energy . the b(gt ) strength for the @xmath3ni(@xmath37 ) @xmath29 @xmath3co(@xmath44 ) transition displays a dominant transition at 937 kev and a set of transitions at energies between 3.3 and 6 mev . as mentioned above , in the shell model calculations a truncation to a maximum of four nucleon excitations from the @xmath45 shell to the rest of the @xmath10 orbitals was necessary due to computational limitations . the b(gt ) strength distribution obtained employing the kb3 g interaction in the truncated space fails to reproduce the experimental data . on the other hand , the calculated b(gt ) obtained with the gxpf1a interaction depict the main elements observed in the experiments . the intense low energy transition is present , although at a slightly lower energy , and two transitions around 4 mev resemble the centroid of the observed ones . the sum of b(gt ) strength naturally follows that same pattern . the results from the kb3 g interaction do not resemble the observed distribution , while those associated to the gxpf1a interaction are in good agreement with experimental data even with truncated calculation . due to the huge matrix dimensions we calculated only ten transitions from ground state of @xmath3ni(@xmath37 ) to @xmath3co(@xmath35 ) . in table [ espm ] , the total sum of the b(gt ) strength is presented for the transitions measured in the four nuclei . the third and fourth columns show the measured valued for the @xmath4-decay and the charge exchange ( @xmath6he,@xmath7 ) reactions , respectively . both experimental results are of the same order , their differences can be ascribed to the different energy regions accessible with these techniques . the last three columns show the calculated results obtained employing the kb3 g interaction , the gxpf1a interaction and the extreme single particle model ( espm ) , respectively . .[tab : table1]comparison between the experimental , sm calculation , and espm summed b(gt ) strengths . [ cols="^,^,^,^,^,^,^ " , ] in the extreme single particle model ( espm ) the @xmath37 ground state of the even - even parent nuclei is described filling the @xmath45 orbital with the appropriate number of valence protons and neutrons . the final @xmath35 states in the odd - odd daughter nuclei are built as a hole in the proton @xmath45 shell , and a neutron particle in any of the @xmath10 orbitals . the gamow - teller strengths are calculated in the espm as @xmath46 in this expression @xmath47 is the number of valence protons in the @xmath45 shell , @xmath48 the number of valence neutron holes in the i - th orbital , which in this case can only be the @xmath45 ( non - spin flip transition ) and the @xmath49 ( spin flip transition ) . @xmath50 is single - particle matrix element connecting the proton state @xmath45 and the neutron state @xmath51 . it is clear from the table that the extreme single particle summed b(gt ) strengths are much larger than the observed ones . those obtained in the sm calculations are closer to the experimental intensities , while still larger . the exception is @xmath3ni , were the strong truncation generates calculated summed b(gt ) strengths which are smaller than the experimental ones . in the present work we have presented a comprehensive shell model calculation for gamow - teller transition strengths in @xmath0ti , @xmath1cr , @xmath2fe and @xmath3ni , employing the effective interactions kb3 g and gxpf1a . they provide a theoretical description of the experimental gamow - teller transition strength distributions measured via @xmath4 decay of these @xmath5=-1 nuclei , produced in fragmentation and also with ( @xmath6he,@xmath7 ) charge - exchange ( ce ) reaction . in the study of the gt transitions in @xmath0ti , @xmath1cr , @xmath2fe , the configuration space of the full @xmath10 shell was employed . both interactions provided a qualitative description of the observed transitions , and were able to closely reproduce the summed b(gt ) strength . in the case of @xmath3ni it was necessary to impose a truncation in the number of excitations allowed from the @xmath45 level . as a consequence only the b(gt ) strengths calculated employing the gxpf1a interaction resembled the observed ones , and the calculated added intensities were smaller than the measured ones . y. fujita _ et al . _ , prog . part . 66 * , 549 ( 2011 ) , and references therein . rolfs , w. rodney , _ cauldrons in the cosmos _ university of chicago press ( 1988 ) . k. langanke and g. martnez - pinedo , rev . phys . * 75 * , 819 ( 2003 ) . balasi , k. langanke , g. martnez - pinedo , progress in particle and nuclear physics * 85 * , 33 ( 2015 ) y. fujita _ et al . _ , lett . * 95 * , 212501 ( 2005 ) . s.e.a . orrigo _ et al . _ , rev . lett . * 112 * , 222501 ( 2014 ) . e. caurier , k. langanke , g. martnez - pinedo , and f. nowacki , nucl . a * 653 * , 439 ( 1999 ) . t. suzuki _ et al . _ , c * 83 * , 044619 ( 2011 ) . a.l . cole _ et al . _ , c * 86 * , 015809 ( 2012 ) . f. molina _ et al . _ , c * 91 * , 014301 ( 2015 ) . f. g. molina ph.d thesis `` beta decay of @xmath52 nuclei and comparison with charge exchange reaction experiments '' , ( 2011 ) . t. c * 73 * , 024311 ( 2006 ) . j. a * 25 * , ( s01 ) 499 ( 2004 ) . b. a. brown , w. d. m. rae , e. mcdonald , and m. horoi , nushellx@msu . a. poves _ et al . a * 694 * , 157 ( 2001 ) . et al . _ , c * 65 * , 061301(r ) ( 2002 ) . g. martinez - pinedo _ et al . c * 53 * , r2602(r ) ( 1996 ) .

a systematic shell model description of the experimental gamow - teller transition strength distributions in @xmath0ti , @xmath1cr , @xmath2fe and @xmath3ni is presented . these transitions have been recently measured via @xmath4 decay of these @xmath5=-1 nuclei , produced in fragmentation reactions at gsi and also with ( @xmath6he,@xmath7 ) charge - exchange ( ce ) reactions corresponding to @xmath8 to @xmath9 carried out at rcnp - osaka . the calculations are performed in the @xmath10 model space , using the gxpf1a and kb3 g effective interactions . qualitative agreement is obtained for the individual transitions , while the calculated summed transition strengths closely reproduce the observed ones . gt - transition , shell model 21.60.cs , 23.40.hc , 25.55.kr

as described in the main article , graphene membranes were prepared by micro - mechanical cleavage and transfer to tem grids @xcite . aberration - corrected hrtem imaging was carried out in an fei titan 80300 , equipped with an objective - side image corrector . the microscope was operated at 80 kev and 100 kev for hrtem imaging , and at 300 kev for irradiation . the extraction voltage of the field emission source was set to a reduced value of 2 kv in order to reduce the energy spread . for both 80 kev and 100 kev imaging , the spherical aberration was set to 20 @xmath2 m and images were obtained at scherzer defocus ( ca . @xmath3 nm ) . at these conditions , dark contrast can be directly interpreted in terms of the atomic structure . image sequences were recorded on the ccd camera with exposure times ranging from 1 s to 3 s and intervals between 4 s and 8 s , and a pixel size of @xmath21 . the effect of slightly uneven illumination is removed by normalization ( division ) of the image to a strongly blurred copy of the same image , effectively removing long - range variations . drift - compensation is done using the stackreg plugin for the imagej software @xcite . we show individual exposures as well as averages of a few frames ( up to 10 ) . this is because we have used different beam current densities , in order to test for possible dose rate effects ( within the 100 kev , simultaneous imaging and defect generation experiment ) . the configurations described here can be discerned in individual exposures if ca . @xmath22 counts per pixel at @xmath21 pixel size are used . with a high beam current density , this is possible in 1 s exposures ( corresponding to a total dose of ca . @xmath23 per image , and a dose rate of @xmath24 ) . at lower current densities , the same dose was spread over several exposures , so that sample drift could be compensated ( individual exposures could not be longer than 3 s due to sample drift ) . the lowest dose rate was ca . @xmath25 . within our dose rate range , density and shape of defects appears to depend only on the total dose . formation energy of a structure in our dft calculations was defined in the usual way as @xmath26 where @xmath27 and @xmath28 are the total energies of the structure with the defect ( @xmath29 missing atoms ) and the same supercell without the defect , respectively , and @xmath30 is the chemical potential of a carbon atom in pristine graphene . the semi - conducting features of the defects were consistently observed with varying number of @xmath5-kpoints used in calculating the electronic density of states . for clarity we show here the tem images of the main article without stucture overlay , and also a few additional tem images . fig . [ px::dv ] shows several di - vacancy ( dv ) defects in linear alignment . in particular , the carbon tetragon is always reproduced when the dvs are aligned along the armchair direction of the graphene lattice . [ px::dv2 ] shows a comparison between a hrtem image simulation based on a dft - optimized structure and an actual hrtem image for the armchair alignment of di - vacancies , which contains the carbon tetragons . in fig . [ px::sfig2 ] , we show the panels a d from the figure 3 of the main article with and without overlays . these images are also contained in supplementary video s2 . in fig . [ px::sfig3 ] we show panels a h from the figure 4 of the main article with and without overlay ( the corresponding time series is shown in supplementary video s3 ) . in fig . [ px::300kv ] , we show two additional images where a defect with a rotated hexagon kernel was generated from clusters of multiple vacancies . these configurations were created by a short 300 kev exposure and subsequent imaging at 80 kev . it should be noted that the rotated hexagon kernels appear in the larger vacancy clusters in both of our experiments ( 100 kev irradiation with simultaneous imaging as in figure 4 of the main article , and the 300 kev/80 kev combination as shown in fig . [ px::300kv ] ) . _ supplementary video s1 _ : generation and transformation of defects under 100 kev irradiation . each frame is an average of 2 ccd exposures . note the continuous increase in defect density and partial amorphization of the membrane . + _ supplementary video s2 _ : further transformation of graphene to a 2d amorphous membrane under 100 kev irradiation . in terms of total dose , it can be considered as a continuation of video s4 ( although sample region is different ) . each frame shows an individual ccd exposure . note that the entire clean graphene area becomes amorphous while remaining one - atom thick , and beam - generated holes make up only a small fraction of the area . + _ supplementary video s3 _ : two isolated di - vacancies , generated by brief 300 kev irradiation and observed at 80 kev . the di - vacancies migrate under the beam and transform between the @xmath11(5 - 8 - 5 ) , @xmath11(555 - 777 ) and @xmath11(5555 - 6 - 7777 ) configurations multiple times . the video shows individual exposures in each frame . + _ supplementary video s4 _ : a cluster of several vacancy defects , generated by brief 300 kev irradiation and observed at 80 kev . the video shows individual exposures in each frame . the configuration changes continuously until the final , linear aligned di - vancy configuration is observed and stable throughout several exposures ( frames 2229 ) . frames 3036 are duplicates of frame 29 in order to show the final configuration as a still image at the end . + _ supplementary video s5 _ : generation and transformation of defects under 100 kev irradiation . each frame is an average of 10 ccd exposures ( recorded at a lower current density ) . this video shows the generation of the rotated hexagon kernel as in figure 4 of the main article . the rotated hexagon kernel is highlighted as overlay in frames 2831 .

while crystalline two - dimensional materials have become an experimental reality during the past few years , an amorphous 2-d material has not been reported before . here , using electron irradiation we create an @xmath0-hybridized one - atom - thick flat carbon membrane with a _ random _ arrangement of polygons , including four - membered carbon rings . we show how the transformation occurs step - by - step by nucleation and growth of low - energy multi - vacancy structures constructed of rotated hexagons and other polygons . our observations , along with first - principles calculations , provide new insights to the bonding behavior of carbon and dynamics of defects in graphene . the created domains possess a band gap , which may open new possibilities for engineering graphene - based electronic devices . hexagonal rings serve as the building blocks for the growing number of @xmath1bonded low - dimensional carbon structures such as graphene @xcite and carbon nanotubes @xcite . non - hexagonal rings usually lead to the development of non - zero curvature , _ e.g. _ , in fullerenes @xcite and carbon nanohorns @xcite , where the arrangement of other polygons can be geometrically deduced via the isolated pentagon rule ( ipr ) @xcite and euler s theorem @xcite . aberration corrected high - resolution transmission electron microscopy ( ac - hrtem ) has recently allowed atomic - resolution imaging of regular carbon nanostructures and identification of defects in these materials @xcite . point defects , mostly vacancies , are naturally created by the energetic electrons of a tem . however , the possibility for selectively creating topological defects representing agglomerations of non - hexagonal rings could be more desirable in the context of carbon - based electronics . @xcite in fact , despite the recent advances , the precise microscopic picture of the response of graphene to electron irradiation remains incomplete . earlier experiments on curved carbon nanosystems @xcite have shown that they avoid under - coordinated atoms under irradiation at high temperatures via vacancy migration and coalescence . recent experiments on graphene @xcite reported only the development of holes . theoretical studies have also predicted the appearance of small holes or formation of hckelite - like configurations @xcite or dislocations @xcite . in this letter , we report the transformation of graphene into a two - dimensional random arrangement of polygons due to continuous exposure to the electron beam with an energy just above the knock - on threshold . by carefully choosing the electron energy , we selectively enhance and suppress the underlying mechanisms of defect production . a combination of experiments and density - functional theory ( dft ) calculations allows us to show that the transformation is driven by two simple mechanisms : atom ejection and bond rotation . the created defects tend to have a low formation energy and exhibit an electronic band gap . we also discover other unexpected configurations , such as stable carbon tetragons @xcite which appear upon linear arrangement of di - vacancies . our graphene membranes were prepared by micro - mechanical cleavage and transfer to tem grids @xcite . aberration - corrected hrtem imaging was carried out in an fei titan 80300 , equipped with an objective - side image corrector . the microscope was operated at 80 kev and 100 kev for hrtem imaging , and at 300 kev for irradiation . the extraction voltage of the field emission source was set to a reduced value of 2 kv in order to reduce the energy spread . for both 80 kev and 100 kev imaging , the spherical aberration was set to 20 @xmath2 m and images were obtained at scherzer defocus ( ca . @xmath3 nm ) . at these conditions , dark contrast can be directly interpreted in terms of the atomic structure . the dft calculations were carried out with the vasp simulation package @xcite using projector augmented wave potentials @xcite to describe core electrons , and the generalized gradient approximation @xcite for exchange and correlation . kinetic energy cutoff for the plane waves was 500 ev , and all structures were relaxed until atomic forces were below 0.01 ev / . the initial structure consisted of 200 c atoms , and brillouin zone sampling scheme of monkhorts - pack @xcite with up to @xmath4 mesh was used to generate the @xmath5-points . barrier calculations were carried out using the nudged elastic band method as implemented in vasp @xcite . we started our experiments by monitoring _ in situ _ the behavior of graphene under a continuous exposure to electron irradiation using ac - hrtem imaging with an electron energy of 100 kev , _ i.e. _ , just above the threshold for knock - on damage ( @xmath6 ) in @xmath0-bonded carbon structures @xcite . fig . [ fgr : amorph]a shows a graphene structure after an electron dose of @xmath7 @xmath8 . contrary to the expectations , the structure does not predominantly consist of holes or collapse into a 3d object . instead , it has remained as a coherent single - layer membrane composed of a random patch of polygons . holes have also formed , but only on a small fraction of the area . the fourier transform of the image shows that the resulting structure is completely amorphous ( fig [ fgr : amorph]b ) . in order to understand the mechanisms behind the transformation , we separated them by varying the electron beam energy . to observe how a defected graphene sheet reacts to an electron beam when atomic ejections are prohibited by a low enough electron energy , we created initial damage in a graphene sheet by brief 300 kev irradiation , and then studied the generated structures at 80 kev . now only under - coordinated atoms can be ejected ( @xmath6 for a @xmath1bonded c is about 1820 ev @xcite , whereas dft calculations predict a @xmath6 of @xmath914 ev for a two - coordinated c ) . however , bond re - organization is possible , as activation energies for bond rotations in @xmath0-bonded carbon structures are in the range of 410 ev @xcite , depending on the local atomic configuration . correspondingly , in pristine graphene , bond rotations are occasionally observed under 80 kev irradiation @xcite ( fig . [ fgr : elements]a , b ) , resulting in the formation of the stone - wales defect @xcite . in all observed cases continued exposure reversed this transition in pristine graphene . however , defect structures , _ e.g. _ , di - vacancies , can convert between different configurations ( fig . [ fgr : elements]d - f ) via bond rotations . according to our dft simulations , the barriers for bond rotations in these structures are 56 ev , which excludes thermally activated migration . in fig . [ fgr : dv]a d we present evolution of a more complex defect structure . the defects created by an electron beam are predominantly mono - vacancies , which quickly convert to di - vacancies due to a higher probability for under - coordinated atoms to be ejected , as noted above . here ( fig . [ fgr : dv]a ) , a brief exposure to a 300 kev beam ( dose @xmath10 @xmath8 ) has created an isolated @xmath11(555 - 777 ) and a defect with 4 missing carbon atoms ( two connected di - vacancies ) . during the image sequence , the 80 kev electron beam causes the structure to re - organize via bond rotations . the @xmath11(555 - 777 ) turns first into a @xmath11(5 - 8 - 5 ) ( fig . [ fgr : dv]b and then a dislocation dipole ( fig . [ fgr : dv]c ) , before forming a defect composed of clustered di - vacancies ( fig . [ fgr : dv]d ) . fig . [ fgr : dv]a d also present two frequently observed linear arrangements of di - vacancies . because atom ejection occurs at random positions , vacancies initially appear randomly in the area exposed to the electron beam . however , during lower - energy exposure ( _ e.g. _ , 80 kev ) , these defects travel via a re - bonding mechanism , which is illustrated in fig . [ fgr : dv]e . each migration step is initiated by a single electron impact from which the atom obtains energy slightly below @xmath6 . in other words , electron irradiation provides the activation energy to drive the system from a local energy minimum into another one , in our case predominantly via ( reversible ) bond rotations ( fig . [ fgr : elements]d - f , fig . [ fgr : dv]a - d ) . this can be clearly seen in video s4 in ref . ( partially shown in fig . [ fgr : dv]a d ) , where the configuration changes frequently until it arrives in the more stable configuration composed of three aligned double - vacancies . individual transitions can also lead to higher energy structures . for example , the intermediate states for the di - vacancy migration [ dislocation dipole ( fig . [ fgr : dv]c ) and @xmath11(555 - 777 ) ( fig . [ fgr : dv]a ) ] have formation energies @xmath12 which differ from that of the @xmath11(5 - 8 - 5 ) by @xmath13 ev and @xmath14 ev , respectively . aligning di - vacancies along the zigzag direction of the lattice ( fig . [ fgr : dv]d ) , @xmath15 ev of energy is gained per a di - vacancy pair , as compared to isolated di - vacancies . when the alignment appears along the armchair direction ( fig . [ fgr : dv]a c ) , the energy gain is 2.01 ev . in this case , a tetragon is formed where the two pentagons of the adjacent di - vacancies would overlap . hrtem simulation of the dft - optimized structure of the defect is in excellent agreement with the experimental image @xcite . note that @xmath0-bonded carbon tetragons in molecules , as in cyclobutadiene , can be stabilized only at low temperatures and when the molecules are embedded into a matrix @xcite . in our case they are stabilized by the surrounding graphene lattice , as theoretically predicted for nanotubes @xcite . under 100 kev irradiation , atom ejection occurs at a very slow rate , so that changes in the atomic network are sufficiently slow to be precisely resolved . therefore , the reconstruction of vacancy defects via bond rotations can be monitored immediately after a vacancy is generated . an example is presented in fig . [ fgr : mvac ] . the initial configuration ( fig . [ fgr : mvac]a ) consists of three di - vacancies in the armchair orientation ( formed prior to recording the first image ) . in the recorded images , the structure loses atoms until 24 atoms are missing . several remarkable configurational changes are found in this image sequence . fig . [ fgr : mvac]a c shows a collapse of linearly clustered defects into an apparently less defective structure with a dislocation dipole . this corresponds to the prediction of jeong _ et al . _ @xcite that the dislocation dipole is favored over a large multi - vacancy . however , this requires a linear arrangement of vacancies . fig . [ fgr : mvac]c d shows the loss of four additional atoms . two of them gave rise to an additional di - vacancy ; the other two contributed to the separation of the dislocation cores ( the rotated hexagon , clustered with a stone - wales defect , constitutes a dislocation core ) . during continued irradiation , we see the formation of a cluster of rotated hexagons surrounded by a chain of alternating pentagons and heptagons ( fig . [ fgr : mvac]e h ) . such configurations , rotated by @xmath16 with respect to the original lattice and matched by pentagons and heptagons to the zigzag lattice direction , appear to be the preferred way to incorporate missing atoms in the graphene structure . due to the matching numbers of pentagons and heptagons and hence cancellation of negative and positive curvature these structures remain flat . to understand the driving force for the transformations , we calculated @xmath12 for the simplest defect structures matching the obverved trend of forming a rotated hexagon kernel . the lowest - energy tetra - vacancy ( four missing atoms ) can be created by combining two @xmath11(5555 - 6 - 7777 ) di - vacancies , whereas the hexa - vacancy ( six missing atom ) requires three of these defects ( fig . [ fgr : dft]a ) . remarkably , these configurations have the lowest @xmath12 of any reported vacancy structures with equal number of missing atoms in graphene ( @xmath12 per missing atom multiplied by the number of missing atoms are @xmath173.14 ev and @xmath182.50 ev ) . a hole with six missing atoms has a formation energy of @xmath193.15 ev , while for a dislocation and a hckelite - like structure values of @xmath193.72 ev and @xmath182.64 ev have been reported @xcite , respectively . evidently , the rotated hexagon defects spawn the family of lowest energy multi - vacancies in graphene . in contrast to what was recently shown for the zigzag - oriented di - vacancies @xcite , these structures open a band gap in graphene , as can be seen from the density of states ( fig . [ fgr : dft]b ) . the calculated band gap is in the order of 200 mev . this value is possibly underestimated within the used gga approximation , and advanced dft methods are likely to give a higher value @xcite . to conclude , we have shown how an electron beam can be used to selectively suppress and enhance bond rotations and atom removal in graphene . we demonstrated that irradiation at electron energies just above the threshold for atom displacement turns graphene not into a `` perforated graphene '' but a two - dimensional coherent amorphous membrane composed of @xmath1hybridized carbon atoms . this membrane grows through nucleation and expansion of defects which result from electron beam driven di - vacancy migration and agglomeration . these defect configurations predominantly consist of a 30@xmath20 rotated kernel of hexagons surrounded by a chain of alternating pentagons and heptagons . these defects are the energetically favored way for the graphene lattice to accommodate missing atoms , and have a semi - conducting nature . since several of the presented examples of the two - dimensional @xmath0hybridized defect configurations violate the ipr , due to increased reactivity @xcite , they may be exploited for functionalization of graphene . we also showed unambiguous evidence for four - membered carbon rings in graphitic structures . clearly , despite of the large amount of research , the richness of carbon chemistry continues to provide surprises . more examples of the observed structures and videos of the complete tem image series are presented in ref . @xcite . we acknowledge support by the german research foundation ( dfg ) and the german ministry of science , research and the arts ( mwk ) of the state baden - wuerttemberg within the salve ( sub angstrom low voltage electron microscopy ) project and by the academy of finland through several projects . we are grateful for the generous grants of computer time provided by csc finland . 10 k. s. novoselov , a. k. geim , s. v. morozov , d. jiang , y. zhang , s. v. dubonos , i. v. grigorieva , and a. a. firsov , science * 306 * , 666 ( 2004 ) . k. s. novoselov , d. jiang , f. shedin , t. j. booth , v. v. khotkevich , s. v. morozov , and a. k. geim , pnas * 102 * , 10451 ( 2005 ) . s. iijima , nature * 354 * , 56 ( 1991 ) . h. w. kroto , nature * 329 * , 529 ( 1987 ) . s. iijima , m. yudasaka , r. yamada , s. bandow , k. suenaga , f. kokai , and k. takahashi , chemical physics letters * 309 * , 165 ( 1999 ) . t. g. schmalz , w. a. seitz , d. j. klein , and g. e. hite , j. am . chem . soc . * 110 * , 1113 ( 1988 ) . h. terrones and a. l. mackay , carbon * 30 * , 1251 ( 1992 ) . a. hashimoto , k. suenaga , a. gloter , k. urita , and s. iijima , nature ( london ) * 430 * , 870 ( 2004 ) . j. c. meyer , c. kisielowski , r. erni , m. d. rossell , m. f. crommie , and a. zettl , nano letters * 8 * , 3582 ( 2008 ) . m. h. gass , u. bangert , a. l. bleloch , p. wang , r. r. nair , and g. k. , nat nano * 3 * , 676 ( 2008 ) . c. o. girit _ et al . _ , science * 323 * , 1705 ( 2009 ) . j. h. warner , m. h. rummeli , l. ge , t. gemming , b. montanari , n. m. harrison , b. buchner , and g. a. d. briggs , nat nano * 4 * , 500 ( 2009 ) . a. h. castro neto , f. guinea , n. m. r. peres , k. s. novoselov , and a. k. geim , rev . mod . phys . * 81 * , 109 ( 2009 ) . j. lahiri , y. lin , p. bozkurt , i. i. oleynik , and m. batzill , nat nano * 5 * , 326 ( 2010 ) . o. v. yazyev and s. g. louie , nat mater * 9 * , 806 ( 2010 ) . l. sun , f. banhart , a. v. krasheninnikov , j. a. rodrguez - manzo , m. terrones , and p. m. ajayan , science * 312 * , 1199 ( 2006 ) . l. sun , a. v. krasheninnikov , t. ahlgren , k. nordlund , and f. banhart , phys . rev . lett . * 101 * , 156101 ( 2008 ) . h. terrones , m. terrones , e. hernndez , n. grobert , j .- c . charlier , and p. m. ajayan , phys . rev . lett . * 84 * , 1716 ( 2000 ) . b. w. jeong , j. ihm , and g .- d . lee , phys . rev . b * 78 * , 165403 ( 2008 ) . y .- m . legrand , a. v. d. lee , and m. barboiu , science * 329 * , 299 ( 2010 ) . j. c. meyer , c. o. girit , m. f. crommie , and a. zettl , appl . phys . lett . * 92 * , 123110 ( 2008 ) . g. kresse and j. furthmller , comput . mat . sci . * 6 * , 15 ( 1996 ) . g. kresse and j. furthmller , phys . rev . b * 54 * , 11169 ( 1996 ) . p. e. blchl , phys . rev . b * 50 * , 17953 ( 1994 ) . j. p. perdew , k. burke , and m. ernzerhof , phys . rev . lett . * 77 * , 3865 ( 1996 ) . h. j. monkhorst and j. d. pack , phys . rev . b * 13 * , 5188 ( 1976 ) . g. henkelman , b. p. uberuaga , and h. jnsson , j. chem . phys . * 113 * , 9901 ( 2000 ) . f. banhart , rep . . phys . * 62 * , 1181 ( 1999 ) . b. w. smith and d. e. luzzi , j. appl . phys . * 90 * , 3509 ( 2001 ) . c. p. ewels , m. i. heggie , and p. r. briddon , chem . phys . lett . * 351 * , 178 ( 2002 ) . l. li , s. reich , and j. robertson , phys . rev . b * 72 * , 184109 ( 2005 ) . a. j. stone and d. j. wales , chem . . lett . * 128 * , 501 ( 1986 ) . see epaps document no . to - be - inserted for experimental videos and additional images . for more information on epaps , see http://www.aip.org/pubservs/epaps.html . w. orellana , phys . rev . b * 80 * , 075421 ( 2009 ) . d. j. appelhans , z. lin , and m. t. lusk , phys . rev . b * 82 * , 073410 ( 2010 ) . y .- z . tan , s .- y . xie , r .- b . huang , and l .- s . zheng , nat chem * 1 * , 450 ( 2009 ) . p. thvenaz , u. ruttimann , and m. unser , ieee transactions on image processing * 7 * , 27 ( 1998 ) .

semiconductors exhibit a multitude of nonlinear optical responses for resonant as well as non - resonant excitation.@xcite one of the most prominent nonlinear features is the generation of higher harmonics of the exciting frequency . when the frequency of the incoming field is tripled one speaks of third harmonic generation ( thg ) . such thg can be employed in spectroscopy and provides important insights into biological processes@xcite or even for palaeontology.@xcite in semiconductors , thg has , for example , been studied in coupled quantum wells,@xcite quantum cascade structures,@xcite quantum wires and dots,@xcite while it is also of interest in newly developed materials like graphene@xcite and atomically thin semiconductors.@xcite in order to understand thg one requires a description of the optical fields and the material which is excited by them and generates the nonlinear interaction . here we focus on the photointeraction of semiconductor quantum wells ( qw ) with ultrashort light pulses . to this end , we employ an auxiliary differential equation finite difference time domain ( fdtd ) approach to describe the dynamics of the light field along with the dynamics of the carriers in the qw . this approach goes beyond rotating wave approximation and slowly - varying envelope approximation , allowing to treat fundamental and third harmonic on the same footing and describe photonic structures that vary on scales much smaller than the wavelength . the combination of fdtd with density matrix models through auxiliary differential equations includes not only the effect of the field on the material but also self - consistently describes the effect of the material on the field . this feature allows , for example , to describe propagation of sit solitons in 1 and 2 dimensions@xcite and to study loss compensation and lasing dynamics in metamaterials@xcite or plasmonic stopped - light lasers.@xcite however , the few - level models employed in those studies can not describe the complicated behavior of an interacting electron gas excited in semiconductor qws.@xcite on the other hand more complex wave - vector resolved semiconductor models have been developed that also consider coulomb interaction between excited carriers within different levels of approximation@xcite or spatially resolved quantum kinetics calculations.@xcite such models have been used to investigate various non linear effects such as the two - band mollow triplet in thin gaas films,@xcite the carrier - wave rabi flopping in bulk gaas@xcite and thg from carbon nanotubes both in the perturbative and non - perturbative regime.@xcite these approaches , however , do not include the self - consistent , spatially resolved resolution of electromagnetic fields . combining a spatially dependent full time - domain ( fdtd ) approach with a description of semiconductor qws containing a wave - vector resolved , many - level density matrix description of the qw in a two - band approximation , has been pioneered in @xcite to describe the spatio - temporal dynamics of semiconductor lasers and recently to describe lasing of semiconductor nanowires.@xcite here , we extend the previous description by taking into account coulomb interaction in hartree - fock approximation , which allows us to describe the excitonic nature of the qw absorption . in this work , we are going to consider specifically the ultrashort pulse excitation of a qw embedded in a bragg mirror structure typical for a semiconductor saturable absorber mirror ( sesam ) . we obtain the carrier dynamics associated with excitation of the qw exciton and study the intensity dependence of thg in this qw . we find that the power - law exponent of the intensity dependence of the thg strongly varies with excitation frequency . for far off - resonant pulses the expected cubic behavior is found , while for pulses resonant with the exciton energy the exponent is reduced due to saturation effects . similar findings have been reported in theoretical and experimental studies on the excitation of carbon nanotubes with ultrashort laser pulses.@xcite the hamiltonian describing the semiconductor structure is given by the three parts@xcite @xmath0 with the free carrier part @xmath1 , the carrier - carrier interaction @xmath2 and the carrier - light field interaction @xmath3 . we assume a two - band structure with one conduction and one valence band , such that the free carrier hamiltonian reads @xmath4.\ ] ] @xmath5 and @xmath6 are the electron / hole creation and annihilation operators with wave - vector @xmath7 and @xmath8 are the corresponding energies . we consider a qw , where the energy is quantized in the @xmath9-direction with a fixed @xmath10 , while @xmath7 always refers to the two - dimensional inplane wave vector @xmath11 . the confinement along @xmath9 is included by applying the envelope function approximation,@xcite while the inplane bands are assumed to have parabolic dispersion . the electron and hole energies are @xmath12 with the effective masses @xmath13 and the band gap @xmath14 . the carrier - carrier interaction is given by the coulomb potential @xmath15,\end{aligned}\ ] ] with the coulomb matrix elements @xmath16 obtained by multiplying the ideal @xmath17 coulomb matrix elements by a band - dependent form - factor obtained from the envelope function approximation . we consider the plasmon - pole@xcite approximation to the screening of the coulomb potential , where the inverse screening length is kept constant at the initial value , @xmath18 , as screening typically builds up on timescales longer than those considered here.@xcite we treat the carrier - light field interaction in dipole approximation resulting in @xmath19,\ ] ] with dipole matrix element @xmath20 for the transition from valence to conduction band . the classical light field @xmath21 is assumed to be spatially constant over the region of the qw , denoted by the parametric dependence of the light field on @xmath22 . to calculate the dynamics of the system we set up the equations of motion for the occupations @xmath23 and @xmath24 and the polarization @xmath25 via the heisenberg equation of motion @xmath26 - \gamma_{p } p_{{\textbf{k } } } , \nonumber \\ \partial_{t } n^{e}_{{\textbf{k } } } & = & i \ , [ \omega_{{\textbf{k } } } p^{*}_{{\textbf{k } } } - \omega^{*}_{{\textbf{k } } } p_{{\textbf{k } } } ] , \nonumber \\ \partial_{t } n^{h}_{{\textbf{k } } } & = & i \ , [ \omega_{{\textbf{k } } } p^{*}_{{\textbf{k } } } - \omega^{*}_{{\textbf{k } } } p_{{\textbf{k}}}].\end{aligned}\ ] ] here @xmath27 is the transition frequency , @xmath28 is a phenomenological dephasing rate and @xmath29 is the rabi frequency beyond rotating wave approximation , i.e. , calculated with the time dependent electric field @xmath30 . due to the homogeneity of the problem , we only take into account the @xmath7-diagonal elements of the density matrix . the off - diagonal element are known to play a crucial role for spatially inhomogeneous problems.@xcite the equations of motion ( eq . [ eq : bloch ] ) already include coulomb interaction under hartree - fock approximation , which is justified for ultra short time scales . within this approximation the interaction leads to a renormalization of the transition energies @xmath31 with the coulomb hole self - energy @xmath32 and the bare ( unscreened ) coulomb potential @xmath33 . also the light - matter coupling becomes renormalized due to the coulomb interaction leading to the renormalized rabi frequency @xmath34 the integration of the equations of motion ( eq . [ eq : bloch ] ) is performed on a grid of @xmath35 @xmath36-points , homogeneously distributed between @xmath37 and @xmath38 , where @xmath39 is the bohr radius in the bulk material . the integration algorithm is runge - kutta of order 4 , where the rabi frequency at the midpoints is obtained by interpolation of the electric field.@xcite in the simulation we are not only interested in the light field acting on the carriers in the qw , but also on the back - action on the field itself . we model the dynamics of the electric field @xmath40 in the whole structure as well as the ingoing and outgoing field through a one dimensional fdtd simulation.@xcite in the one - dimensional case , with the field propagating along @xmath9 , maxwell equations can be reduced to @xmath41,\end{aligned}\ ] ] where @xmath42 and @xmath43 are the electric and magnetic fields and @xmath44 is the background permittivity . the dynamic material polarization @xmath45 is zero everywhere but at the position of the qw . @xmath46 can be calculated from the microscopic polarizations as @xmath47 the spatial grid used to describe the system has a step of @xmath48 . due to fdtd stability constraints this results in a time step of @xmath49 , which has been used for the simultaneous resolution of the semiconductor equations of motion and maxwell equations . the injection of field inside the simulation domain is performed through the total field scattered field ( tfsf ) technique.@xcite the open boundaries of the system are simulated through perfectly matched layers ( pml ) boundary conditions . to test our model we will start by investigating a qw in a homogeneous background . we will then study the field and semiconductor dynamics for a qw embedded in a multilayered structure . in the simulation different passive materials are defined by a constant refractive index and different structures can be modeled by defining a space dependent refractive index profile . the active medium we chose to investigate with our model is a @xmath50 qw . the parameters required for the simulation are listed in table [ tab : qw_parameters ] . the system is probed with pulses having a hyperbolic secant shape @xmath51 with the pulse energy @xmath52 . the full width half maximum ( fwhm ) of the pulse is @xmath53 . as function of energy @xmath54 of a single qw immersed in an infinitely extended background of gaas with ( black ) and without ( red ) coulomb interaction.[fig : onlyqwabsorption ] ] .parameters for the carrier dynamics in the qw , with the free electron mass @xmath55.[tab : qw_parameters ] [ cols="<,^,^ " , ] after studying the electron dynamics of an isolated qw in a homogeneous background we proceed by introducing a more realistic optical environment . for this we choose a sesam , which is a well established structure for ultra short pulse generation.@xcite the whole structure is included in our simulations as a spatially varying background permittivity @xmath56 , as shown in fig . [ fig : structure ] , and is surrounded by @xmath57 of air on each side . table [ tab : refractive_indices ] contains the refractive index and background permittivity of all materials included in the structure . in order for the structure to be effective , most of the layer thickness need to be proportional to the central wavelength of the incoming pulse , @xmath58 . we coated both ends of the structure with a sin layer of optical length @xmath59 which minimizes reflection of the pulse coming from air . this allows for a more efficient in - coupling of the light . the mirror is composed by a set of alternating gaas and alas layers , each with an optical length of @xmath59 . this basic two - layered module is repeated @xmath60 times in order to achieve a very high reflectivity around @xmath58 , as shown in fig . [ fig : qw_absorption](a ) , where the pulse is resonant with the exciton energy @xmath61 . due to the presence of the mirror , a standing wave is created inside the gaas layer between the mirror itself and the anti - reflective layer . such an interference pattern has zeros at even integer multiples of @xmath62 and maxima at odd ones . the qw is then located in such a maximum , in order to take advantage of the field enhancement provided by this interference pattern , and is thus at an optical distance @xmath59 from the start of the mirror . two further layers of arbitrary size are used to isolate the anti - reflection coatings from the mirror on one side and the qw on the other . ] we start again by analyzing the linear regime where we use a pulse with @xmath63 and a fwhm of @xmath64 . from the reflected and transmitted field we obtain the spectra of transmittance ( @xmath65 ) and reflectance ( @xmath66 ) of the structure , shown in fig . [ fig : qw_absorption](a ) . we see that the mirror used in the simulation reflects almost perfectly in a broad spectral region around the central wavelength of the pulse . we further calculate the absorption of the structure as @xmath67 , shown in fig . [ fig : qw_absorption](b ) . the absorption is mostly determined by the qw and thus we find a similar behavior as in fig . [ fig : onlyqwabsorption ] with a resonance at the exciton energy . the resonance height is increased by a factor of @xmath68 in comparison with the isolated qw , as a consequence of the almost four - fold enhancement in intensity introduced by the structure , while the in - band absorption is now decreasing the further we go from the band edge . the difference is due to the mirror which is optimized for the central wavelength of the pulse and whose reflectance decreases with the distance from the exciton . [ fig : thg_spectra ] ] next , we shine a set of subsequently stronger pulses on the sesam structure to investigate the non - linear regime . the pulses are resonant with the exciton ( @xmath69 ) and have a fwhm of @xmath70 . figure [ fig : thg_spectra ] shows the spectra of the reflected intensity for the same peak intensities used in fig . [ fig : onlyqwspectra ] , namely @xmath71 . similarly to what happened for the isolated qw , increasing the intensity of the exciting pulse brings a second spectral peak above the background . this is located at three times the pulse energy and is due to thg in the semiconductor layer of the structure . by comparing the spectra with fig . [ fig : onlyqwspectra ] we see that the third harmonic is more intense in the sesam structure than it is for an isolated qw excited with the same pulse . this is due to the structure enhancing the field at the qw position . the oscillations appearing in fig . [ fig : thg_spectra ] , particularly evident in the fundamental peak , are due to the fabry - prot resonance associated with the whole structure . as a function of the spectrally integrated intensity @xmath72 of the incoming pulse on a log - log scale for @xmath69 ( black squares ) and @xmath73 ( red circles ) . the dashed lines are the best fit of the data according to eq . [ eq : intensity_dependence_linear].[fig : thg_intensity ] ] a more quantitative analysis of the intensity is given in fig . [ fig : thg_intensity ] , where we plot the integrated intensity of the thg as function of the incoming pulse integrated intensity . the two sets of data correspond to different excitation energies , where one is obtained with pulses resonant with the exciton @xmath69 ( squares ) and one with off - resonant pulses with @xmath73 ( circles ) . the spectrally integrated intensity of the fundamental and its third harmonic are defined as the integral of the intensity over the corresponding spectral peak , @xmath74 where the integration is carried out over the width of the highest intensity peak . we find that the intensity of the thg has a power law behavior as function of the incoming field , @xmath75 , as @xmath76 in the log - log plot this is seen as a linear curve , where we can get the value of the exponent by performing a linear fit of the logarithm of the data according to @xmath77 which gives the dashed lines in fig . [ fig : thg_intensity ] . the values obtained for the exponents ( slope of the lines ) are @xmath78 for the off - resonant configuration and @xmath79 for the pulses resonant with the exciton energy . , obtained by fitting eq . [ eq : intensity_dependence_linear ] , as function of the pulse energy @xmath80 . the inset shows the residual density as a function of the spectrally integrated intensity for resonant and off - resonant excitations.[fig : thg_pulse ] ] figure [ fig : thg_pulse ] shows the value of the exponent @xmath81 as a function of the central wavelength of the pulse . similarly to fig . [ fig : thg_intensity ] we have performed a linear fit of eq . [ eq : intensity_dependence_linear ] to different sets of data . we see that when the excitation is enough off - resonant , i.e. , the pulse energy lies in the band gap of the semiconductor , the value of @xmath81 approaches an asymptotic value of @xmath82 which is consistent with a phenomenological description in terms of the non - linear susceptibility of third order , @xmath83.@xcite conversely , as the excitation gets closer to be resonant we observe a decrease in @xmath81 down to a value of about @xmath84 . a similar non cubic dependence has been observed by @xcite while performing four wave mixing experiments on znse qws with the central laser energy close to resonance with the exciton . we want to stress that we are able to uncover this unusual behavior only due to a self - consistent combination of the bloch equations with a description of the light field using a full time - domain ( fdtd ) code with spatial resolution on sub - wavelength scales . to further investigate the origin of this subcubic dependence , we analyzed the density of carriers generated by pulses with different detuning from the excitonic resonance . the inset of fig . [ fig : thg_pulse ] shows the density remaining in the semiconductor after the pulse has left the simulation domain as a function of the spectrally integrated intensity of the exciting pulse . we see that the amount of population generated in the qw is significantly higher under resonant excitation , even for the lowest intensity generating a thg signal , than the population for the off - resonant excitation via more intense pulses . also the amount of population excited in the qw rises linearly with the pulse intensity for off - resonant excitation , while it shows a marked saturation behavior under resonant excitation . in order to test whether the power law changes for smaller intensities our numerical simulation allows us to repeat the same analysis analyzing the transmitted field which has a lower level of background noise . we observe a crossover from @xmath85 for low intensities , to @xmath86 at higher intensities . we find that the minimum intensity for which an exponent different from 3 is obtained is @xmath87 . we have also checked that the crossover position is independent of the dephasing time . because of this and the correlation with the density of carriers in the qw , we attribute the change of the power law exponent @xmath81 to the presence of optically excited carriers and to the saturation of the total density in the semiconductor . in summary , we have studied the emergence of third harmonic signals in semiconductor quantum wells ( qw ) , photo - excited by intense femtosecond optical pusles . for this , we have introduced a general model combining a full time and space dependent finite - difference time - domain ( fdtd ) description of the light field , i.e. , a discretization of maxwell s equations without the inherent limitations of the slowly - varying envelope approximation , with a wave - vector resolved many level and many - body density matrix approach for the charge carrier dynamics . for a qw embedded in a homogeneous background we studied the interplay of light field dynamics and carrier dynamics , demonstrating the emergence of non - linear optical effects such as third harmonic generation ( thg ) . we further analyzed the intensity dependence of the generated non - linear response for a qw embedded in a many - layer semiconductor saturable absorber mirror ( sesam ) structure and show that the intensity dependence of the thg signal strongly varies with excitation frequency . for an excitation well below the band gap of the qw , we found that the intensity of the thg signal follows a cubic dependence on the intensity of the exciting pulse . this is in direct agreement with a description based on an expansion in powers of the field with non - linear susceptibilities as constant coefficients . for a resonant excitation at the excitonic frequency , however , the intensity dependence still follows a power - law , now with an exponent that is reduced to @xmath84 , clearly deviating from the cubic behavior . although a non - cubic dependence can also be obtained with a more phenomenological approach of an intensity dependent @xmath83 coefficient,@xcite this can only be fit to existing data rather than emerge from a more fundamental model . the simultaneous description of the light field and carrier dynamics not only allows for a deeper understanding of non - linear optical effects but is also readily expandable to other 2-dimensional semiconductor systems such as graphene , transition metal dichalcogenides@xcite or more complex structures like combined plasmonic - semiconductor structures.@xcite der gratefully acknowledges support from the german academic exchange service ( daad ) within the p.r.i.m.e . this study was partially support by the air force office of scientific research ( afosr ) , and the european office of aerospace research and development ( eoard ) is also acknowledged . 45ifxundefined [ 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 10.1103/revmodphys.70.145 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1038/ncomms5972 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , vol . ( , ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1063/1.4821158 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.92.217403 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.87.057401 [ * * , ( ) ] link:\doibase 10.1016/j.carbon.2006.02.035 [ * * , ( ) ] link:\doibase 10.1063/1.1521508 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , nanoscience and technology ( , , ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , ) @noop _ _ ( , ) @noop * * , ( ) http://dx.doi.org/10.1103/physrevb.90.245423 [ * * , ( ) ] http://dx.doi.org/10.1103/physrevb.92.235307 [ * * , ( ) ] link:\doibase 10.1126/science.1258479 [ * * , ( ) ]

non - linear phenomena in optically excited semiconductor structures are of high interest . we here develop a model capable of studying the dynamics of the photoexcited carriers , including coulomb interaction on a hartree - fock level , on the same footing as the dynamics of the light field impinging on an arbitrary photonic structure . applying this method to calculate the third harmonic generation in a semiconductor quantum well embedded in a bragg mirror structure , we find that the power - law exponent of the intensity dependence of the third harmonic generation depends on the frequency of the exciting pulse . off - resonant pulses follow the expected cubic dependence , while the exponent is smaller for resonant pulses due to saturation effects in the induced carrier density . our study provides a detailed understanding of the carrier and light field dynamics during non - linear processes .

a great deal of work in algebraic topology has exploited the generalised cohomology theory @xmath8 ( for spaces @xmath9 ) , which is known as complex cobordism ; good entry points to the literature include @xcite . this theory is interesting because of its connection with the theory of formal group laws ( fgl s ) , starting with quillen s fundamental theorem @xcite that @xmath10 is actually the universal example of a ring equipped with an fgl . suppose that we have a graded ring @xmath11 equipped with an fgl . in the cases discussed below , the fgl involved will generally be the universal example of an fgl with some interesting property . examples include the rings known to topologists as @xmath12 , @xmath13 , @xmath14 and @xmath15 ; see section [ sec - statements ] for the definitions . it is natural to ask whether there is a generalised cohomology theory @xmath16 whose value on a point is the ring @xmath11 , and a natural transformation @xmath17 , such that the resulting map @xmath18 carries the universal fgl over @xmath10 to the given fgl over @xmath11 . this question has a long history , and has been addressed by a number of different methods for different rings @xmath11 . the simplest case is when @xmath11 is obtained from @xmath10 by inverting some set @xmath19 of nonzero homogeneous elements , in other words @xmath20 . in that case the functor @xmath21 is a generalised cohomology theory on finite complexes , which can be extended to infinite complexes or spectra by standard methods . for example , given a prime @xmath22 one can invert all other primes to get a cohomology theory @xmath23 . cartier had previously introduced the notion of a @xmath22-typical fgl and constructed the universal example of such a thing over @xmath12 , which is a polynomial algebra over @xmath24 on generators @xmath25 in degree @xmath26 for @xmath27 . it was thus natural to ask our `` realisation question '' for @xmath28 . quillen @xcite constructed an idempotent self map @xmath29 , whose image is a subring , which we call @xmath30 . he showed that this is a cohomology theory whose value on a point is the ring @xmath12 , and that the fgl s are compatible in the required manner . this cohomology theory was actually defined earlier by brown and peterson @xcite ( hence the name ) , but in a less structured and precise way . it is not hard to check that we again have @xmath31 when @xmath9 is finite . this might tempt us to just define @xmath21 for any @xmath11 , but unfortunately this does not usually have the exactness properties required of a generalised cohomology theory . another major advance was landweber s determination @xcite of the precise conditions under which @xmath32 does have the required exactness properties , which turned out to be natural ones from the point of view of formal groups . however , there are many cases of interest in which landweber s exactness conditions are not satisfied , and for these different methods are required . many of them are of the form @xmath33 for set @xmath19 of homogeneous elements and some homogeneous ideal @xmath34 . for technical reasons things are easier if we assume that @xmath35 is generated by a regular sequence , in other words @xmath36 and @xmath37 is not a zero - divisor in @xmath38 . if @xmath11 arises in this way , we say that it is a _ localised regular quotient ( lrq ) _ of @xmath10 . if @xmath39 we say that @xmath11 is a _ regular quotient _ of @xmath10 . the first advance in this context was the baas - sullivan theory of cobordism of manifolds with singularities @xcite . given a regular quotient @xmath11 of @xmath10 , this theory constructed a cohomology theory @xmath16 , landing in the category of @xmath10-modules , and a map @xmath17 . unfortunately , the details were technically unwieldy , and it was not clear whether @xmath16 was unique or whether it had a natural product structure , and if so whether it was commutative or associative . some of these questions were addressed by shimada and yagita @xcite , mironov @xcite and morava @xcite , largely using the geometry of cobordisms . another idea was ( in special cases , modulo some technical details ) to calculate the group of all natural transformations @xmath40 and then see which of them are commutative , associative and unital . this was the approach of wrgler @xcite ; much more recently , nassau has corrected some inaccuracies and extended these results @xcite . baas - sullivan theory eventually yielded satisfactory answers for rings of the form @xmath41 , but the work involved in handling ideals with more than one generator remained rather hard . the picture changed dramatically with the publication of @xcite by elmendorf , kriz , mandell and may ( hereafter referred to as ekmm ) , which we now explain . firstly , the natural home for our investigation is not really the category of generalised cohomology theories , but rather boardman s homotopy category of spectra @xcite , which we call @xmath42 . there is a functor @xmath43 from finite complexes to @xmath42 , and any cohomology theory @xmath16 on finite complexes is represented by a spectrum @xmath44 in the sense that @xmath45 $ ] for all @xmath46 and @xmath9 . the representing spectrum @xmath47 is unique up to isomorphism @xcite , and the isomorphism is often unique . there have been many different constructions of categories equivalent to @xmath42 . the starting point of @xcite was ekmm s construction of a topological model category @xmath48 with a symmetric monoidal smash product , whose homotopy category is equivalent to @xmath42 . this was previously feared to be impossible , for subtle technical reasons @xcite . ekmm were also able to construct a version of @xmath0 which was a strictly commutative monoid in @xmath48 , which allowed them to define the category @xmath49 of @xmath0-modules . they showed how to make this into topological model category , and thus defined an associated homotopy category @xmath50 . this again has a symmetric monoidal smash product , which should be thought of as a sort of tensor product over @xmath0 . they showed that the problem of realising lrq s of @xmath10 becomes very much easier if we work in @xmath50 ( and then apply a forgetful functor to @xmath42 if required ) . in fact their methods work when @xmath0 is replaced by any strictly commutative monoid @xmath51 in @xmath48 such that @xmath52 is concentrated in even degrees . they show that if @xmath11 is an lrq of @xmath52 and @xmath6 is invertible in @xmath11 and @xmath11 is concentrated in degrees divisible by @xmath4 , then @xmath47 can be realised as a commutative and associative ring object in @xmath53 . in the present work , we will start by sharpening this slightly . the main point here is that ekmm notice an obstruction to associativity in @xmath54 , so they assume that these groups are zero . motivated by a parallel result in baas - sullivan theory @xcite , we show that the associativity obstructions are zero even if the groups are not ( see remark [ rem - ass ] ) . we deduce that if @xmath11 is an lrq of @xmath52 and @xmath6 is invertible in @xmath11 then @xmath47 can be realised as a commutative and associative ring in @xmath53 , in a way which is unique up to unique isomorphism ( theorem [ thm - odd - general ] ) . we also prove a number of subsidiary results about the resulting ring objects . the more substantial part of our work is the attempt to remove the condition that @xmath6 be invertible in @xmath11 , without which the results become somewhat more technical . we show that the obstruction to defining a commutative product on @xmath55 is given by @xmath56 for a certain power operation @xmath57 . this was again inspired by a parallel result of mironov @xcite . we deduce that if @xmath58 is an lrq of @xmath52 without @xmath6-torsion and @xmath59 then @xmath11 is again uniquely realisable ( theorem [ thm - even - general ] ) . when @xmath11 has @xmath6-torsion we have no such general result and must proceed case by case . again following mironov , we show that when @xmath60 , the operation @xmath61 can be computed using formal group theory . we considerably extend and sharpen mironov s calculations , using techniques which i hope will be useful in more general work on power operations . using these results , we show that many popular lrq s of @xmath62 have almost unique realisations as associative , almost commutative rings in @xmath50 . see theorems [ thm - muin ] and [ thm - pn ] for precise statements . the major exceptions are the rings @xmath63 and @xmath15 , but we show that even these become uniquely realisable as commutative rings in @xmath50 if we allow ourselves to modify the usual definition slightly . we call the resulting spectra @xmath64 and @xmath65 ; they are acceptable substitutes for @xmath66 and @xmath67 in almost all situations . we use the category @xmath48 of @xmath19-modules as constructed in @xcite ; we recall some details in section [ sec - found ] . the main point is that @xmath48 is a symmetric monoidal category with a closed model structure whose homotopy category is boardman s homotopy category of spectra . we shall refer to the objects of @xmath48 simply as spectra . because @xmath48 is a symmetric monoidal category , it makes sense to talk about strictly commutative ring spectra ; these are essentially equivalent to @xmath68 ring spectra in earlier foundational settings . let @xmath51 be such an object , such that @xmath69 is even ( by which we mean , concentrated in even degrees ) . we also assume that @xmath51 is @xmath70-cofibrant in the sense of ( * ? ? ? * chapter vii ) ( if not , we replace @xmath51 by a weakly equivalent cofibrant model ) . the main example of interest to us is @xmath60 . there are well - known constructions of @xmath0 as a spectrum in the earlier sense of lewis and may @xcite , with an action of the @xmath68 operad of complex linear isometries . thus , the results of ( * ? ? ? * chapter ii ) allow us to construct @xmath0 as a strictly commutative ring spectrum . one can define a category @xmath71 of @xmath51-modules in the evident way , with all diagrams commuting at the geometric level . after inverting weak equivalences , we obtain a homotopy category @xmath72 , referred to as the derived category of @xmath71 . we shall mainly work in this derived category , and the category @xmath73 of ring objects in @xmath74 ( referred to in @xcite as @xmath51-ring spectra ) . all our ring objects are assumed to be associative and to have a two - sided unit . thus , an object @xmath75 has an action @xmath76 which makes various diagrams commute at the geometric level , and a product @xmath77 that is geometrically compatible with the @xmath51-module structures , and is homotopically associative and unital . we also write @xmath78 for the category of algebras over the discrete ring @xmath79 . we write @xmath80 for the category of even @xmath79-algebras , and @xmath81 for the commutative ones , and similarly @xmath82 , @xmath83 , @xmath84 and @xmath85 . let @xmath86 be an even commutative @xmath79-algebra without @xmath6-torsion . a _ strong realisation _ of @xmath86 is a commutative ring object @xmath87 with a given isomorphism @xmath88 , such that the resulting map @xmath89 is an isomorphism whenever @xmath90 and @xmath91 has no @xmath6-torsion . we say that @xmath86 is _ strongly realisable _ if such a realisation exists . it is easy to see that the category of strongly realisable @xmath79-algebras is equivalent to the category of those @xmath87 for which @xmath92 is strongly realisable . in particular , any two strong realisations of @xmath86 are canonically isomorphic . our main aim is to prove that certain @xmath79-algebras are strongly realisable , and to prove some more _ ad hoc _ results for certain algebras over @xmath93 . a _ localised regular quotient ( lrq ) _ of @xmath79 is an algebra @xmath86 over @xmath79 that can be written in the form @xmath94 , where @xmath19 is any set of ( homogeneous ) elements in @xmath79 and @xmath35 is an ideal which can be generated by a regular sequence . we say that @xmath86 is a _ positive localised regular quotient ( plrq ) _ if it can be written in the form @xmath95 as above , where @xmath35 can be generated by a regular sequence of elements of nonnegative degree . if @xmath86 is an lrq of @xmath79 and @xmath91 is an arbitrary @xmath79-algebra then @xmath96 has at most one element . suppose that @xmath47 is a commutative ring object in @xmath87 with a given isomorphism @xmath88 . it follows that @xmath47 is a strong realisation of @xmath86 if and only if : whenever there is a map @xmath97 of @xmath79-algebras , there is a unique map @xmath98 in @xmath85 . [ rem - localisation ] let @xmath19 be a set of homogeneous elements in @xmath79 . using the results of ( * section viii.2 ) one can construct a strictly commutative ring spectrum @xmath99 and a map @xmath100 inducing an isomorphism @xmath101 . results of wolbert show that @xmath102 is equivalent to the subcategory of @xmath53 consisting of objects @xmath103 such that each element of @xmath19 acts invertibly on @xmath104 . using this it is easy to check that any algebra over @xmath105 is strongly realisable over @xmath51 if and only if it is strongly realisable over @xmath99 . for more discussion of this , see section [ sec - realise ] . we start by stating a result for odd primes , which is relatively easy . [ thm - odd - general ] if @xmath86 is an lrq of @xmath79 and @xmath6 is a unit in @xmath86 then @xmath86 is strongly realisable . this will be proved as theorem [ thm - odd - proof ] . our main contribution is the extension to the case where @xmath6 is not inverted . our results involve a certain `` commutativity obstruction '' @xmath106 , which is defined in section [ sec - prod - rx ] . in section [ sec - pow - op ] , we show that when @xmath107 this arises from a power operation @xmath108 . this result was inspired by a parallel result of mironov in baas - sullivan theory @xcite . the restriction @xmath107 is actually unneccessary but the argument for the case @xmath109 is intricate and we have no applications so we have omitted it . in section [ sec - formal ] we show how to compute this power operation using formal group theory , at least in the case @xmath60 . the first steps in this direction were also taken by mironov @xcite , but our results are much more precise . by remark [ rem - localisation ] we also have a power operation @xmath110 . this is in fact determined algebraically by the power operation on @xmath111 , as we will see in section [ sec - formal ] . our result for the case where @xmath86 has no @xmath6-torsion is quite simple and similar to the case where @xmath6 is inverted . [ thm - even - general ] let @xmath94 be a plrq of @xmath79 which has no @xmath6-torsion . suppose also that @xmath112 maps to @xmath113 in @xmath114 . then @xmath86 is strongly realisable . this will be proved as theorem [ thm - even - proof ] . we next recall the definitions of some algebras over @xmath7 which one might hope to realise as spectra using the above results . first , we have the rings @xmath115 \qquad\qquad |u| = 2 \\ ku _ * & { : = } { { \mathbb{z}}}[u^{\pm 1 } ] \\ h _ * & { : = } { { \mathbb{z}}}_{\hphantom{p } } \qquad\text{(in degree zero)}\\ h{{\mathbb{f } } } _ * & { : = } { { \mathbb{f}_p}}\qquad\text{(in degree zero)}.\end{aligned}\ ] ] these are plrq s of @xmath7 in well - known ways . next , we consider the brown - peterson ring @xmath116 \qquad\qquad |v_k|=2(p^k-1).\ ] ] we take @xmath117 as usual . there is a unique @xmath22-typical formal group law @xmath118 over this ring such that @xmath119_f(x ) = \exp_f(px ) + _ f \sum^f_{k>0 } v_k x^{p^k}.\ ] ] ( thus , our @xmath25 s are hazewinkel s generators rather than araki s . ) we use this fgl to make @xmath120 into an algebra over @xmath7 in the usual way . we define @xmath121 \\ b(n ) _ * & { : = } v_n^{-1}bp_*/(v_i{\;|\;}i < n ) = v_n^{-1}{{\mathbb{f}_p}}[v_j{\;|\;}j\ge n]\\ k(n ) _ * & { : = } bp_*/(v_i{\;|\;}i\neq n ) = { { \mathbb{f}_p}}[v_n ] \\ k(n ) _ * & { : = } v_n^{-1}bp_*/(v_i{\;|\;}i\neq n ) = { { \mathbb{f}_p}}[v_n^{\pm 1 } ] \\ { bp\langle n\rangle}_*&{:=}bp_*/(v_i{\;|\;}i > n ) = { { \mathbb{z}_{(p)}}}[v_1,\ldots , v_n ] \\ e(n ) _ * & { : = } v_n^{-1}bp_*/(v_i{\;|\;}i > n ) = { { \mathbb{z}_{(p)}}}[v_1,\ldots , v_{n-1},v_n^{\pm 1}]\end{aligned}\ ] ] these are all plrq s of @xmath120 , and it is not hard to check that @xmath120 is a plrq of @xmath122 , and thus that all the above rings are plrq s of @xmath122 . we also let @xmath123 denote the bordism class of a smooth hypersurface @xmath124 of degree @xmath22 in @xmath125 . it is well - known that @xmath126 is the smallest ideal modulo which the universal formal group law over @xmath7 has height @xmath46 , and that the image of @xmath127 in @xmath120 is the ideal @xmath128 . in fact , we have @xmath129 x^m dx = [ p]_f(x ) d\log_f(x ) = [ p]_f(x ) \sum_{m\geq 0 } [ { { \mathbb{c}p}}^m ] x^m dx.\ ] ] moreover , the sequence of @xmath130 s is regular , so that @xmath131 is a plrq of @xmath7 . one can also define plrq s of @xmath132_*$ ] giving rise to various versions of elliptic homology , but we refrain from giving details here . if we do not invert @xmath133 then the relevant rings seem not to be lrq s of @xmath7 . if we take @xmath134 then we can make @xmath135 $ ] into an lrq of @xmath79 in such a way that the resulting formal group law is of the ( non-@xmath22-typical ) type considered by lubin and tate in algebraic number theory . we can also take @xmath136 and consider @xmath137 as an lrq of @xmath79 via the ando orientation @xcite rather than the more usual @xmath22-typical one . we leave the details of these applications to the reader . the following proposition is immediate from theorem [ thm - odd - general ] . if @xmath138 and @xmath60 or @xmath139 then @xmath140 , @xmath141 , @xmath142 , @xmath143 , @xmath120 , @xmath144 , @xmath145 , @xmath146 , @xmath147 , @xmath148 and @xmath131 are all strongly realisable . after doing some computations with the power operation @xmath61 , we will also prove the following . [ prop - mu - omni ] if @xmath60 then @xmath149 , @xmath150 , @xmath151 and @xmath152 are strongly realisable . if @xmath153 then @xmath154 , @xmath155 , @xmath156 and @xmath120 are strongly realisable . the situation is less satisfactory for the rings @xmath147 and @xmath148 at @xmath157 . for @xmath158 , they can not be realised as the homotopy rings of commutative ring objects in @xmath74 . however , if we kill off a slightly different sequence of elements instead of the sequence @xmath159 , we get a quotient ring that is realisable . the resulting spectrum serves as a good substitute for @xmath66 in almost all arguments . [ prop - bpn ] if @xmath153 and @xmath160 , there is a quotient ring @xmath161 of @xmath120 such that 1 . the evident map @xmath162 { \xrightarrow{}}bp _ * { \xrightarrow{}}{bp\langle n\rangle}'_*\ ] ] is an isomorphism . @xmath161 is strongly realisable . we have @xmath163 as @xmath7-algebras . moreover , the ring @xmath164 is also strongly realisable . if @xmath165 then we can take @xmath166 . this is proved in section [ sec - mu ] . the situation for @xmath93 and algebras over it is also more complicated than for odd primes . throughout this paper , we write @xmath167 for the twist map @xmath168 , for any object @xmath9 for which this makes sense . we say that a ring map @xmath169 in @xmath170 is _ central _ if @xmath171 where @xmath172 is the product . we say that @xmath173 is a _ central @xmath47-algebra _ if there is a given central map @xmath98 . [ thm - muin ] when @xmath153 , there is a ring @xmath174 with @xmath175 , and derivations @xmath176 for @xmath177 . if @xmath178 is the product on @xmath179 we have @xmath180 this is proved in section [ sec - mu ] . there are actually many non - isomorphic rings with these properties . we will outline an argument that specifies one of them unambiguously . we get a sharper statement for algebras over @xmath144 . [ thm - pn ] when @xmath153 , there is a central @xmath1-algebra @xmath181 and an isomorphism @xmath182 . this has derivations @xmath183 for @xmath177 . if @xmath178 is the product on @xmath184 we have @xmath185 if @xmath173 is another central @xmath1-algebra such that @xmath186 then either there is a unique map @xmath187 of @xmath1-algebras , or there is a unique map @xmath188 . analogous statements hold for @xmath189 , @xmath190 and @xmath2 with @xmath1 replaced by @xmath191 , @xmath64 and @xmath65 respectively . this is also proved in section [ sec - mu ] . related results were announced by wrgler in @xcite , but there appear to be some problems with the line of argument used there . a correct proof on similar lines has recently been given by nassau @xcite . suppose that @xmath192 is not a zero - divisor ( so @xmath193 is even ) . we then have a cofibre sequence in the triangulated category @xmath74 : @xmath194 because @xmath195 is not a zero divisor , we have @xmath196 . in particular , @xmath197 ( because @xmath198 is odd ) , and thus @xmath199\simeq[r , r / x]$ ] . it follows that @xmath55 is unique up to unique isomorphism as an object under @xmath51 . we next set up a theory of products on objects of the form @xmath55 . apart from the fact that all such products are associative , our results are at most minor sharpenings of the those in ( * ? ? ? * chapter v ) . observe that @xmath200 is a cell @xmath51-module with one @xmath113-cell , two @xmath201-cells and one @xmath202-cell . we say that a map @xmath203 is a _ product _ if it agrees with @xmath204 on the bottom cell , in other words @xmath205 . the main result is as follows . [ prop - rx ] 1 . all products are associative , and have @xmath204 as a two - sided unit . 2 . the set of products on @xmath55 has a free transitive action of the group @xmath206 ( in particular , it is nonempty ) . 3 . there is a naturally defined element @xmath207 such that @xmath55 admits a commutative product if and only if @xmath208 . if so , the set of commutative products has a free transitive action of @xmath209 . if @xmath107 there is a power operation @xmath210 such that @xmath211 for all @xmath195 . part ( 1 ) is proved as lemma [ lem - unital ] and proposition [ prop - ass ] . in part ( 2 ) , the fact that products exist is ( * ? ? ? * theorem v.2.6 ) ; we also give a proof in corollary [ cor - products - exist ] , which is slightly closer in spirit with our other proofs . parts ( 3 ) and ( 4 ) form corollary [ cor - comm ] . part ( 5 ) is explained in more detail and proved in section [ sec - pow - op ] . from now on we will generally state our results in terms of @xmath56 instead of @xmath212 , as that is the form in which the results are actually applied . [ lem - x - zero ] the map @xmath213 is zero . using the cofibration @xmath214 and the fact that @xmath215 , we find that @xmath199_d{\xrightarrow}{}[r , r / x]_d=\pi_d(r / x)$ ] is injective . it is clear that @xmath195 gives zero on the right hand side , so it is zero on the left hand side as claimed . [ cor - products - exist ] there exist products on @xmath55 . there is a cofibration @xmath216 . the lemma tells us that the first map is zero , so @xmath217 is a split monomorphism , and any splitting is clearly a product . [ lem - unital ] if @xmath203 is a product then @xmath204 is a two - sided unit for @xmath178 , in the sense that @xmath218 by hypothesis , @xmath219 is the identity on the bottom cell of @xmath55 . we observed earlier that @xmath220\simeq[r , r / x]$ ] , and it follows that @xmath221 . similarly @xmath222 . ekmm study products for which @xmath204 is a one - sided unit , and our definition of products is _ a priori _ even weaker . it follows from the lemma that ekmm s products are the same as ours and have @xmath204 as a two - sided unit . [ lem - a - split ] let @xmath223 be such that @xmath224 is zero . then the diagram @xmath225 induces a left - exact sequence @xmath226 { \xrightarrow{}}[{(r / x)^{(2)}},a ] { \xrightarrow } { } [ r / x{\vee}r / x , a].\ ] ] similarly , the diagram @xmath227 gives a left - exact sequence @xmath228 { \xrightarrow{}}[{(r / x)^{(3)}},a ] { \xrightarrow } { } [ { ( r / x)^{(2)}}{\vee}{(r / x)^{(2)}}{\vee}{(r / x)^{(2)}},a].\ ] ] consider the following diagram : @xmath229 we now apply the functor @xmath230 $ ] and make repeated use of the cofibration @xmath231 the conclusion is that all maps involving @xmath232 become monomorphisms , all maps involving @xmath204 become epimorphisms , and the bottom row and the middle column become short exact . the first claim follows by diagram chasing . for the second claim , consider the diagram @xmath233 we apply the same logic as before , using the first claim ( with @xmath47 replaced by @xmath234 ) to see that the middle column becomes left exact . we next determine how many different products there are on @xmath55 . [ lem - uni - obs ] if @xmath178 is a product on @xmath55 and @xmath235 $ ] then @xmath236 is another product . moreover , this construction gives a free transitive action of @xmath237 on the set of all products . let @xmath238 be the set of products . as @xmath239 , it is clear that the above construction gives an action of @xmath237 on @xmath238 . now suppose that @xmath240 . we need to show that there is a unique @xmath241 such that @xmath236 . using the unital properties of @xmath178 and @xmath242 given by lemma [ lem - unital ] , we see that @xmath243 because of lemma [ lem - x - zero ] , we can apply lemma [ lem - a - split ] to see that @xmath244 for a unique element @xmath245 , as claimed . [ prop - ass ] any product on @xmath55 is associative . let @xmath178 be a product , and write @xmath246 so the claim is that @xmath247 is nullhomotopic . using the unital properties of @xmath178 we see that @xmath248 using lemma [ lem - a - split ] , we conclude that @xmath249 for a unique element @xmath250=\pi_{3d+3}(r)/x=0 $ ] ( because @xmath251 is odd ) . thus @xmath252 as claimed . the corresponding result in baas - sullivan theory was already known ( this is proved in @xcite in a form which is valid when @xmath79 need not be concentrated in even degrees , for example for @xmath253 ) . [ rem - ass ] the ekmm approach to associativity is essentially as follows . they note that @xmath55 has cells of dimension @xmath113 and @xmath198 , so @xmath254 has cells in dimensions @xmath113 , @xmath198 , @xmath255 and @xmath251 . the map @xmath247 vanishes on the zero - cell and @xmath256 so the only obstruction to concluding that @xmath252 lies in @xmath257 . ekmm work only with lrq s that are concentrated in degrees divisible by @xmath4 , so the obstruction goes away . we instead use lemma [ lem - a - split ] to analyse the attaching maps in @xmath254 ; implicitly , we show that the obstruction is divisible by @xmath195 and thus is zero . we now discuss commutativity . [ lem - comm ] there is a natural map @xmath258 from the set of products to @xmath259 such that @xmath260 if and only if @xmath178 is commutative . moreover , @xmath261 let @xmath262 be the twist map . clearly , if @xmath178 is a product then so is @xmath263 . thus , there is a unique element @xmath264 such that @xmath265 we define @xmath266 . next , recall that the twist map on @xmath267 is homotopic to @xmath268 , because @xmath198 is odd . it follows by naturality that @xmath269 . consider a second product @xmath236 . we now see that @xmath270 thus @xmath271 as claimed . [ cor - comm ] there is a naturally defined element @xmath207 such that @xmath55 admits a commutative product if and only if @xmath208 . if so , the set of commutative products has a free transitive action of the group @xmath272 . in particular , if @xmath273 has no @xmath6-torsion then there is a unique commutative product . we choose a product @xmath178 on @xmath55 and define @xmath274 . this is well - defined , by the lemma . if @xmath275 then @xmath276 for all @xmath242 , so there is no commutative product . if @xmath208 then @xmath277 , say , so that @xmath278 is a commutative product . in this case , the commutative products are precisely the products of the form @xmath279 where @xmath280 , so they have a free transitive action of @xmath281 . next , we consider the bockstein operation : @xmath282 let @xmath75 be a ring , with product @xmath283 . we say that a map @xmath284 is a _ derivation _ if we have @xmath285 [ prop - der ] the map @xmath286 is a derivation with respect to any product @xmath178 on @xmath55 . write @xmath287 , so the claim is that @xmath252 . it is easy to see that @xmath288 , so by lemma [ lem - a - split ] we see that @xmath247 factors through a unique map @xmath289 . this is an element of @xmath290 , which is zero because @xmath198 is odd . we end this section by analysing maps out of the rings @xmath55 . [ prop - maps - rx ] let @xmath291 be an even ring . if @xmath195 maps to zero in @xmath292 then there is precisely one unital map @xmath293 , and otherwise there are no such maps . if @xmath294 exists and @xmath178 is a product on @xmath55 , then there is a naturally defined element @xmath295 such that * @xmath296 if and only if @xmath294 is a ring map with respect to @xmath178 . * @xmath297 . * if @xmath47 is commutative then @xmath298 . the statement about the existence and uniqueness of @xmath294 follows immediately from the cofibration @xmath299 , and the fact that @xmath300 . suppose that @xmath294 exists ; it follows easily using the product structure on @xmath47 that @xmath224 is zero . now let @xmath301 be the given product on @xmath47 , and let @xmath178 be a product on @xmath55 . consider the map @xmath302 by the usual argument , we have @xmath303 for a unique map @xmath304 . we define @xmath305 . it is obvious that this vanishes if and only if @xmath294 is a ring map , and that @xmath297 . now suppose that @xmath47 is commutative , so @xmath306 . on the one hand , using the fact that @xmath307 we see that @xmath308 . on the other hand , from the definition of @xmath247 and the fact that @xmath309 , we see that @xmath310 because @xmath311 $ ] is a split monomorphism , we conclude that @xmath312 in @xmath313 . in this section we assemble the products which we have constructed on the @xmath51-modules @xmath55 to get products on more general @xmath79-algebras . we will work entirely in the derived category @xmath74 , rather than the underlying geometric category . all the main ideas in this section come from ( * ? ? ? * chapter v ) . we start with some generally nonsensical preliminaries . given a diagram @xmath314 in @xmath170 , we say that @xmath294 commutes with @xmath315 if and only if we have @xmath316 note that this can be false when @xmath317 ; in particular @xmath47 is commutative if and only if @xmath318 commutes with itself . the next three lemmas become trivial if we replace @xmath74 by the category of modules over a commutative ring , and the smash product by the tensor product . the proofs in that context can easily be made diagrammatic and thus carried over to @xmath74 . [ lem - ring - smash ] if @xmath47 and @xmath173 are rings in @xmath170 , then there is a unique ring structure on @xmath319 such that the evident maps @xmath320 are commuting ring maps . moreover , with this product , @xmath321 is the universal example of a commuting pair of maps out of @xmath47 and @xmath173 . [ lem - comm - test ] a map @xmath322 commutes with itself if and only if @xmath323 commutes with itself and @xmath324 commutes with itself . in particular , @xmath319 is commutative if and only if @xmath325 and @xmath326 commute with themselves . [ lem - comm - smash ] if @xmath47 and @xmath173 are commutative , then so is @xmath319 , and it is the coproduct of @xmath47 and @xmath173 in @xmath84 . [ cor - tensor ] suppose that * @xmath47 and @xmath173 are strong realisations of @xmath86 and @xmath91 . * the ring @xmath327 has no @xmath6-torsion . * the natural map @xmath328 is an isomorphism . then @xmath319 is a strong realisation of @xmath327 . we next consider the problem of realising @xmath105 , where @xmath19 is a set of homogeneous elements of @xmath79 . if @xmath19 is countable then we can construct an object @xmath329 by the method of ( * ? ? ? * section v.2 ) ; this has @xmath330 . if we want to allow @xmath19 to be uncountable then it seems easiest to construct @xmath99 as the finite localisation of @xmath51 away from the @xmath51-modules @xmath331 ; see @xcite or ( * ? ? ? * theorem 3.3.7 ) . in either case , we note that @xmath99 is the bousfield localisation of @xmath51 in @xmath74 with respect to @xmath99 . we may thus use ( * ? ? ? * section viii.2 ) to construct a model of @xmath99 which is a strictly commutative algebra over @xmath51 in the underlying topological category of spectra . the localisation functor involved here is smashing , so results of wolbert @xcite ( * ? ? ? * section viii.3 ) imply that @xmath102 is equivalent to the full subcategory of @xmath53 consisting of @xmath51-modules @xmath103 for which @xmath104 is a module over @xmath105 . this makes the following result immediate . [ prop - s - inv ] let @xmath19 be a set of homogeneous elements of @xmath79 , and let @xmath86 be an algebra over @xmath105 . then @xmath86 is strongly realisable over @xmath51 if and only if it is strongly realisable over @xmath99 . this allows us to reduce everything to the case @xmath39 . now consider a sequence @xmath332 in @xmath79 , with products @xmath333 on @xmath334 . write @xmath335 , and make this into a ring as in lemma [ lem - ring - smash ] . there are evident maps @xmath336 , so we can form the telescope @xmath337 . [ lem - lim - one ] if @xmath338 and @xmath339 acts trivially on @xmath103 and @xmath340 then @xmath341={\operatornamewithlimits{\underset{\longleftarrow}{lim}}}_i[a^{(r)}_i , m]$ ] . this will follow immediately from the milnor sequence if we can show that @xmath342_*=0 $ ] . for this , it suffices to show that the map @xmath343{\xrightarrow}{}[b , m]$ ] is surjective for all @xmath173 . this follows from the cofibration @xmath344 and the fact that @xmath345 acts trivially on @xmath103 . [ prop - coprod ] let @xmath332 be a sequence in @xmath79 , and @xmath333 a product on @xmath334 for each @xmath325 . let @xmath47 be the homotopy colimit of the rings @xmath335 , and let @xmath346 be the evident map . then there is a unique associative and unital product on @xmath47 such that maps @xmath347 are ring maps , and @xmath347 commutes with @xmath348 when @xmath349 . this product is commutative if and only if each @xmath347 commutes with itself . ring maps from @xmath47 to any ring @xmath173 biject with systems of ring maps @xmath350 such that @xmath351 commutes with @xmath352 for all @xmath349 . because @xmath334 admits a product , we know that @xmath345 acts trivially on @xmath334 . because @xmath47 has the form @xmath353 , we see that @xmath345 acts trivially on @xmath47 . thus @xmath35 acts trivially on @xmath47 , and lemma [ lem - lim - one ] assures us that @xmath354={\operatornamewithlimits{\underset{\longleftarrow}{lim}}}_i[a_i^{(r)},a]$ ] . let @xmath355 be the product on @xmath356 . by the above , there is a unique map @xmath357 which is compatible with the maps @xmath355 . it is easy to check that this is an associative and unital product , and that it is the only one for which the @xmath347 are commuting ring maps . it is also easy to check that @xmath301 is commutative if and only if each of the maps @xmath358 commutes with itself , if and only if each @xmath347 commutes with itself . now let @xmath173 be any ring in @xmath170 . we may assume that each @xmath345 maps to zero in @xmath359 , for otherwise the claimed bijection is between empty sets . as @xmath173 is a ring , this means that each @xmath345 acts trivially on @xmath173 , so that @xmath360={\operatornamewithlimits{\underset{\longleftarrow}{lim}}}_i[a_i^{(r)},b]$ ] . we see from lemma [ lem - ring - smash ] that ring maps from @xmath356 to @xmath173 biject with systems of ring maps @xmath361 for @xmath362 such that @xmath352 commutes with @xmath363 for @xmath364 . the claimed description of ring maps @xmath98 follows easily . [ cor - coprod ] if each @xmath334 is commutative , then @xmath47 is the coproduct of the @xmath334 in @xmath84 . if the sequence @xmath332 is regular , then it is easy to see that @xmath365 . note also that ring maps out of @xmath55 were analysed in proposition [ prop - maps - rx ] . we now restate and prove theorems [ thm - odd - general ] and [ thm - even - general ] . of course , the former is a special case of the latter , but it seems clearest to prove theorem [ thm - odd - general ] first and then explain the improvements necessary for theorem [ thm - even - general ] . [ thm - odd - proof ] if @xmath86 is an lrq of @xmath79 and @xmath6 is a unit in @xmath86 then @xmath86 is strongly realisable . we can use proposition [ prop - s - inv ] to reduce to the case where @xmath366 where @xmath6 is invertible in @xmath79 and @xmath35 is generated by a regular sequence @xmath367 . we know from proposition [ prop - rx ] that there is a unique commutative product @xmath333 on @xmath334 . if @xmath368 and @xmath369 in @xmath370 then in the notation of proposition [ prop - maps - rx ] we have @xmath371 and thus @xmath372 , so the unique unital map @xmath373 is a ring map . it follows that @xmath334 is a strong realisation of @xmath374 , and thus that @xmath335 is a strong realisation of @xmath375 . using proposition [ prop - coprod ] , we get a ring @xmath47 which is a strong realisation of @xmath376 . we next address the case where @xmath6 is not a zero - divisor , but is not invertible either . [ thm - even - proof ] let @xmath94 be a plrq of @xmath79 which has no @xmath6-torsion . suppose also that @xmath112 maps to @xmath113 in @xmath114 , where @xmath210 is the power operation defined in section [ sec - pow - op ] . then @xmath86 is strongly realisable . after using proposition [ prop - s - inv ] , we may assume that @xmath39 . choose a regular sequence @xmath332 generating @xmath35 . as @xmath377 , we can choose a product @xmath333 on @xmath334 such that @xmath378 . we let @xmath47 be the `` infinite smash product '' of the @xmath334 , as in proposition [ prop - coprod ] , so that @xmath379 . because @xmath380 maps to zero in @xmath92 , we see easily that the map @xmath381 commutes with itself . by proposition [ prop - coprod ] , we conclude that @xmath47 is commutative . let @xmath90 be an even commutative ring , and that @xmath359 has no @xmath6-torsion . the claim is that @xmath382 . the right hand side has at most one element , and if it is empty , then the left hand side is also . thus , we may assume that there is a map @xmath97 of @xmath79-algebras , and we need to show that there is a unique ring map @xmath98 . by proposition [ prop - coprod ] , we know that ring maps @xmath98 biject with systems of ring maps @xmath383 ( which automatically commute as @xmath173 is commutative ) . there is a unique unital map @xmath384 , and proposition [ prop - maps - rx ] tells us that the obstruction to @xmath294 being a homomorphism satisfies @xmath385 . because @xmath359 has no @xmath6-torsion , we have @xmath386 , so there is a unique ring map @xmath383 , and thus a unique ring map @xmath98 as required . the following result is also useful . [ prop - free ] let @xmath86 be a strongly realisable @xmath79-algebra , and let @xmath387 be a map of @xmath79-algebras that makes @xmath91 into a free module over @xmath86 . then @xmath91 is strongly realisable . first , observe that if @xmath118 and @xmath103 are @xmath47-modules , there is a natural map @xmath388 which is an isomorphism if @xmath118 is a wedge of suspensions of @xmath47 ( in other words , a free @xmath47-module ) . choose a homogeneous basis @xmath389 for @xmath91 over @xmath86 , where @xmath390 has degree @xmath391 . define @xmath392 , so that @xmath173 is a free @xmath47-module with a given isomorphism @xmath393 of @xmath86-modules . define @xmath394 and @xmath395 and @xmath396 the product map @xmath397 gives rise to evident maps @xmath398 which in turn give isomorphisms @xmath399 of @xmath86-modules . the multiplication map @xmath400 corresponds under the isomorphism @xmath401 to a map @xmath402 . after composing this with @xmath403 , we get a product map @xmath404 . a similar procedure gives a unit map @xmath98 . we next prove that this product is associative . each of the two associated products @xmath405 factors as @xmath406 followed by a map @xmath407 , corresponding to a map @xmath408 . the two maps @xmath408 in question are just the two possible associated products , which are the same because @xmath91 is associative . it follows that @xmath173 is associative . similar arguments show that @xmath173 is commutative and unital . now consider an object @xmath409 equipped with a map @xmath410 ( and thus a map @xmath411 ) . as @xmath47 is a strong realisation of @xmath86 , there is a unique map @xmath412 compatible with the map @xmath411 . this makes @xmath413 into an @xmath47-module , and thus gives an isomorphism @xmath414 . there is thus a unique @xmath47-module map @xmath415 inducing the given map @xmath410 . it follows easily that @xmath91 is a strong realisation of @xmath91 . we will need to consider certain @xmath79-algebras that are not strongly realisable . the following result assures us that weaker kinds of realisation are not completely uncontrolled . [ prop - weak - unique ] let @xmath86 be an lrq of @xmath79 , and let @xmath416 be rings ( not necessarily commutative ) such that @xmath417 . then there is an isomorphism @xmath418 ( not necessarily a ring map ) that is compatible with the unit maps @xmath419 . we may as usual assume that @xmath39 , and write @xmath36 . let @xmath47 be the infinite smash product of the @xmath334 s , so that @xmath379 . it will be enough to show that there is a unital isomorphism @xmath98 . moreover , any unital map @xmath98 is automatically an isomorphism , just by looking at the homotopy groups . there is a unique unital map @xmath420 . write @xmath335 , and let @xmath351 be the map @xmath421 where the second maps is the product . because @xmath173 is a ring and each @xmath345 goes to zero in @xmath359 , we can apply lemma [ lem - lim - one ] to get a unital map @xmath422 as required . we conclude this section by investigating @xmath51-module maps @xmath423 for various @xmath51-algebras @xmath75 . [ prop - self - maps ] let @xmath424 be a regular sequence in @xmath79 , let @xmath333 be a product on @xmath334 , and let @xmath47 be the infinite smash product of the rings @xmath334 . let @xmath425 be obtained by smashing the bockstein map @xmath426 with the identity map on all the other @xmath427 s . then @xmath428 is isomorphic as an algebra over @xmath86 to the completed exterior algebra on the elements @xmath429 . it is not hard to see that @xmath430 , with a sign coming from an implicit permutation of suspension coordinates . we also have @xmath431 and thus @xmath432 . given any finite subset @xmath433 of the positive integers , we define @xmath434 where @xmath435 . the claim is that one can make sense of homogeneous infinite sums of the form @xmath436 with @xmath437 , and that any graded map @xmath423 of @xmath51-modules is uniquely of that form . write @xmath438 , and let @xmath439 be the evident map . it is easy to check that @xmath440 if @xmath441 , and a simple induction shows that @xmath442 is a free module over @xmath86 generated by the maps @xmath443 for which @xmath444 . moreover , lemma [ lem - lim - one ] implies that @xmath445 . the claim follows easily . the above result relies more heavily than one would like on the choice of a regular sequence generating the ideal @xmath446 . we will use the following construction to make things more canonical . [ cons - dq ] let @xmath291 be an even ring , with unit @xmath447 , and let @xmath35 be the kernel of @xmath448 . given a derivation @xmath284 , we define a function @xmath449 as follows . given @xmath450 , we have a cofibration @xmath451 as usual . here @xmath195 may be a zero - divisor in @xmath79 , so we need not have @xmath452 . nonetheless , we see easily that there is a unique map @xmath453 such that @xmath454 . as @xmath455 is a derivation , one checks easily that @xmath456 , so @xmath457 , so @xmath458 for some @xmath459 . because @xmath195 acts as zero on @xmath47 , we see that @xmath460 is unique . we can thus define @xmath461 . [ prop - dq ] let @xmath291 be such that @xmath462 , where @xmath35 can be generated by a regular sequence . let @xmath463 be the set of derivations @xmath423 . then construction [ cons - dq ] gives rise to a natural monomorphism @xmath464 ( with degrees shifted by one ) . choose a regular sequence @xmath424 generating @xmath35 . write @xmath438 , and let @xmath465 be the map @xmath466 it is easy to see that @xmath47 is the homotopy colimit of the objects @xmath467 ( although there may not be a ring structure on @xmath467 for which @xmath465 is a homomorphism ) . we also write @xmath468 for the smash product of the @xmath427 for which @xmath469 and @xmath470 , and @xmath471 for the evident map @xmath472 . consider a derivation @xmath284 , and write @xmath473 . because @xmath455 is a derivation , we see that @xmath474 is a sum of @xmath46 terms , of which the @xmath325th is @xmath475 times the composite @xmath476 now consider an element @xmath477 of @xmath35 . it is easy to see that there is a unique unital map @xmath478 , and that @xmath479 . now consider the following diagram . @xmath480 the left hand square commutes because the terms @xmath481 for @xmath470 become zero in @xmath482 . it follows that there exists a map @xmath483 making the whole diagram commute . however , @xmath484 is the _ unique _ map making the middle square commute , so the whole diagram commutes as drawn . thus @xmath485 ( thinking of @xmath486 as an element of @xmath92 ) . as @xmath487 , we conclude that @xmath488 . thus @xmath489 . this shows that @xmath490 is actually a homomorphism @xmath491 . it is easy to check that the whole construction gives a homomorphism @xmath492 . if @xmath493 then all the elements @xmath475 are zero , so @xmath494 . as @xmath47 is the homotopy colimit of the objects @xmath467 , we conclude from lemma [ lem - lim - one ] that @xmath495 . thus , @xmath193 is a monomorphism . the meaning of the proposition is elucidated by the following elementary lemma . [ lem - i - sq ] if @xmath424 is a regular sequence in @xmath79 , and @xmath35 is the ideal that it generates , then @xmath496 is freely generated over @xmath376 by the elements @xmath345 . it is clear that @xmath496 is generated by the elements @xmath345 . suppose that we have a relation @xmath497 in @xmath35 ( not @xmath496 ) . we claim that @xmath498 for all @xmath325 . indeed , it is clear that @xmath499 so by regularity we have @xmath500 say ; in particular , @xmath501 . moreover , @xmath502 , so by induction we have @xmath503 for @xmath504 , and thus @xmath498 as required . now suppose that we have a relation @xmath505 , say @xmath506 . we then have @xmath507 , so by the previous claim we have @xmath508 , so @xmath498 . this shows that the elements @xmath345 generate @xmath496 freely . [ cor - self - map ] in the situation of proposition [ prop - self - maps ] the map @xmath509 is an isomorphism , and @xmath428 is the completed exterior algebra generated by @xmath463 . it is easy to see that @xmath429 is a derivation and that @xmath510 ( kronecker s delta ) . this shows that @xmath193 is surjective , and the rest follows . in this section , we take @xmath60 , and let @xmath118 be the usual formal group law over @xmath7 . in places it will be convenient to use cohomological gradings ; we recall the convention @xmath511 . we will write @xmath70 for the usual map @xmath512 , and note that @xmath513 . a well - known construction gives a power operation @xmath514 which is natural for spaces @xmath9 and strictly commutative ring spectra @xmath51 . a good reference for such operations is @xcite ; in the case of @xmath0 , the earliest source is probably @xcite . in the case @xmath515 there is an element @xmath516 such that @xmath517/(2{\epsilon},{\epsilon}^2)$ ] . more generally , the even - dimensional part of @xmath518 is @xmath519/(2{\epsilon},{\epsilon}^2)$ ] , and @xmath520 for a uniquely determined operation @xmath521 . we also have the following properties : @xmath522 to handle the nonadditivity of @xmath238 , we make the following construction . for any @xmath10-algebra @xmath11 , we define @xmath523/(2,{\epsilon}^2){\;|\;}s = r^2\pmod{{\epsilon}}\}.\ ] ] given @xmath524 ( with @xmath525 ) we define @xmath526{:=}(a , a^2+{\epsilon}b)\in t(a^*)$ ] . we make @xmath527 into a ring by defining @xmath528 or equivalently @xmath529 + [ c , d ] & { : = } [ a+c , b+d+w_1 a c ] \\ [ a , b ] . [ c , d ] & { : = } [ a c , a^2 d + b c^2 ] . \end{aligned}\ ] ] note that @xmath530=[0,w_1 a^2]$ ] and @xmath531=0 $ ] , so @xmath532 . if we define @xmath533 $ ] , then @xmath455 gives a ring map @xmath534 . [ defn - induced ] suppose that @xmath11 is a plrq of @xmath10 , and let @xmath535 be the unit map . we say that @xmath11 has an induced power operation ( ipo ) if there is a ring map @xmath536 making the following diagram commute : @xmath537 because @xmath11 is an lrq , we know that such a map is unique if it exists . if @xmath20 then we know that @xmath538 can be constructed as a strictly commutative @xmath0-algebra and thus an @xmath68 ring spectrum , and the power operation coming from this @xmath68 structure clearly gives an ipo on @xmath539 . for a more elementary proof , it suffices to show that when @xmath540 the image of @xmath541 in @xmath542 is invertible . however , the element @xmath543 is trivially invertible in @xmath544 and @xmath541 differs from this by a nilpotent element , so it too is invertible . it is now easy to reduce the following result to theorem [ thm - even - general ] . let @xmath86 be a plrq of @xmath7 which has no @xmath6-torsion and admits an ipo . then @xmath86 is strongly realisable . we now give a formal group theoretic criterion for the existence of an ipo . [ defn - zx ] let @xmath118 be a formal group law over a ring @xmath11 . given an algebra @xmath545 over @xmath11 and an element @xmath546 , we define @xmath547 . ( we need @xmath195 to be topologically nilpotent in a suitable sense to interpret this , but we leave the details to the reader . ) thus , if @xmath9 is a space , @xmath548 , @xmath549 and @xmath195 is the euler class of a complex line bundle over @xmath9 then @xmath550 . [ prop - ipo - fgl ] let @xmath11 be a lrq of @xmath10 , and let @xmath118 be the obvious formal group law over @xmath11 . then a ring map @xmath536 is an ipo if and only if we have @xmath551\!]}}).\ ] ] let @xmath552 be the universal fgl over @xmath10 and put @xmath553 . let @xmath535 be the unit map , so that @xmath554 . using the universality of @xmath552 , we see that @xmath555 is an ipo if and only if we have @xmath556\!]}}.\ ] ] the left hand side is of course @xmath557 . there is an evident map @xmath558\ ! ] } } { \xrightarrow } { } t({{a^ * [ \ ! [ x , y ] \!]}}),\ ] ] sending @xmath9 to @xmath559 and @xmath560 to @xmath561 , and one can check that this is injective . thus , @xmath555 is an ipo if and only if @xmath562\!]}}).\ ] ] the right hand side here is @xmath563 and @xmath564 so the proposition will follow once we prove that @xmath565\!]}})$ ] . to do this , we use the usual isomorphism @xmath566\!]}}=mu^*({{\mathbb{c}p^\infty}}{\times}{{\mathbb{c}p^\infty}})$ ] , so that @xmath195 , @xmath460 and @xmath567 are euler classes , so @xmath568 and @xmath569 and @xmath570 . as @xmath455 is a natural multiplicative operation we also have @xmath571 , which gives the desired equation . we now use this to show that there is an ipo on @xmath572 . in this case the real reason for the ipo is that the todd genus gives an @xmath573 map @xmath574 , but we give an independent proof as a warm - up for the case of @xmath12 . [ prop - ipo - ku ] let @xmath575 $ ] be the todd genus . then there is an induced power operation on @xmath572 , given by @xmath576 $ ] . thus , @xmath572 and @xmath577 are strongly realisable . the fgl over @xmath572 coming from @xmath294 is just the multiplicative fgl @xmath578 , so @xmath579 $ ] . if we put @xmath580 $ ] then @xmath581 . we thus need only verify that @xmath582 . this is a straightforward calculation ; some steps are as follows . @xmath583 \\ z(x)z(y ) & = [ xy , xy(x+y ) ] \\ u\,z(x)z(y ) & = [ uxy , u^2xy(x+y)+u^3x^2y^2 ] \\ z(x+y+uxy ) & = [ x+y+uxy , x+y+u(x^2+xy+y^2)+u^3x^2y^2 ] . \end{aligned}\ ] ] we now turn to the case of @xmath12 . for the moment we prove only that an ipo exists ; in the next section we will calculate it . [ prop - ipo - bp ] there is an ipo on @xmath12 , so @xmath12 is strongly realisable . this is proved after lemma [ lem - zxqf ] . [ defn - z ] for the rest of this section , we will write @xmath584 [ \ ! [ x ] \!]}}.\ ] ] note that @xmath585 so @xmath586 [ lem - invdif ] we have @xmath587 [ \ ! [ x ] \!]}}/(2{\epsilon},{\epsilon}^2).\ ] ] working rationally and modulo @xmath588 , we have @xmath589 so @xmath590 note that @xmath591 is integral and its constant term is @xmath592 , so the above equation is between integral terms and we can sensibly reduce it modulo @xmath6 . we next recall the formula for @xmath593 given in ( * ? ? ? * section 4.3 ) . we consider sequences @xmath594 with @xmath595 and @xmath596 for each @xmath326 . we write @xmath597 and @xmath598 . we also write @xmath599 where @xmath600 the formula is @xmath601 the only terms which contribute to @xmath591 modulo @xmath6 are those for which @xmath602 , so @xmath603 for all @xmath326 . if @xmath35 has this form and @xmath604 then @xmath605 . thus @xmath606 as remarked in definition [ defn - z ] , we have @xmath607 , so @xmath608 as claimed . [ lem - zxqf ] in @xmath609\!]}})$ ] we have @xmath610 + _ { qf } [ 0,y ] = [ 0,x+y]\ ] ] and @xmath611 = [ x,0 ] + _ { qf } { \left[}0,\sum_{k\ge 0 } ( v_1 x)^{2(2^k-1 ) } y{\right]}.\ ] ] in particular , we have @xmath612 + _ { qf } [ 0,z / v_1 ] = [ x,0 ] + _ { qf } { \left[}0,\sum_{k\ge 0 } v_1^{2^k-1}x^{2^k}{\right]}.\ ] ] the first statement is clear , just because @xmath613[0,y]=0 $ ] . for the second statement , write @xmath614 $ ] and @xmath615 and @xmath616 $ ] . let @xmath617 be the coefficient of @xmath618 in @xmath619 . because @xmath620 we have @xmath621 , and @xmath622[x^j,0][0,w ] = [ 0,a^2_{1j}x^{2j}w].\ ] ] this expression is to be interpreted in @xmath609\!]}})$ ] , so we need to interpret @xmath623 in @xmath624 . thus lemma [ lem - invdif ] tells us that @xmath625 and @xmath626 and all other @xmath623 s are zero . thus @xmath627 = \\ { \left[}x , w\left(1+\sum_{k\ge 0 } ( v_1 x)^{2^{k+1}}\right){\right]}= [ x , w(1+z^2 ) ] = [ x , y ] \end{gathered}\ ] ] as claimed . for the last statement , lemma [ lem - invdif ] gives @xmath628.\ ] ] by the previous paragraph , this can be written as @xmath629+_{qf}[0,x(1+z)/(1+z)^2]=[x,0]+_{qf}[0,z / v_1]$ ] . to show that @xmath555 exists , it is enough to show that the formal group law on @xmath630 obtained from the map @xmath631 is @xmath6-typical . let @xmath22 be an odd prime , so the associated cyclotomic polynomial is @xmath632 . we need to show that @xmath633 [ \ ! [ x ] \!]}}/\phi_p({\omega}).\ ] ] ( this is just the definition of @xmath6-typicality for formal groups over rings which may have torsion . ) consider the ring @xmath634 [ \ ! [ x ] \!]}}/\phi_p({\omega}))$ ] . as @xmath635 we have @xmath636 , and by looking at the coefficient of @xmath637 we find that @xmath638 . now write @xmath639 $ ] and @xmath614 $ ] , so that @xmath640 . we find that @xmath641 = { \left[}\phi_p({\omega}),v_1\sum_{0\le i < j < p}{\omega}^{i+j}{\right]}= 0.\ ] ] this gives us a ring map @xmath642 ; we claim that this is injective . indeed , it is easy to see that @xmath643 is a basis for @xmath644/\phi_p({\omega})$ ] , and that @xmath645 is a permutation of this basis . suppose that we have @xmath646{\omega}^ix^j=0 \text { in } c^*.\ ] ] using the evident map @xmath647 [ \ ! [ x ] \!]}}/(2,\phi_p({\omega}))$ ] , we see that @xmath648 for all @xmath649 . as @xmath650{\omega}^ix^j=[0,{\omega}^{2i}x^{2j}b]$ ] , we see that @xmath651 as the elements @xmath652 are a permutation of the elements @xmath653 , we see that @xmath654 for all @xmath649 . we may thus regard @xmath545 as a subring of @xmath655 . next , we know that @xmath656 because @xmath118 is @xmath6-typical over @xmath120 . by lemma [ lem - zxqf ] , we also know that @xmath657,\ ] ] where @xmath658 . it is easy to see that @xmath659[0,w_j]=0 $ ] , so that @xmath659+_{qf}[0,w_j]=[0,w_i+w_j]$ ] . we also have @xmath660 for all @xmath661 . this means that @xmath662 = [ 0,\sum_k v_1^{2^k-1}x^{2^k } \sum_{i=0}^{p-1}{\omega}^{2^ki}]=0.\ ] ] by combining equations ( [ eqn - ind - bp - a ] ) to ( [ eqn - ind - bp - c ] ) , we see that @xmath663 as required . we now give explicit formulae for the ipo on @xmath12 . [ defn - u ] given a subset @xmath664 , we define @xmath665 and @xmath666 , where @xmath667 runs over subsets of @xmath668 that contain @xmath46 . by separating out the case @xmath669 and putting @xmath670 in the remaining cases we obtain a recurrence relation @xmath671 [ prop - q - formula ] the induced power operation on @xmath12 is given by @xmath672 & \text { if } n=0 \\ { } [ v_1,v_2 ] & \text { if } n=1 \\ { } [ v_n , v_1 v_n^2 + u_n ] & \text { if } n>1 \end{cases}\ ] ] moreover , we have @xmath673 . this is proved after corollary [ cor - exp - qf ] . we will reuse the notation of definition [ defn - z ] . [ lem - exp - f ] we have @xmath674 in @xmath675\!]}}/4 $ ] . using ravenel s formulae as in the proof of lemma [ lem - invdif ] , we have @xmath676 when @xmath677 we have @xmath678 , with equality only when @xmath679 or @xmath680 . it follows easily that @xmath681 by inverting this , we find that @xmath682 and thus that @xmath683 . because @xmath630 is a torsion ring , the formal group law @xmath684 has no @xmath685 series . nonetheless , @xmath686 is a power series over @xmath12 , so we can apply @xmath455 to the coefficients to get a power series over @xmath630 which we call @xmath687 . this makes perfect sense even though @xmath688 does not . [ cor - exp - qf ] in @xmath689\!]}}$ ] , we have @xmath690 x^{2^k}.\ ] ] by taking @xmath691\!]}})$ ] , we get @xmath692}= [ 0 , z / v_1 + x ] .\ ] ] because @xmath693 in @xmath630 , it follows immediately from the lemma that @xmath694 . using @xmath695 $ ] , we see that @xmath696 $ ] , and the first claim follows . if we now put @xmath697 $ ] then @xmath698x^{2^k}=[0,v_1^{2^{k+1}-1}x^{2^{k+1}}]$ ] , and the second claim follows . let @xmath699 denote the image of @xmath700 in @xmath624 and write @xmath701\in t(bp^*)$ ] . recall that the hazewinkel generators @xmath25 are characterised by the formula @xmath702_f(x)=\exp_f(2x)+_f\sum^f_{k>0}v_kx^{2^k } \in{{bp^ * [ \ ! [ x ] \!]}}.\ ] ] by applying the ring map @xmath555 and putting @xmath703 we obtain @xmath702_{qf}(z(x ) ) = \exp_{qf}(2z(x ) ) + _ { qf } \sum^{qf}_{k>0 } v_k z(x)^{2^k } \in t({{bp^ * [ \ ! [ x ] \!]}}).\ ] ] the first term can be evaluated using corollary [ cor - exp - qf ] . for the remaining terms , we have @xmath704[x^{2^k},0 ] = [ v_k x^{2^k},p_k x^{2^{k+1}}].\ ] ] we can use lemma [ lem - zxqf ] to rewrite this as @xmath705 + _ { qf } { \left[}0,\sum_{l>0}(v_1v_k x^{2^k})^{2^l-2}p_k x^{2^{k+1}}{\right]}\\ & = [ v_k x^{2^k},0 ] + _ { qf } { \left[}0,\sum_{l>0 } ( v_1v_k)^{2^l-2 } p_k x^{2^{k+l}}{\right]}. \end{aligned}\ ] ] after using the formula @xmath650+_{qf}[0,c]=[0,b+c]$ ] to collect terms , we find that @xmath706_{qf}(z(x ) ) = { \left[}0,\sum_{l>0 } v_1^{2^l-1}x^{2^l } + \sum_{k , l>0 } ( v_1v_k)^{2^l-2 } p_k x^{2^{k+l}}{\right]}+_{qf } \sum^{qf}_{k>0 } [ v_k x^{2^k},0].\ ] ] on the other hand , we know that @xmath707_{qf}(z(x ) ) & = z([2]_f(x ) ) \\ & = z\left(\exp_f(2x)+_f\sum^f_{k>0}v_kx^{2^k}\right)\\ & = z(\exp_f(2x))+_{qf}\sum^{qf}_{k>0}z(v_kx^{2^k } ) . \end{aligned}\ ] ] the first term is zero because @xmath708 is divisible by @xmath6 . for the remaining terms , lemma [ lem - zxqf ] gives @xmath709 + _ { qf } [ 0,\sum_{j\ge 0 } v_1^{2^j-1 } v_k^{2^j } x^{2^{k+j}}].\ ] ] thus , we have @xmath710_{qf}(z(x ) ) = { \left[}0,\sum_{k>0}\sum_{l\ge 0 } v_1^{2^l-1 } v_k^{2^l } x^{2^{k+l}}{\right]}+_{qf } \sum^{qf}_{k>0 } [ v_k x^{2^k},0].\ ] ] by comparing this with equation ( [ eqn - q - formula - a ] ) and equating coefficients of @xmath711 , we find that @xmath712 after some rearrangement and reindexing , this becomes @xmath713 in particular , we have @xmath714 . we now define @xmath715 the claim of the proposition is just that @xmath716 for all @xmath717 . using the recurrence relation given in definition [ defn - u ] , one can check that for all @xmath160 we have @xmath718 in particular , we have @xmath719 , and it follows inductively that @xmath716 for all @xmath160 . we also have @xmath720+[1,0]=[0,v_1]\ ] ] so @xmath721 . [ rem - pvk ] the first few cases are @xmath722 in particular , we find that @xmath723 , which shows that there is no commutative product on @xmath724 , considered as an object of @xmath74 . this problem does not go away if we replace the hazewinkel generator @xmath25 by the corresponding araki generator , or the bordism class @xmath725 of a smooth quadric hypersurface in @xmath726 . however , it is possible to choose a more exotic sequence of generators for which the problem does go away , as indicated by the next result . [ prop - ideal - j ] fix an integer @xmath160 . there is an ideal @xmath727 such that 1 . the evident map @xmath162 { \xrightarrow } { } bp^ * { \xrightarrow } { } bp^*/j\ ] ] is an isomorphism . 2 . @xmath728 . 3 . @xmath729 . the proof will construct an ideal explicitly , but it is not the only one with the stated properties . if @xmath165 we can take @xmath730 , but for @xmath158 this violates condition ( 2 ) . the subring @xmath731 $ ] of @xmath12 is the same as the subring generated by all elements of degree at most @xmath732 ; it is thus defined independently of the choice of generators for @xmath120 . first consider the case @xmath165 , and write @xmath733 . by inspecting definition [ defn - u ] , we see that @xmath734 for all @xmath158 , and thus proposition [ prop - q - formula ] tells us that @xmath728 . we may thus assume that @xmath158 . write @xmath735 $ ] , thought of as a subring of @xmath12 . we will recursively define a sequence of elements @xmath736 for @xmath737 such that * @xmath738 * @xmath739 if @xmath740 . it is clear that we can then take @xmath741 . we start by putting @xmath742 . suppose that we have defined @xmath743 with the stated properties . there is an evident map @xmath744 { \xrightarrow}{f } bp_*/(2,x_{n+1},\ldots , x_k),\ ] ] which is an isomorphism in degree @xmath745 . let @xmath746 be the image of @xmath747 in @xmath748 , and write @xmath749 . we can lift this to get an element @xmath750 of @xmath751 $ ] such that @xmath752 and every coefficient in @xmath750 is @xmath113 or @xmath592 . it is easy to see that condition ( b ) is satisfied , and that @xmath753 . however , we still need to show that @xmath754 is divisible by @xmath755 . by assumption we have @xmath756 for some @xmath757 . recall from proposition [ prop - q - formula ] that @xmath758 . it follows after a small calculation that @xmath759 also . moreover , we have @xmath760 , so @xmath761 . it follows easily that @xmath762 , as required . we give one further calculation , closely related to proposition [ prop - q - formula ] . [ prop - pwk ] recall that @xmath763 , where @xmath130 is the bordism class of a smooth quadric hypersurface in @xmath764 . we have @xmath765 , and @xmath766 . if @xmath680 we have @xmath767 and @xmath768 , so @xmath769 , as required . thus , we may assume that @xmath770 , and it follows easily from the formulae for @xmath771 and @xmath772 that @xmath238 induces a ring map @xmath773/{\epsilon}^2 $ ] . note that @xmath774_f(x)=w_kx^{2^k}+o(x^{2^k+1})$ ] over @xmath545 . write @xmath775 [ \ ! [ x ] \!]}}/(i_k,{\epsilon}^2)$ ] . arguing in the usual way , we see that @xmath702_{p_*f}(x)= [ 2]_f(x)([2]_f(x)+_f{\epsilon } ) = { \epsilon}w_k x^{2^k } + o(x^{2^k+1}).\ ] ] it follows easily that we must have @xmath702_{p_*f}(x ) = { \epsilon}w_k x^{2^{k-1 } } + o(x^{2^{k-1}+1}).\ ] ] it follows that @xmath776 for @xmath777 , and that @xmath778 , as required . the claims involving @xmath779 and @xmath780 follow from proposition [ prop - ipo - ku ] , and those for @xmath1 follow from proposition [ prop - ipo - bp ] . the claim @xmath781 follows from theorem [ thm - even - general ] , as the condition @xmath782 is trivially satisfied for dimensional reasons . the claim for @xmath783 can be proved in the same way as theorem [ thm - even - proof ] after noting that all the obstruction groups are trivial . choose an ideal @xmath667 as in proposition [ prop - ideal - j ] and set @xmath784 . everything then follows from theorem [ thm - even - general ] . we now take @xmath153 and turn to the proof of theorem [ thm - pn ] . as previously , we let @xmath785 denote the bordism class of the quadric hypersurface @xmath786 in @xmath787 . recall that the image of @xmath725 in @xmath120 is @xmath25 modulo @xmath788 , and thus @xmath789 . we next choose a product @xmath790 on @xmath791 for each @xmath661 . for @xmath679 we choose there are two possible products , and we choose one of them randomly . ( it is possible to specify one of them precisely using baas - sullivan theory , but that would lead us too far afield . ) for @xmath27 , we recall from proposition [ prop - pwk ] that @xmath792 . it follows easily that there is a product @xmath790 such that @xmath793 , and that this is unique up to a term @xmath794 with @xmath795 . from now on , we take @xmath790 to be a product with this property . it is easy to see that the resulting product @xmath796 is independent of the choice of @xmath790 s ( except for @xmath797 ) . [ defn - muin ] we write @xmath798 made into a ring as discussed above . for @xmath504 , we define @xmath799 by smashing the bockstein map @xmath800 with the identity on the other factors . we also define @xmath801 it is clear that @xmath175 and @xmath802 and @xmath803 . condition ( 2 ) in proposition [ prop - bpn ] assures us that @xmath804 and @xmath805 as well . as @xmath1 and @xmath64 are commutative , it is easy to see that @xmath184 , @xmath189 , @xmath190 and @xmath2 are central algebras over @xmath1 , @xmath191 , @xmath64 and @xmath65 respectively . the derivations @xmath429 on @xmath179 clearly induce compatible derivations on @xmath184 , @xmath189 , @xmath190 and @xmath2 . [ prop - twist - q ] the product @xmath178 on @xmath179 satisfies @xmath806 similarly for @xmath184 , @xmath189 , @xmath190 and @xmath2 . this follows easily from the fact that @xmath807 , given by proposition [ prop - pwk ] . [ prop - pn - map ] let @xmath47 be a central @xmath1-algebra such that @xmath808 , @xmath809 and @xmath810 for @xmath811 . then either there is a unique map @xmath812 of @xmath1-algebras , or there is a unique map @xmath813 ( but not both ) . analogous statements hold for @xmath189 , @xmath190 and @xmath2 with @xmath1 replaced by @xmath191 , @xmath64 and @xmath65 respectively . we treat only the case of @xmath184 ; the other cases are essentially identical . any ring map @xmath814 commutes with the given map @xmath815 , because the latter is central . it follows that maps @xmath812 of @xmath1-algebras biject with maps @xmath814 of rings , which biject with systems of commuting ring maps @xmath816 for @xmath177 . for @xmath817 we have @xmath818 , so proposition [ prop - maps - rx ] tells us that the unique unital map @xmath819 is a ring map . this remains the case if we replace the product @xmath301 on @xmath47 by @xmath820 , or in other words replace @xmath47 by @xmath821 . there is an obstruction @xmath822 which may prevent @xmath823 from being a ring map . if it is nonzero , we have @xmath824 this shows that @xmath825 is a ring homomorphism . after replacing @xmath47 by @xmath821 if necessary , we may thus assume that all the @xmath819 are ring maps . the obstruction to @xmath347 commuting with @xmath348 lies in @xmath826 . if @xmath325 and @xmath326 are different then at least one is strictly less than @xmath827 ; it follows that @xmath828 and thus that the obstruction group is zero . thus @xmath347 commutes with @xmath348 when @xmath349 , and we get a unique induced map @xmath814 , as required . in order to analyse the commutativity obstruction @xmath212 more closely and relate them to power operations , we need to recall some internal details of the ekmm category . ekmm use the word `` spectrum '' in the sense defined by lewis and may @xcite , rather than the sense we use elsewhere in this paper . they construct a category @xmath829 of `` @xmath830-spectra '' . this depends on a universe @xmath831 , but the functor @xmath832 gives a canonical equivalence of categories from @xmath830-spectra over @xmath831 to @xmath830-spectra over @xmath833 , so the dependence is only superficial . ( here @xmath834 is the space of linear isometries from @xmath831 to @xmath833 . ) we therefore take @xmath835 . ekmm show that @xmath829 has a commutative and associative smash product @xmath836 , which is not unital . however , there is a sort of `` pre - unit '' object @xmath19 , with a natural map @xmath837 . they then define the subcategory @xmath838 of `` @xmath19-modules '' , and prove that @xmath839 so that @xmath840 is an @xmath19-module for any @xmath9 . we write @xmath841 for the restriction of @xmath836 to @xmath48 . we next give a brief outline of the properties of @xmath48 . let @xmath842 be the category of based spaces ( all spaces are assumed to be compactly generated and weakly hausdorff ) . we write @xmath113 for the one - point space , or for the basepoint in any based space , or for the trivial map between based spaces . we give @xmath842 the usual quillen model structure for which the fibrations are serre fibrations . we write @xmath843 for the category with hom sets @xmath844 , and @xmath845 for the category obtained by inverting the weak equivalences . we refer to @xmath843 as the strong homotopy category of @xmath842 , and @xmath845 as the weak homotopy category . the category @xmath48 is a topological category : the hom sets @xmath846 are based spaces , and there are continuous composition maps @xmath847 we again have a strong homotopy category @xmath848 , with @xmath849 ; when we have defined homotopy groups , we will also define a weak homotopy category @xmath850 in the obvious way . @xmath48 is a closed symmetric monoidal category , with smash product and function objects again written as @xmath851 and @xmath852 . both of these constructions are continuous functors of both arguments . the unit of the smash product is @xmath19 . there is a functor @xmath853 , such that @xmath854 ( for the last of these , see @xcite . ) the last equation shows that @xmath43 is a full and faithful embedding of @xmath842 in @xmath48 , so that all of unstable homotopy theory is embedded in the strong homotopy category @xmath848 . in particular , @xmath848 is very far from boardman s stable homotopy category @xmath42 . however , it turns out that the weak homotopy category @xmath850 is equivalent to @xmath42 . the definition of this weak homotopy category involves certain `` cofibrant sphere objects '' which we now discuss . it will be convenient for us to give a slightly more flexible construction than that used in @xcite , so as to elucidate certain questions of naturality . let @xmath831 be a universe . there is a natural way to make the lewis - may spectrum @xmath855 into a @xmath830-spectrum , using the action of @xmath856 on @xmath857 as well as on the suspension coordinates . one way to see this is to observe that @xmath858 , where the @xmath859 on the right hand side refers to the sphere spectrum indexed on the universe @xmath831 . we then define @xmath860 . this gives a contravariant functor @xmath861 , and it is not hard to check that @xmath862 . moreover , for any finite - dimensional subspace @xmath863 , there is a natural subobject @xmath864 and a canonical isomorphisms @xmath865 this indicates that the objects @xmath866 are in some sense stable . they can be defined as follows : take the lewis - may spectrum @xmath867 indexed on @xmath831 , and then take the twisted half smash product with the space @xmath857 to get a lewis - may spectrum indexed on @xmath868 which is easily seen to be an @xmath830-spectrum in a natural way . we then apply @xmath869 to get @xmath866 . for any @xmath160 and @xmath107 we write @xmath870 we will also allow ourselves to write @xmath871 for @xmath872 where @xmath873 is a subspace of @xmath874 of dimension @xmath875 and @xmath661 and @xmath873 are clear from the context . any object of the form @xmath876 is non - canonically isomorphic to @xmath877 , where @xmath878 , but when one is interested in the naturality or otherwise of various constructions it is often a good idea to forget this fact . there are isomorphisms @xmath879 that become canonical and coherent in the homotopy category . the homotopy groups of an object @xmath880 are defined by @xmath881 we say that a map @xmath882 is a weak equivalence if it induces an isomorphism @xmath883 , and we define the weak homotopy category @xmath850 by inverting weak equivalences . we define a cell object to be an object of @xmath48 that is built from the sphere objects @xmath884 in the usual sort of way ; the category @xmath850 is then equivalent to the category of cell objects and homotopy classes of maps . in subsequent sections we will consider various spaces of the form @xmath885 . this is weakly equivalent to @xmath886 but not homeomorphic to it ; the functor @xmath887 is not representable and has rather poor behaviour . for this and many related reasons it is preferable to replace @xmath9 by @xmath888 and thus work with ekmm s `` mirror image '' category @xmath889 rather than the equivalent category @xmath890 . however , our account of these considerations is still in preparation so we have used @xmath890 in the present work . now let @xmath51 be a commutative ring object in @xmath48 , in other words an object equipped with maps @xmath891 making the relevant diagrams geometrically ( rather than homotopically ) commutative . ( the term `` ring '' is something of a misnomer , as there is no addition until we pass to homotopy . ) we let @xmath71 denote the category of module objects over @xmath51 in the evident sense . this is again a topological model category with a closed symmetric monoidal structure . the basic cofibrant objects are the free modules @xmath892 for @xmath893 . the weak homotopy category @xmath894 obtained by inverting weak equivalences is also known as the derived category of @xmath51 , and written @xmath72 ; it is equivalent to the strong homotopy category of cell @xmath51-modules . it is not hard to see that @xmath74 is a monogenic stable homotopy category in the sense of @xcite ; in particular , it is a triangulated category with a compatible closed symmetric monoidal structure . in the previous sections we worked in the derived category @xmath74 of ( strict ) @xmath51-modules . in this section we sharpen the picture slightly by working with modules with strict units . these are not cell @xmath51-modules , so we need to distinguish between @xmath895 $ ] and @xmath896 . note that the latter need not have a group structure ( let alone an abelian one ) . however , most of the usual tools of unstable homotopy theory are available in @xmath897 , because @xmath71 is a topological category enriched over pointed spaces . in particular , we will need to use puppe sequences . as previously , we let @xmath195 be a regular element in @xmath898 , so @xmath193 is even . we regard @xmath195 as an @xmath51-module map @xmath899 , and we write @xmath55 for the cofibre . there is thus a pushout diagram @xmath900 as @xmath51 is not a cell @xmath51-module , the same is true of @xmath55 . however , the map @xmath901 is a @xmath70-cofibration . one can also see that @xmath902 is a cell @xmath51-module which is the cofibre in @xmath74 of the map @xmath903 , so it has the homotopy type referred to as @xmath55 in the previous section . moreover , the map @xmath904 is a weak equivalence . it follows that our new @xmath55 has the same weak homotopy type as in previous sections . let @xmath905 be defined by the following pushout diagram : @xmath906 there is a unique map @xmath907 such that @xmath908 , and there is an evident cofibration @xmath909 here @xmath910 we define a _ strictly unital product _ on @xmath55 to be a map @xmath203 of @xmath51-modules such that @xmath911 . let @xmath238 be the space of strictly unital products , and let @xmath912 be the set of products on @xmath55 in the sense of section [ sec - prod - rx ] . [ prop - strictly - unital ] the evident map @xmath913 is a bijection . the cofibration @xmath914 gives a fibration @xmath915 of spaces . the usual theory of puppe sequences and fibrations tells us that the image of @xmath916 is the union of those components in @xmath917 that map to zero in @xmath918 , so @xmath916 is surjective . in particular , we find that @xmath919 is nonempty . similar considerations then show that the @xmath781-space @xmath920 acts on @xmath238 , and that for any @xmath921 the action map @xmath922 gives a weak equivalence @xmath923 . this shows that @xmath924 acts freely and transitively on @xmath925 . this is easily seen to be compatible with our free and transitive action of @xmath257 on @xmath912 ( lemma [ lem - uni - obs ] ) , and the claim follows . [ rem - associativity ] these ideas also give another proof of associativity . let @xmath560 be the union of all cells except the top one in @xmath926 , so there is a cofibration @xmath927 . let @xmath178 be a product on @xmath55 ; by the proposition , we may assume that it is strictly unital . it is easy to see that @xmath928 and @xmath929 have the same restriction to @xmath560 ( on the nose ) . it follows using the puppe sequence that they only differ ( up to homotopy ) by the action of the group @xmath930 . thus , @xmath178 is automatically associative up to homotopy . we end this section with a more explicit description of the element @xmath931 . define @xmath932 ; this is a space with @xmath933 . the twist map @xmath167 of @xmath934 gives a self - map of @xmath9 , which we also call @xmath167 . let @xmath460 be the map @xmath935 considered as a point of @xmath9 . as @xmath51 is commutative , this is fixed by @xmath167 . next , let @xmath936 be the obvious nullhomotopy of @xmath195 , and consider the map @xmath937 this is adjoint to a path @xmath938 with @xmath939 and @xmath940 . we could do a similar thing using @xmath941 to get another map @xmath942 , but it is easy to see that @xmath943 . we now define a map @xmath944 by @xmath945 we can use the pushout description of @xmath55 to get a pushout description of @xmath200 . using this , we find that strictly unital products are just the same as maps @xmath946 that extend @xmath797 . let @xmath178 be such an extension . let @xmath947 be the twist map ; we find that @xmath948 also extends @xmath797 and corresponds to the opposite product on @xmath55 . let @xmath949 be the space @xmath950 , where @xmath951 if @xmath952 ; clearly this is homeomorphic to @xmath953 . define @xmath954 by @xmath955 and @xmath956 . it is not hard to see that the class in @xmath957 corresponding to @xmath301 is just @xmath958 , and thus that the image in @xmath959 is @xmath212 . another way to think about this is to define a map @xmath960 by @xmath961 , and to think of @xmath962 as the image of @xmath963 in @xmath949 . we can then say that @xmath301 is the unique @xmath167-equivariant extension of @xmath178 . in this section , we identify the commutativity obstruction @xmath212 of proposition [ prop - rx ] with a kind of power operation . this is parallel to a result of mironov in baas - sullivan theory , although the proofs are independent . we assume for simplicity that @xmath964 . because @xmath52 is concentrated in even degrees , we know that the atiyah - hirzebruch spectral sequence converging to @xmath965 collapses and thus that @xmath51 is complex orientable . we choose a complex orientation once and for all , taking the obvious one if @xmath51 is ( a localisation of ) @xmath0 . this gives thom classes for all complex bundles . we write @xmath966 for the even - degree part of @xmath967 , so that @xmath968/(2{\epsilon},{\epsilon}^2)$ ] . ( in the interesting applications the ring @xmath52 has no @xmath6-torsion and so @xmath969 has no odd - degree part . ) we will need notation for various twist maps . we write @xmath970 for the twist map of @xmath971 , or for anything derived from that by an obvious functor . similarly , we write @xmath972 for the twist map of @xmath973 , and @xmath974 for that of @xmath975 . we can thus factor the twist map @xmath167 of @xmath976 as @xmath977 . we will need to consider the bundle @xmath978 over @xmath979 . here @xmath980 is acting on @xmath981 by @xmath970 , and antipodally on @xmath953 ; the thom space is @xmath982 . as @xmath193 is even , we can regard @xmath983 as @xmath984 , so we have a thom class in @xmath985 which generates @xmath986 as a free module over @xmath987/(2{\epsilon},{\epsilon}^2)$ ] . suppose that @xmath988 . recall that @xmath195 is represented by a map @xmath989 . by smashing this with itself and using the product structure of @xmath51 we obtain a map @xmath990 . as @xmath51 is commutative we have @xmath991 . because @xmath992 is a continuous contravariant functor of @xmath831 , we have a map @xmath993 and thus a map @xmath994 . if we let @xmath995 be the twist map and let @xmath980 act on @xmath996 by @xmath997 then @xmath996 is a model for @xmath998 and thus @xmath999 . as @xmath991 we see that our map factors through @xmath1000 . for any cw complex @xmath47 , the spectrum @xmath1001 is a cofibrant approximation to @xmath1002 , so we can regard this map as an element of @xmath1003 . by restricting to @xmath1004 and using the thom isomorphism , we get an element of @xmath1005 ; we define @xmath1006 to be this element . we also recall that @xmath968/(2{\epsilon},{\epsilon}^2)$ ] and define @xmath56 to be the coefficient of @xmath1007 in @xmath1006 , so @xmath1008 . if @xmath47 is a cw complex with only even - dimensional cells then we can replace @xmath51 by @xmath1009 to get power operations @xmath1010 and @xmath1011 . it is not hard to check that this is the same as the more classical definition given in @xcite and thus to deduce the properties listed at the beginning of section [ sec - formal ] . we also need a brief remark about the process of restriction to @xmath1004 . the space of maps @xmath1012 such that @xmath1013 is easily seen to be contractible . choose such a map @xmath1014 . we then have @xmath1015 , and @xmath1006 is represented by the composite @xmath1016 we call this map @xmath1017 . let @xmath103 be the monoid @xmath1018 . this acts contravariantly on @xmath1019 , giving a map @xmath1020 here we use the action of @xmath980 on @xmath103 given by @xmath997 . there is also a homotopically unique map @xmath1021 such that @xmath1022 for all @xmath1023 . by combining this with the above map , we get a map @xmath1024 we call this map @xmath1025 . recall that in the homotopy category there is a canonical isomorphism @xmath1026 , so @xmath1025 again represents an element of @xmath1027 . we claim that this is the same as @xmath1006 . to see this , choose an isomorphism @xmath1028 , giving a map @xmath1029 and a map @xmath1030 . take @xmath1031 as our choice of @xmath1014 , and use @xmath1032 as a representative of the canonical equivalence @xmath1026 in the homotopy category ; under these identifications , @xmath1025 becomes @xmath1017 . we leave the rest of the details to the reader . define @xmath1033 . the twist maps @xmath970 , @xmath1034 and @xmath167 induce commuting involutions of @xmath560 with @xmath1035 . we can think of @xmath990 as a point of @xmath560 , which is fixed under @xmath167 . the contravariant action of @xmath103 on @xmath1019 gives a covariant action on @xmath560 , which commutes with @xmath970 . using this and our map @xmath1021 we can define a map @xmath1036 by @xmath1037 . if we let @xmath980 act on @xmath560 by @xmath970 then one finds that this is equivariant . we can think of @xmath1038 as an adjoint of @xmath1025 and thus a representative of @xmath1006 . we are really only interested in the image of @xmath56 in @xmath959 . to understand this , we reintroduce the space @xmath932 as in section [ sec - strict - unit ] . the unit map @xmath901 induces an equivariant map @xmath1039 . we define @xmath1040 . note that @xmath1041 , and @xmath1038 lands in the component corresponding to @xmath1042 , and @xmath1043 , so @xmath1044 lands in the base component . moreover , we have @xmath1045 and @xmath980 acts freely on @xmath1046 so by equivariant obstruction theory we can extend @xmath1044 over the cofibre of the inclusion @xmath1047 to get a map @xmath1048 say . this cofibre is equivariantly equivalent to @xmath1049 and @xmath1050^{c_2}\simeq\pi_2(x)=\pi_{2d+2}(r)/x.\ ] ] it is not hard to see that the element of @xmath959 coming from @xmath1048 is just the image of @xmath56 . we now set up an abstract situation in which we have a space @xmath9 and we can define two elements @xmath1051 and prove that they are equal ; later we apply this to show that @xmath1052 . while this involves some repetition of previous constructions , we believe that it makes the argument clearer . let @xmath103 be a @xmath6-connected topological monoid , containing an involution @xmath1034 . let @xmath1053 be the group of order two , and define @xmath1054 . let @xmath9 be a space with basepoint @xmath113 and another distinguished point @xmath460 in the base component . suppose that @xmath1055 acts on @xmath9 , the whole group fixes @xmath113 , and @xmath167 fixes @xmath460 . suppose also that @xmath1056 and @xmath1057 . write @xmath1058 . given @xmath1059 we can let @xmath873 act on @xmath1060 by @xmath1061 and @xmath1062 so @xmath1063 . we write @xmath1064 for this representation of @xmath873 , and @xmath1065 for the sphere in @xmath1064 , so that @xmath1066 nonequivariantly . [ defn - alpha ] define a @xmath167-equivariant map @xmath1067 by @xmath1068 and @xmath1069 . using the evident @xmath167-equivariant cw structure on @xmath1070 and the fact that @xmath1057 we find that there is an equivariant extension of @xmath1071 over @xmath1070 , which is unique modulo @xmath1072 . nonequivariantly we have @xmath1073 and @xmath1074 so we get a homotopy class of maps @xmath1075 , which is unique modulo @xmath6 . we write @xmath1071 for the corresponding element of @xmath1076 . [ defn - lambda ] define @xmath1077 by @xmath1078 and @xmath1079 ; this is equivariant with respect to the evident right action of @xmath970 on @xmath103 . as @xmath1034 acts freely on @xmath1070 and @xmath103 is @xmath6-connected , we see that there is a @xmath970-equivariant extension @xmath1080 , which is unique up to equivariant homotopy . [ defn - beta ] define @xmath1081 by @xmath1082 . as @xmath1083 is @xmath1034-equivariant and @xmath460 is fixed by @xmath1035 and @xmath970 commutes with @xmath103 we find that @xmath1084 . we next claim that @xmath232 can be extended over the cofibre of the inclusion of @xmath1085 in @xmath1070 in such a way that we still have @xmath1086 . this follows easily from the fact that @xmath1034 acts freely on @xmath1085 and @xmath460 lies in the base component of @xmath9 and @xmath1057 . the cofibre in question can be identified @xmath1034-equivariantly with @xmath1087 . by composing with the inclusion @xmath1088 we get an element of @xmath1089 . this can be seen to be unique modulo @xmath1090 but by hypothesis @xmath1056 so we get a well - defined element of @xmath1076 , which we also call @xmath232 . consider the following picture of @xmath1092 . @xmath1093 the axes are set up so that @xmath1094 and @xmath1095 , so @xmath1096 we write @xmath1097 and @xmath1098 for the upper and lower hemispheres and @xmath1099 for the unit disc in the plane @xmath1100 . thus @xmath1101 and @xmath1102 , so @xmath1103 can be identified with the cofibre of the inclusion @xmath1104 . note also that @xmath1105 is @xmath167-equivariantly homeomorphic to @xmath1070 ( by radial projection from the @xmath167-fixed point @xmath1106 , say ) . let @xmath1107 be the closed disc of radius @xmath1108 centred at @xmath1109 and let @xmath47 be the closure of @xmath1110 . define @xmath1111 by @xmath1112 on @xmath1070 and @xmath1113 on @xmath1107 . we see by obstruction theory that @xmath1114 can be extended @xmath167-equivariantly over the whole of @xmath1115 . moreover , if we identify @xmath1105 with @xmath1070 as before then the restriction of @xmath1114 to @xmath1105 represents the same homotopy class @xmath1071 as considered in definition [ defn - alpha ] , as one sees directly from the definition . next , note that @xmath1116 retracts @xmath1034-equivariantly onto @xmath1070 , so we can extend our map @xmath1080 over @xmath1116 equivariantly . as @xmath103 is @xmath592-connected , we can extend it further over the whole of @xmath1115 , except that we have no equivariance on @xmath1107 . now define @xmath1117 by @xmath1118 . we claim that @xmath1119 . away from @xmath1107 this follows easily from the equivariance of @xmath1083 and @xmath1114 , and on @xmath1107 it holds because both sides are zero . using this and our identification of @xmath1115 with the cofibre of @xmath1104 we see that the restriction of @xmath1120 to @xmath1105 represents the class @xmath232 in definition [ defn - beta ] . now observe that @xmath1115 is @xmath6-dimensional and @xmath103 is @xmath6-connected , so our map @xmath1121 is nonequivariantly homotopic to the constant map with value @xmath592 . this implies that @xmath1114 is homotopic to @xmath1120 , so @xmath1091 as claimed . we now prove that @xmath1122 . we take @xmath932 and @xmath1123 as before , and define involutions @xmath970 , @xmath1034 and @xmath167 as in section [ subsec - defn - p ] . we also define @xmath460 as in section [ subsec - adjunction ] . it is then clear that the map @xmath1048 of section [ subsec - rx ] represents the class @xmath232 of definition [ defn - beta ] , so that @xmath1124 . now consider the constructions at the end of section [ sec - strict - unit ] . it is not hard to see that the space @xmath949 defined there is @xmath167-equivariantly homeomorphic to @xmath1070 , with the two fixed points being @xmath1125 and @xmath1126 . as the map @xmath954 is equivariant and @xmath1127 and @xmath1128 , we see that @xmath301 represents the class @xmath1071 of definition [ defn - alpha ] , so @xmath1129 . it now follows from proposition [ prop - alpha - beta ] that @xmath1130 , as claimed .

elmendorf , kriz , mandell and may have used their technology of modules over highly structured ring spectra to give new constructions of @xmath0-modules such as @xmath1 , @xmath2 and so on , which makes it much easier to analyse product structures on these spectra . unfortunately , their construction only works in its simplest form for modules over @xmath3_*$ ] that are concentrated in degrees divisible by @xmath4 ; this guarantees that various obstruction groups are trivial . we extend these results to the cases where @xmath5 or the homotopy groups are allowed to be nonzero in all even degrees ; in this context the obstruction groups are nontrivial . we shall show that there are never any obstructions to associativity , and that the obstructions to commutativity are given by a certain power operation ; this was inspired by parallel results of mironov in baas - sullivan theory . we use formal group theory to derive various formulae for this power operation , and deduce a number of results about realising @xmath6-local @xmath7-modules as @xmath0-modules .

when optical , electromagnetic or acoustic signals are measured , often the measurement apparatus records an intensity , the magnitude of the signal amplitude , while discarding phase information . this is the case for x - ray crystallography @xcite , many optical and acoustic systems @xcite , and also an intrinsic feature of quantum measurements @xcite . phase retrieval is the procedure of determining missing phase information from suitably chosen intensity measurements , possibly with the use of additional signal characteristics @xcite . many of these instances of phase retrieval are related to the fourier transform @xcite , but it is also of interest to study this problem from an abstract point of view , using the magnitudes of any linear measurements to recover the missing information . next to infinite dimensional signal models @xcite , the finite dimensional case has received considerable attention in the past years @xcite . in this case , the signals are vectors in a finite dimensional hilbert space @xmath2 and one chooses a frame @xmath3 to obtain for each @xmath4 the magnitudes of the inner products with the frame vectors , @xmath5 . when recovering signals , we allow for a remaining undetermined global phase factor , meaning we identify vectors in the hilbert space @xmath2 that differ by a unimodular factor @xmath6 , @xmath7 , in the real or complex complex case . accordingly , we associate the equivalence class @xmath8=\mathbb t x = \{\omega x : |\omega|=1 \}$ ] with a representative @xmath4 and consider the quotient space @xmath9 as the domain of the magnitude measurement map @xmath10)=(|\langle x , f_j\rangle|^2)_{j=1}^n$ ] . the map @xmath11 is well defined , because @xmath12 does not depend on the choice of @xmath13 . the metric on @xmath14 relevant for the accuracy of signal recovery is the quotient metric @xmath15 , which assigns to elements @xmath8 $ ] and @xmath16 $ ] with representatives @xmath17 the distance @xmath18,[y])=\min_{|\omega|=1}\|x-\omega y\|$ ] . the case of signals in real hilbert spaces is now fairly well understood @xcite , while complex signals still pose many open problems . when the number of measured magnitudes is allowed to grow at a sufficient rate , then techniques from low - rank matrix completion are applicable to phase retrieval @xcite , providing stable recovery from noisy measurements . other methods also achieve stability by a method that locally patches the phase information together @xcite . recently , it was shown that for a vector in a complex @xmath1-dimensional hilbert space , a generic choice of @xmath19 linear measurements is sufficient to recover the vector up to a unimodular factor from the magnitudes @xcite , complementing an earlier result on a deterministic choice of @xmath19 vectors @xcite . nevertheless , fully quantitative stability estimates were missing in this case of lowest redundancy known to be sufficient for recovery . a main objective of this paper is to find frames @xmath3 for the @xmath1-dimensional complex hilbert space such that @xmath20 is small and the magnitude measurement map is injective on @xmath9 with explicit error bounds for the approximate recovery when the magnitude measurements are affected by noise . more precisely , we find a left inverse of @xmath11 which extends to a neighborhood of the range of @xmath11 and is lipschitz continuous for all input signals whose signal - to - noise ratio is sufficiently large . we show that the recovery is implemented with an explicit algorithm that restores the signal from measurements to a given accuracy in a number of operations that is polynomial in the dimension of the hilbert space . the algorithm can be chosen to be either phase propagation or what we call the kernel method , a special case of semidefinite programming . the smallest number of frame vectors for which we could provide an algorithm with explicit error bounds is @xmath0 , as presented here . to formulate the main result , it is convenient to take the hilbert space as a space of polynomials @xmath21 of maximal degree @xmath22 equipped with the standard inner product , see section [ sec : poly ] for details . with this choice of hilbert space , the magnitude measurements we use are expressed in terms of point evaluations . we let @xmath23 denote the primitive @xmath24-st root of unity and @xmath25 the primitive @xmath1-th root of unity . for a polynomial @xmath26 , the noiseless magnitude measurements are @xmath27 the noisy magnitude measurements @xmath28 are obtained from perturbing the noiseless magnitudes with a vector @xmath29 , @xmath30 , @xmath31 . our main theorem states that for all measurement errors with a sufficiently small maximum noise component @xmath32 , the noisy magnitude measurements determine an approximate reconstruction of @xmath33 with an accuracy @xmath34 . to state the theorem precisely involves several auxiliary quantities that all depend solely on the dimension @xmath1 . we let @xmath35 and choose a slack variable @xmath36 as well as @xmath37 . let @xmath38 and @xmath39 be as above . for any nonzero analytic polynomial @xmath40 and @xmath41 with @xmath42 , an approximation @xmath43 can be constructed from the perturbed magnitude measurements @xmath44 , such that if @xmath45 then the recovery error is bounded by @xmath46,[\tilde p])\le \left(\frac{2+\sqrt2}{\beta^2(1-\alpha)}\frac{d - d\tilde c-1+\tilde c^d}{1-\tilde c}\sqrt d+\frac{1-\tilde c^d}{2\beta\sqrt{\frac1{\sqrt d}(1-\alpha)}}\right)\frac{d(2d-1)}{(1-\tilde c)}\frac{\|\eps\|_\infty}{\|p\|_2 } \ , .\ ] ] the proof and the construction of approximate recovery proceeds in several steps : step 1 . : : first , we augment the finite number of magnitude measurements to an infinite family of such measurements . to this end , the dirichlet kernel is used to interpolate the perturbed measurements to functions on the entire unit circle . in the noiseless case , the magnitude measurements @xmath47 determine the values @xmath48 , @xmath49 , and @xmath50 for each @xmath51 , because these are trigonometric polynomials of degree at most @xmath22 . in the noisy case , the interpolation using values from @xmath44 , yields trigonometric polynomials that differ from the unperturbed ones by at most @xmath52 , uniformly on the unit circle . : : we select a suitable set of non - zero magnitude measurements from the infinite family . a lemma will show that there exists a @xmath53 on the unit circle such that the distance between any element of @xmath54 and any roots of any non - zero truncation of the polynomial @xmath33 is at least @xmath55 . the reason why we need to consider all non - zero truncations of the polynomial is that the influence of the noise prevents us from determining the true degree of @xmath33 . however , when the coefficients of leading powers are sufficiently small compared to the noise , we can replace @xmath33 with a truncated polynomial without losing the order of approximation accuracy . as a consequence , we show that for this @xmath53 , @xmath56 with some @xmath57 that only depends on the dimension @xmath1 and the norm of @xmath33 . thus , if the noise is sufficiently small compared to the norm of the vector , then there is a similar lower bound on the real trigonometric polynomials that interpolate the noisy magnitude measurements . : : in the last step , the reconstruction evaluates the trigonometric approximations at the sample points @xmath54 and recovers an approximation to the equivalence class @xmath58 $ ] . it is essential for this step that the sample values are bounded away from zero in order to achieve a unique reconstruction . there are two algorithms considered for this , phase propagation , which recovers the phase iteratively using the phase relation between sample points , and the kernel method , which computes a vector in the kernel of a matrix determined by the magnitude measurements . the error bound is first derived for phase propagation and then related to that of the kernel method . both algorithms are known to be polynomial time , either from the explicit description , or from results in numerical analysis @xcite . the nature of the main result has also been observed in simulations ; assuming an a priori bound on the magnitude of the noise results in a worst - case recovery error that grows at most inverse proportional to the signal - to - noise ratio . outside of this regime , the error is not controlled in a linear fashion . to illustrate this , we include two plots of the typical behavior for the recovery error for @xmath59 . the range of the plots is chosen to show the behavior of the worst - case error in the linear regime and also for errors where this linear behavior breaks down . we tested the algorithm on more than 4.5 million randomly generated polynomials with norm 1 . when errors were graphed for a fixed polynomial , the linear bound for the worst - case error was confirmed , although the observed errors were many orders of magnitude less than the error bound given in this paper . a small number of polynomials we found exhibited a max - min value that is an order of magnitude smaller than that of all the other randomly generated polynomials . we chose the polynomial with the worst max - min value out of the 4.5 million that had been tried , and applied a random walk to its coefficients , with steps of decreasing size that were accepted only if the max - min value decreased . the random walk terminated at a polynomial which provided an error bound that is an order of magnitude worse than any other polynomials that had been tested before . this numerically found , local worst - case polynomial is given by @xmath60 . the accuracy of the coefficients displayed here is sufficient to reproduce the results initially obtained with floating point coefficients of double precision . the errors resulting for this polynomial in the linear and transition regimes are shown in figures [ fig1 ] and [ fig2 ] . as a function of the maximal noise magnitude in the linear regime.,width=345 ] as a function of the maximal noise magnitude beyond the linear regime.,width=345 ] it is instructive to follow the construction of the magnitude measurements and the recovery strategy in the absence of noise . to motivate and prepare the recovery strategy , we compare two recovery methods , phase propagation and the kernel method , a simple form of semidefinite programming , in the absence of noise and under additional non - orthogonality conditions on the input vector . if @xmath61 is a basis for @xmath62 , and @xmath63 such that for all @xmath64 from @xmath65 to @xmath1 , @xmath66 then we call @xmath67 _ full _ with respect to @xmath61 . we recall a well known result concerning recovery of full vectors @xcite . let @xmath4 be full with respect to an orthonormal basis @xmath61 . for any @xmath64 from @xmath65 to @xmath68 , we define the measurement vector @xmath69 as @xmath70 the set @xmath71 of measurement vectors is a frame for @xmath62 because it contains a basis . define the magnitude measurement map @xmath72 by @xmath73 . recovery of full vectors with @xmath68 measurements has been shown in @xcite , and was proven to be minimal in @xcite . we show recovery of full vectors with @xmath68 measurements using the measurement map @xmath74 with two different recovery methods . the first is called phase propagation , the second is a special case of semidefinite programming . the phase propagation method sequentially recovers the components of the vector , similar to the approach outlined in @xcite , see also @xcite . for any vector @xmath63 , if @xmath61 is an orthonormal basis with respect to which @xmath67 is full , then the vector @xmath75 may be obtained by induction on the components of @xmath76 , using the values of @xmath77 . without loss of generality , we assume that @xmath61 is the standard basis , drop the subscript from @xmath74 and abbreviate the components of the vector @xmath67 by @xmath78 for each @xmath79 , and similarly for @xmath76 . to initialize , we let @xmath80 so that @xmath81 . for the @xmath82th inductive step with @xmath83 , we assume that we have constructed @xmath84 with the given information . we then let @xmath85 inserting the values for the magnitude measurements and by a fact similar to the polarization identity , @xmath86 iterating this , we obtain @xmath87 . recovery by the kernel method minimizes the values of a quadratic form subject to a norm constraint , or equivalently , computes an extremal eigenvector for an operator associated with the quadratic form . the operator we use is @xmath88 , with @xmath89 where each @xmath90 denotes the linear functional which is associated with the basis vector @xmath91 . the operator @xmath92 is indeed determined by the magnitude measurements . in particular , the second term in the series is computed via the polarization - like identity as in the proof of the preceding theorem , @xmath93 for any integer @xmath64 from @xmath65 to @xmath22 . by construction , the rank of @xmath94 is at most equal to @xmath22 , because the range of @xmath92 is in the span of @xmath95 . in the next theorem , we show that indeed the kernel of @xmath92 , or equivalently , the kernel of @xmath94 , is one dimensional , consisting of all multiples of @xmath67 . for any vector @xmath63 , if @xmath61 is a basis with respect to which @xmath67 is full , then the null space of the operator @xmath92 is given by all complex multiples of @xmath67 . as in the preceding proof , we let @xmath61 denote the standard basis . thus , using the measurements provided , we may obtain the quantity @xmath96 . with respect to the basis @xmath97 , let @xmath98 be the left shift operator @xmath99 where we extend the vector @xmath76 with the convention @xmath100 . we also define the multiplication operator @xmath101 by the map @xmath102 , where again by convention we let @xmath103 . similarly , we define the multiplication operator @xmath104 by the map @xmath105 . note that @xmath106 is invertible if and only if @xmath107 for all @xmath64 from @xmath65 to @xmath1 , which is true by assumption . in terms of these operators , the operator @xmath108 is expressed as @xmath109 . then for any @xmath110 and @xmath111 , @xmath112 so any complex multiple of @xmath67 is in the null space of this operator . conversely , assume that @xmath76 is in the null space of @xmath92 . we use an inductive argument to show that for any @xmath64 from @xmath65 to @xmath1 , @xmath113 . the base case @xmath114 is trivial . for the inductive step , note that for any @xmath64 from @xmath65 to @xmath22 , @xmath115 thus , @xmath116 , and because @xmath117 and @xmath118 , we obtain @xmath119 we conclude that for any @xmath64 from @xmath65 to @xmath1 , @xmath120 , so the vector @xmath76 is a complex multiple of @xmath67 . since the frame vectors used for the magnitude measurements contain an orthonormal basis , @xmath74 determines the norm of @xmath67 . this is sufficient to recover @xmath8 $ ] . if the vector @xmath4 is full with respect to the orthonormal basis @xmath61 , then the equivalence class @xmath8 $ ] is the solution of the problem @xmath121 because the solution to the phase retrieval problem is obtained from the kernel of the linear operator @xmath92 , or equivalently of @xmath94 , we may use methods from numerical linear algebra such as a rank - revealing qr factorization @xcite to recover the equivalence class @xmath8 $ ] . one of the main tools for the recovery procedure is that an entire family of magnitude measurements is determined from the initial choice . we call this an augmentation of the measured values . from this family a suitable subset is chosen which corresponds to a measurement of the form @xmath74 related to an orthonormal basis as explained in the previous section . to describe the augmentation procedure we represent the @xmath1-dimensional vector to be recovered as an element of @xmath21 , the space of complex analytic polynomials of degree at most @xmath22 on the unit circle . this space is used to represent the vector because @xmath21 is a reproducing kernel hilbert space , and the magnitude squared of any element of @xmath21 is an element of @xmath122 , the space of trigonometric polynomials of degree at most @xmath22 on the unit circle , which is itself a reproducing kernel hilbert space . the space @xmath21 is equipped with the scaled @xmath123 inner product on the unit circle such that for any @xmath124 , @xmath125}p(e^{it})\overline{q(e^{it})}dt \ , .\ ] ] for any @xmath126 in @xmath21 , let @xmath127 be the vector of coefficients @xmath128 of @xmath33 . then by orthogonality , the norm induced by the inner product satisfies @xmath129}p(e^{it})\overline{p(e^{it})}dt = \frac1{2\pi}\int_{[0,2\pi]}\sum_{j=0}^{d-1}|c_j|^2dt = \|c\|_2 ^ 2 \ , .\ ] ] if @xmath130 is defined such that @xmath131 , then for any @xmath40 and any @xmath53 on the unit circle , @xmath132 thus , these polynomials @xmath133 correspond to point evaluations , and linear combinations of these polynomials may be used as measurement vectors for the recovery procedure . if @xmath23 is the @xmath134-st root of unity and @xmath25 is the @xmath1-th root of unity , then for any @xmath64 from @xmath65 to @xmath135 , we define the measurement vector @xmath136 as @xmath137 then the magnitude measurement map @xmath138 defined in the introduction satisfies @xmath139 . the measurements are grouped into three subsets , corresponding to magnitudes of point evaluations , magnitudes of differences , and magnitudes of differences between complex multiples of point values . each of these subsets can be interpolated to a family of measurements from which suitable representatives are chosen . in order to simplify the recovery , we recall that for any @xmath140 with @xmath141 and @xmath142 , @xmath143 and @xmath144 are orthogonal because the series given by the inner product @xmath145 simply sums all the @xmath1-th roots of unity . for any polynomial @xmath40 , the measurements @xmath47 determine the values of @xmath146 with @xmath147 such that @xmath148 is an orthonormal basis with respect to which @xmath33 is full . let @xmath149 be the normalized dirichlet kernel of degree @xmath22 , so that for any @xmath140 in the unit circle @xmath150 . then the set of functions @xmath151 is orthonormal with respect to the @xmath123 inner product on the unit circle , and any @xmath152 can be interpolated as @xmath153 . note that if @xmath154 , then @xmath155 . thus , each of the functions @xmath48 , @xmath49 , and @xmath50 , are in @xmath122 , and using the dirichlet kernel these functions may be interpolated from the values of @xmath47 . so the values of each of these functions are known at all points on the unit circle , not just the points that were measured explicitly . let @xmath156 be the zeros of the polynomial @xmath33 . then the set @xmath157 has finitely many elements , so we may choose a point @xmath53 on the unit circle such that @xmath158 . then for all @xmath82 from @xmath65 to @xmath1 , @xmath159 . additionally , the set @xmath160 is an orthonormal basis for @xmath21 , so @xmath33 is a full vector with respect to this basis and the values of @xmath161 , @xmath162 , and @xmath163 at the points @xmath164 correspond to the measurements @xmath165 . because the recovery procedure for full vectors has been established in section [ sec : full ] , the phase - retrieval problem may now be solved using either procedure outlined there to obtain the equivalence class @xmath58 $ ] . for any polynomial @xmath40 , the measurements @xmath47 determine @xmath58 $ ] . moreover , the number of points that need to be tested in order to find @xmath53 such that @xmath166 for all @xmath167 is at most @xmath168 , quadratic in @xmath1 . this means , in the noiseless case either of the two equivalent methods for recovery requires a number of steps that is polynomial in the dimension @xmath1 . for the purpose of recovery from noisy measurements , we consider the perturbed magnitude measurement map @xmath169 by @xmath170 . in the presence of noise , choosing a basis such that @xmath67 is full with respect to that basis is not sufficient to establish a bound on the error of the recovered polynomial . a lower bound on the magnitude of the entries of @xmath67 in a such a basis is needed . once such a lower bound is obtained , recovery may proceed as in the noiseless case , by reduction of the phase retrieval problem to full vector recovery , with quantitative bounds on stability added for each step . obtaining the needed lower bound on the magnitude requires a few lemmas and definitions . to obtain a lower bound on the entries of a full vector originating from a polynomial , a lower bound on the distance between any basis element and any roots of any nonzero truncations of the base polynomial is needed . [ lem : dist - bnd ] for any polynomial @xmath40 , there exists a @xmath53 on the unit circle such that the linear distance between any element of @xmath54 and any roots of any nonzero truncations of @xmath33 is at least @xmath183 . for any @xmath174 from @xmath65 to @xmath1 , let @xmath184 be the number of distinct roots of the @xmath174-th truncation of @xmath33 if that truncation is nonzero , and let @xmath185 if the @xmath174-th truncation of @xmath33 is a zero polynomial . then the number of distinct roots of all nonzero truncations of @xmath33 is @xmath186 then for the set @xmath187 , we know @xmath188 . if the elements of @xmath189 are ordered by their angle around the unit circle , then the average angle between adjacent elements is @xmath190 , and so there is at least one pair of adjacent elements that is separated by at least this amount . thus , if we let @xmath53 be the midpoint between these two maximally separated elements on the unit circle , then the angle between @xmath53 and any element of @xmath189 is at least @xmath191 . thus the linear distance between @xmath53 and any element of the set @xmath192 is at least @xmath183 . [ lem : mag - bnd ] let @xmath193 . for any polynomial @xmath40 , if there exists a @xmath53 on the unit circle such that the linear distance between any element of @xmath54 and any zeros of any nonzero truncations of @xmath33 is at least @xmath55 , then for all @xmath64 from @xmath65 to @xmath1 @xmath194 where @xmath195 , @xmath196 . let @xmath197 be the smallest @xmath174 obtained by applying lemma [ lem : geom - trick ] to @xmath198 and @xmath199 . then if we let @xmath200 , so that @xmath179 , we have @xmath201 and for all @xmath202 @xmath203 let @xmath204 we prove , by induction on @xmath174 from @xmath197 to @xmath1 , that @xmath205 for the @xmath174-th truncation @xmath181 , and for all @xmath64 from @xmath65 to @xmath1 . for the inductive step , assume that we have proven that @xmath205 . then we choose a threshold @xmath211 . if the leading coefficient @xmath212 of @xmath213 satisfies @xmath214 , then @xmath213 is clearly a nonzero truncation of @xmath33 , so by using the factored form of @xmath213 , for all @xmath64 from @xmath65 to @xmath1 , @xmath215 otherwise , if the leading coefficient satisfies @xmath216 , then for all @xmath64 from @xmath65 to @xmath1 @xmath217 either way , for all @xmath64 from @xmath65 to @xmath1 , @xmath218 thus , for all @xmath64 from @xmath65 to @xmath1 , @xmath219 using the @xmath55 from lemma [ lem : dist - bnd ] in the bound in the equation in lemma [ lem : mag - bnd ] gives us the desired lower bound . note that the bound in lemma [ lem : dist - bnd ] was obtained by showing a worst case of equally spaced roots and the bound in lemma [ lem : mag - bnd ] was obtained by showing a worst case of roots that are bunched together . thus , the lower bound on the minimum magnitude obtained by combining lemma [ lem : dist - bnd ] and lemma [ lem : mag - bnd ] will not be achieved for any polynomial of degree greater than @xmath220 and is thus not the greatest lower bound for higher dimensions . [ lem : full - induct - noise ] let @xmath57 . for any vectors @xmath63 and @xmath221 , if @xmath61 is an orthonormal basis such that for all @xmath64 from @xmath65 to @xmath1 , @xmath222 , and @xmath223 , then a vector @xmath76 may be obtained such that for all @xmath82 from @xmath65 to @xmath1 , @xmath224 by using the values of @xmath225 . for the @xmath82th ( with @xmath83 ) inductive step , we assume that we have constructed @xmath84 with the given information such that @xmath232 . we abbreviate @xmath233 and @xmath234 a direct computation shows the error for the approximation of the term used in phase propagation , @xmath235 we use similar identities to simplify the relationship between the vector and approximate recovery , @xmath236 next , we estimate using the triangle inequality @xmath237 finally , recalling that @xmath238 was bounded by the induction assumption , @xmath239 let @xmath57 . for any vectors @xmath243 and @xmath221 , if @xmath61 is an orthonormal basis such that for all @xmath64 from @xmath65 to @xmath1 , @xmath222 , and @xmath244 is the vector recovered from lemma [ lem : full - induct - noise ] , then an operator @xmath245 with null space equal to the set of complex multiples of @xmath244 can be constructed using the values of @xmath225 and the basis @xmath61 . as in lemma [ lem : full - induct - noise ] , we let @xmath246 with respect to the basis @xmath97 , we define the multiplication operator @xmath247 by the map @xmath248 , where as in the noiseless case , we set @xmath249 . similarly , we define the multiplication operator @xmath250 by the map @xmath251 . note that @xmath252 is invertible if and only if @xmath253 for all @xmath64 from @xmath65 to @xmath1 . this is true because @xmath254 let @xmath98 be the left shift operator @xmath255 as before . with these operators , we define the operator @xmath256 as @xmath257 . then for any @xmath110 , @xmath258 so any complex multiple of @xmath244 is in the null space of this operator . conversely , assume that @xmath76 is in the null space of @xmath259 . we will use an inductive argument to show that for any @xmath64 from @xmath65 to @xmath1 , @xmath260 . note that these quotients are well defined , because @xmath261 then the base case @xmath114 is trivial . for the inductive step , note that for any @xmath64 from @xmath65 to @xmath22 , @xmath262 thus , for any @xmath64 from @xmath65 to @xmath22 , @xmath263 which shows that @xmath76 is a complex multiple of @xmath244 . if @xmath67 , @xmath264 , @xmath265 , @xmath266 , and @xmath259 are as in the preceding theorem , @xmath244 is the vector recovered from lemma [ lem : full - induct - noise ] , and @xmath267 is the vector that solves @xmath268 then @xmath269,[x])\le2\left\|\tilde x-\frac{\overline{x_1}}{|x_1|}x\right\|_2+\frac{\sqrt d\|\eps\|_\infty}{2\sqrt m\|x\|_\infty}$ ] by the mean value theorem and the concavity of the square root , there exists a @xmath227 between @xmath270 and @xmath271 ( so that @xmath272 ) such that @xmath273 because the null space of @xmath259 contains only multiples of @xmath244 , we know @xmath274 and thus , using concavity gives the estimate @xmath275 let @xmath277 . for any nonzero polynomial @xmath40 , and any @xmath41 , if there exists a @xmath53 on the unit circle such that @xmath278 , then an approximation @xmath43 can be constructed using the dirichlet kernel and the values of @xmath44 , such that if @xmath279 then for some @xmath280 on the unit circle @xmath281 let @xmath149 be the normalized dirichlet kernel of degree @xmath22 , so that for any @xmath140 in the unit circle @xmath150 . then the set of functions @xmath151 is orthonormal with respect to the @xmath123 inner product on the unit circle , and any @xmath152 can be interpolated as @xmath153 . if an error @xmath282 is present on each of the values @xmath283 , and if we let @xmath284 , then for any @xmath140 on the unit circle @xmath285 if @xmath154 , then @xmath155 . thus , each of the functions @xmath48 , @xmath49 , and @xmath50 , are in @xmath122 , and using the dirichlet kernel these functions may be interpolated from the values of @xmath44 . the error present in the sample values means that approximating trigonometric polynomials are obtained from this interpolation , with a uniform error that is less than @xmath52 for any point on the unit circle . let @xmath286 , @xmath287 , and @xmath288 be these approximating trigonometric polynomials . we find a @xmath53 that satisfies the hypotheses of the theorem by a simple maximization argument on @xmath289 . the set @xmath160 is an orthonormal basis for @xmath21 , and @xmath33 is full in this basis . then because the values of @xmath290 , @xmath291 , and @xmath292 at the points @xmath164 , as well as an error @xmath293 that depends on @xmath294 , @xmath295 , and @xmath296 , with @xmath297 , correspond to the measurements @xmath298 and we know these values on the entire unit circle , we may apply either of the full vector reconstructions given earlier to obtain an approximation @xmath299 for @xmath33 . if lemma [ lem : full - induct - noise ] is applied to these measurements , and we use the equivalence of norms , @xmath300 and @xmath301 then with @xmath302 , and @xmath303 we obtain a vector of coefficients @xmath304 , such that for all @xmath82 from @xmath65 to @xmath1 , @xmath305 and by remark [ rem : incr ] @xmath306 let @xmath307 . then minkowski s inequality gives terms that form geometric series , @xmath308 to obtain a uniform error bound that only assumes bounds on the norms of the vector @xmath33 and on the magnitude of the noise @xmath309 , we use the max - min principle from section [ sec : max - min ] . this provides us with a universally valid lower bound @xmath310 that applies to the above theorem . [ thm : main - uni ] let @xmath35 , and @xmath311 . for any polynomial @xmath40 with @xmath312 , and any @xmath41 , if @xmath313 and @xmath314 , then an approximation @xmath43 can be reconstructed using the dirichlet kernel and the values of @xmath44 , such that if @xmath315 then for some @xmath280 on the unit circle @xmath316 by lemma [ lem : dist - bnd ] we know that there exists a @xmath53 on the unit circle such that the distance between any element of @xmath54 and any roots of any nonzero truncations of @xmath33 is at least @xmath55 . then by lemma [ lem : mag - bnd ] we know that for all @xmath64 from @xmath65 to @xmath1 , @xmath317 . thus , there exists a @xmath53 on the unit circle such that @xmath318 for all @xmath64 from @xmath65 to @xmath1 and we may use @xmath53 and @xmath319 in the preceding theorem . when we apply the above theorem , we get @xmath320 we remark that any @xmath53 that satisfies the claimed max - min bound does not necessarily satisfy lemma [ lem : dist - bnd ] . this means that the above theorem would benefit immediately from an improved lower bound on the minimum magnitude . as the final step for the main result , we remove the normalization condition on the input vector . since the norm of the vector enters quadratically in each component of @xmath321 , the dependence of the error bound on @xmath322 is not linear . instead , we obtain a bound on the accuracy of the reconstruction which is inverse proportional to the signal - to - noise ratio @xmath323 , assuming that @xmath309 is sufficiently small compared to @xmath324 . let @xmath35 , and @xmath311 . for any nonzero polynomial @xmath40 , and any @xmath41 , if @xmath313 and @xmath325 , then an approximation @xmath43 can be reconstructed using the dirichlet kernel and the values of @xmath44 , such that if @xmath326 then for some @xmath280 on the unit circle @xmath327

the main objective of this paper is to find algorithms accompanied by explicit error bounds for phase retrieval from noisy magnitudes of frame coefficients when the underlying frame has a low redundancy . we achieve these goals with frames consisting of @xmath0 vectors spanning a @xmath1-dimensional complex hilbert space . the two algorithms we use , phase propagation or the kernel method , are polynomial time in the dimension @xmath1 . to ensure a successful approximate recovery , we assume that the noise is sufficiently small compared to the squared norm of the vector to be recovered . in this regime , the error bound is inverse proportional to the signal - to - noise ratio . upper and lower bounds on the sample values of trigonometric polynomials are a central technique in our error estimates .

as described in the literature ( e.g. * ? ? ? * ) , the k@xmath0 emission of helium - like ions takes place via four lines , with the designations w , x , y and z : @xmath5 ( w ) , @xmath6 ( x ) , @xmath7 ( y ) , @xmath8 ( z ) . from these four lines , line ratios have been investigated for diagnostic purposes . one of these ratios , @xmath2 , is sensitive to temperature and is defined as @xmath9 where @xmath10 is the intensity of the w line in units of number of photons per unit volume per unit time . for heavier elements , additional lines arising from transitions of the type @xmath11 , where the upper @xmath12 states are autoionizing , tend to complicate this simple spectrum ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . the kll satellite lines ( which arise from configurations of the type @xmath13 ) are designated with the letters a v ( see @xcite ; the most recent treatment is given by @xcite ) . higher satellite lines arising from @xmath12 , @xmath14 , are usually not given separate designations . often , astrophysical spectra can not be measured such that these satellite lines are adequately resolved ; the result is what appears to be a broadened and redshifted w line according to the intensities of the kll lines in toto within the k@xmath0 complex ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? @xcite first proposed a method of analysing emission spectra if the resolution was sufficient to resolve the spectra into two ranges , one corresponding to an energy range around the w line , the other including everything redward of this range . the range about the w line would include not only the w line itself , but also most of the satellite lines arising from the configurations @xmath12 , @xmath15 . the redward range would include the bulk of the kll satellite lines , in addition to the x , y and z lines . thus , they proposed redefining @xmath2 ( referred to as @xmath16 below ) by taking the integral of the flux redward of some specified boundary line and dividing by the integral of the flux blueward of that same line . @xcite considered the same effect by including the intensity of the satellite lines as part of the numerator in their calculation of @xmath2 . soon after , @xcite proposed a new ratio , @xmath1 , which included all the kll satellite lines in the numerator and all the satellite lines arising from higher shells in the denominator . for some elements , a weak kll line is present in the area one would associate with the w line ; additionally , for most heavier elements , some higher lines ( which arise from configurations such as @xmath12 , @xmath4 ) have low enough energies such that they should be included with the x , y and z lines in the numerator @xcite . to improve on these earlier efforts , the @xmath1 line ratio is redefined in the current work as @xmath17 where @xmath18 is the boundary line between the two energy ranges , @xmath19 and @xmath20 denote the energy range of the @xmath21 complex , and @xmath22 is the energy of a particular line , @xmath23 . thus , each sum includes the intensity of each line which has its centroid in the appropriate range . in the low temperature limit , @xmath24 , but at high temperatures , doppler broadening will cause the wings of lines near @xmath18 to appear in the other range when computing @xmath16 . in addition to being a temperature sensitive diagnostic , the @xmath2 and @xmath1 ratios are also sensitive to the ionization state of the plasma @xcite . while plasmas out of coronal equilibrium are not considered in this work , the results presented here have direct implications and utility to modeling those systems . lastly , it should be noted that some recent work @xcite omits the satellite lines from analysis of fe k@xmath0 observations on the basis of the argument that @xcite showed that the contributions from these lines can be neglected above the temperature of he - like maximum abundance . while @xcite reported that @xmath25 in the range @xmath26 k ( * ? ? ? 2 ) , they also showed that the satellites are an important part of the flux in this temperature range ( * ? ? ? 1 ) . in these earlier calculations , the contribution of satellite lines to the denominator and numerator of @xmath1 effectively cancelled each other out , resulting in @xmath25 . however , the calculations of @xcite did not treat the kln ( @xmath4 ) , satellite lines on par with the kll lines . specifically , the kll lines were treated rigorously according to the method of @xcite , while the kln ( @xmath4 ) lines were treated more approximately via a scaling of ratios of autoionization rates . additionally , recent work @xcite has shown that the cascade contribution to the recombination rates @xcite used in @xcite diverges from the corresponding contribution calculated with more modern distorted - wave and r - matrix methods at temperatures above the fe he - like temperature of maximum abundance ( @xmath27 k ) . for these reasons a new study which treats the kln ( @xmath4 ) satellite lines on par with the kll satellite lines , and which is also based on more accurate atomic data , is warranted . the present work employed the general spectral modeling ( gsm ) code ( @xcite , see also @xcite ) . gsm is based on the ground - state - only quasi - static approximation ( e.g. * ? ? ? * ) , a common method for modelling low density plasmas such as found in astrophysics , which assumes that the ionisation balance portion of the model can be separated from a determination of excited - state populations . the rationale for this approximation is two fold : first , the times scales for ionisation and recombination are much longer than the time scales for processes inside an ionisation stage , and second , the populations of the excited states have a negligible effect on ionisation and recombination . ( see * ? ? ? * for a discussion of the validity of this approximation . ) thus , the first step in a gsm calculation is to solve the coupled set of ionisation balance equations given by @xmath28 where @xmath29 is the total population in the @xmath30 ionisation stage , @xmath31 the electron number density , @xmath32 is the electron temperature , @xmath33 is a bulk collisional ionisation rate coefficient , @xmath34 a bulk 3-body recombination rate coefficient , and @xmath0 a bulk recombination rate coefficient ( which includes radiative and dielectronic recombination ) . in general , photoionization and stimulated recombination are included as well , but as this work considers only collisional plasmas , the rate coefficients associated with these processes have been omitted from equation ( [ ionbaleqn ] ) . once the values of @xmath35 have been determined , the ground - state - only quasi - static approximation then allows one to solve for the excited - state populations in a given ionisation stage , with the approximation of treating the ionisation stages adjacent to the ionisation stage of interest as being entirely in the ground state . as the total population in the ionisation stage of interest and the two adjacent ionisation stages are known , the excited - state populations can be determined by solving a modified version of the full set of collisional - radiative equations given by @xmath36 where the variables are defined more or less as before ; @xmath37 is the population in the @xmath38 state of the @xmath39 ionisation stage , @xmath40 is an electron - impact ( de-)excitation effective rate coefficient , @xmath41 is a proton - impact ( de-)excitation effective rate coefficient , @xmath42 is an alpha - particle - impact ( de-)excitation effective rate coefficient , @xmath43 is an effective autoionization rate coefficient , and @xmath44 is an effective radiative decay rate . here we have used the `` eff '' superscript to denote the possible use of effective rate coefficients since gsm offers the option of treating some of the excited states as statistical conduits ( using branching ratios ) and others explicitly ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? explicit states are those that appear in the set of coupled equations presented in equations ( [ excitedformula ] ) and ( [ matcons ] ) . when all states within an ionisation stage are treated explicitly , the `` eff '' superscript is not necessary since all of the rate coefficients represent direct processes only . when the statistical treatment is employed , the rate coefficients associated with the processes passing through statistical states are combined with the direct rate coefficients between explicit levels by summing over all the indirect paths through the statistical states . this process is simplified by the use of the _ collisionless transition matrix _ ( ctm ) , @xmath45 , which can be thought of as the probability that an ion in statistical state @xmath46 will end up in an explicit state @xmath47 , assuming that the time scale for collisions is very long when compared to the time scale for the spontaneous processes of autoionization and radiative decay . if @xmath48 represents the set of explicit states , and @xmath49 a state such that @xmath50 , the ctm can be defined using the recursive expression @xmath51 where @xmath52 is the appropriate type of spontaneous rate ( either radiative decay or autoionization ) to connect states @xmath49 and @xmath53 . it should be noted that if @xmath54 the ctm is not meaningful , and as such is defined to be zero . the effective rate coefficient is then calculated by summing the direct rates , and the fraction ( as determined by the ctm ) of each indirect rate which contributes to an effective rate . for example , effective recombination ( rr+dr ) rate coefficients are calculated as @xmath55 where @xmath56 is a dielectronic capture rate and the sums take into account both radiative recombination and dielectronic capture followed by radiative cascade . it should be noted that there are terms in these sums that would be represented by explicit resonances in r - matrix cross sections . such an approach allows for the inclusion of resonances when perturbative ( e.g. distorted - wave ) cross sections are employed . this approach is sometimes referred to as the independent - process , isolated - resonance ( ipir ) method ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in calculations that consider r - matrix data , care must be taken to exclude these terms from the summations in equation ( [ rceq ] ) in order to avoid double counting the resonance contributions . once excited - state populations have been calculated , the intensity of each line in the spectral region of interest is calculated according to @xmath57 each line is then given a line shape corresponding to a thermal doppler - broadened gaussian profile . the total spectrum ( or emissivity ) , @xmath58 , for a given photon energy , @xmath59 , can be expressed as @xmath60 where @xmath58 is in units of energy per unit volume per unit time per energy interval , @xmath23 ranges over the set of all included transitions in the desired energy range , @xmath61 is the transition energy associated with a given line , and the ion temperature , @xmath62 , is taken to be equal to the electron temperature . as this work is concerned with steady - state plasmas , the solution to the coupled set of ionisation balance equations , eq . ( [ ionbaleqn ] ) , were taken to be those of @xcite for all three elements ( ca , fe , and ni ) considered in this work . furthermore , as the cases considered are well within the low density limit ( @xmath63 @xmath64 ) , the approximation @xcite made in neglecting three - body recombination is valid . the present work considered multiple classes of models for each of the three elements . each model contains a different set of detailed atomic data . the first class , composed mostly of distorted - wave ( dw ) data ( and denoted by ni : dw , fe : dw , and ca : dw for the three elements ) , uses a set of data calculated entirely by the los alamos suite of atomic physics codes ( e.g. * ? ? ? * ; * ? ? ? the cats code was used to calculate the wave functions , energies , and dipole allowed radiative decay rates for all fine - structure levels arising from the configurations @xmath65 , @xmath66 , @xmath67 , @xmath68 , @xmath12 , and @xmath69 with @xmath70 and @xmath71 , which span the h - like , he - like , and li - like ionisation stages . the gipper code was used to calculate all autoionization rates and photoionization cross sections in the distorted - wave approximation , as well as collisional ionisation cross sections using a scaled - hydrogenic approximation which has been shown to agree well with distorted - wave results for highly charged systems . distorted - wave cross sections for all electron - impact excitation transitions out of the lowest seven levels of the helium - like ionisation stage , as well as the @xmath72 complex of the li - like ionisation stage were calculated with the ace code . cross sections for the remaining electron - impact excitation transitions were computed in the more approximate plane - wave born approximation . lastly , the non - dipole @xmath73 values that give rise to the x and z lines , as well as a two - photon decay rate from @xmath74 used in obtaining the populations from equations ( [ excitedformula ] ) and ( [ matcons ] ) , were obtained from @xcite . proton- and alpha - particle - impact excitation rates between the he - like @xmath75 levels were also taken from @xcite . the protons and alpha particles were taken to have the same temperature as the electrons , and to have densities of 0.77 and 0.115 times the electron density respectively ( @xmath76 , @xmath77 ) . the cats level energies for the lowest seven levels of the he - like ionisation stage and the lowest three levels of the li - like ionisation stage were replaced by values taken from the nist atomic spectra database @xcite , as were the energies for the kll autoionizing levels for li - like ni and fe . as the nist database does not contain complete information for the autoionizing kll levels of ca , the level energies calculated by cats were retained for all ca autoionizing states . all of the fine - structure levels arising from the @xmath78 , @xmath79 , @xmath75 , @xmath72 , and @xmath12 configurations with @xmath70 and @xmath71 were treated explicitly when solving for the excited - state populations appearing in equations ( [ excitedformula ] ) and ( [ matcons ] ) . in the second class of models the electron - impact excitation , radiative decay , and both radiative and dielectronic recombination data in the dw model are replaced with data calculated using r - matrix ( rm ) methods , where such data are publicly available . for ni , the radiative decay rates of @xcite and the unified recombination rates of @xcite were used to create the ni : rm data set . as for fe , two sets of r - matrix electron - impact excitation rates are available and are considered here . the first set , fe : rm , includes the electron - impact excitation collision strengths of @xcite , a subset of the radiative decay rates of @xcite ( where the initial state is a fine - structure level arising from the configurations @xmath66 where @xmath80 , @xmath81 or @xmath82 , @xmath83 ) and the corresponding subset of the unified recombination rates of @xcite ( @xmath78 @xmath84 recombining into all fine - structure levels arising from @xmath66 where @xmath80 , @xmath81 or @xmath82 , @xmath83 ) . the second set , fe : rm2 , is identical to fe : rm except that it uses the electron - impact excitation collision strengths of @xcite . lastly , one r - matrix type model is considered for ca , ca : rm , which also incorporates the electron - impact excitation data of @xcite . the last class of models is an expansion of the second class by also incorporating autoionization rates calculated from recombination cross sections ( e.g. * ? ? ? specifically , @xcite provided this type of data for fe and ni . these data have been combined with the fe : rm and ni : rm sets to make the fe : rm+ and ni : rm+ sets . the fe : rm2 data set , which incorporates the collision strengths of @xcite , has not been expanded into a fe : rm2 + data set due to the good agreement ( which is shown in the following section ) between the fe : rm and fe : rm2 data set . as no data of this type are yet available for ca , no model of this class is considered for ca . in addition to constructing the models , the boundary line between the high energy and low energy section of each spectrum had to be chosen . as pointed out by @xcite there is an energy gap that forms between the w line ( and the satellite lines that blend with it ) and the rest of the spectrum . this gap was found by inspection , and the boundary energy , @xmath18 , was chosen to be 3895 ev , 6690 ev , and 7794 ev for ca , fe , and ni respectively . figs . [ ca - gd][ni - gd ] display the calculated values of the @xmath2 and @xmath1 ratios as a function of temperature for each of the models , along with plots of certain ratios that help to illustrate where the differences occur . overall , the present calculations predict @xmath1 ratios that are significantly higher than the corresponding @xmath2 ratios for all the models that are considered . this behaviour is in qualitative agreement with previous studies @xcite ; it should be noted that this more detailed study predicts a significantly greater value of @xmath1 below the temperature of maximum abundance than any of the previous studies . additionally , the impact of satellite lines on the @xmath1 ratio keeps the @xmath1/@xmath2 ratio greater than one over a much broader range than shown in the study of @xcite . the principal reason for this behaviour is that the more approximate treatment of klm and higher lines in @xcite appears to overestimate their importance , especially at higher temperatures ( see * ? ? ? 3 ) . this overestimation leads to a cancelling effect , whereby the klm and higher lines in the denominator of @xmath1 cancel out the effect of the kll satellite lines in the numerator . additionally , the present calculations allow the klm and higher satellite lines to be included within the energy range they actually fall , which is in the redward section ( i.e. the numerator of @xmath1with the x , y , and z lines ) for some of the higher satellite lines . thus , the satellite lines in these new calculations have an impact on the line ratio @xmath1 at temperatures well above the temperature of maximum abundance for the he - like ionisation stage . one practical consequence of this last statement is that essentially any spectral analysis of the he - like k@xmath0 lines requires the satellite lines to be treated in a detailed manner ( unless the measured spectra are sufficiently well resolved so that the satellite lines can be readily distinguished ) . due to the level of detail and improved atomic data included in the present calculations , they are expected to be a significant improvement over previous work . while there are differences between the @xmath2 ratios , as well as the @xmath1 ratios , predicted by each of the data sets , these differences are all less than 15 percent , which is within the typical 1020 percent uncertainty reported for the r - matrix data ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? in order to understand these differences , spectra were examined for a wide range of temperatures . in general , spectra for all the elements and models considered were found to be in excellent agreement with each other , even when comparing results obtained from rm and dw data sets . the differences were all less than 12 percent for strong lines , which include the w , x , y , and z lines , as well as most of the satellite lines . there were larger differences ( up to @xmath8550 percent ) for some weak but barely visible satellite lines ( like c ) , and even larger differences ( up to @xmath85150 percent ) for some weaker satellite lines that do not contribute in any appreciable manner to the spectra . these larger differences have very little impact on the spectra or the line ratios as the corresponding lines are quite weak . two sample spectra for fe , for which the disagreement in the ratios was among the largest , are presented in figs . [ fe-1 ] and [ fe-2 ] . as illustrated in the upper panel of fig . [ fe-1 ] , at an electron temperature of @xmath86 k , the overall agreement between the spectra computed with the various models is excellent . the data in the bottom panel of fig . [ fe-1 ] indicate more precisely where the largest discrepancies occur . one observes that the use of r - matrix data results in an increase of the z line and a decrease in the x line relative to the distorted - wave model . additionally , the fe : rm data set predicts a decrease in the y line , and an increase in the w line relative to the distorted - wave model ; the fe : rm2 data set predicts the same changes , but to a lesser extent . from this inspection one can conclude that the agreement between the @xmath2 ratios calculated from the fe : dw and fe : rm2 data sets is fortuitous because of a cancellation in the quantities that comprise the numerator and denominator of that ratio . on the other hand , the decrease in the x and y lines predicted by the fe : rm versus the fe : dw data set are larger than the corresponding increase in the z line . this overall reduction in the numerator of the @xmath2 ratio , when coupled with the increase in the w line between the fe : rm and fe : dw data sets , results in the reduced @xmath2 ratio calculated from the fe : rm model at low temperatures . fig . [ fe-2 ] , which displays spectra at a much higher electron temperature of @xmath87 k ( which is approximately ten times higher than the temperature of maximum abundance for he - like fe ) again shows excellent agreement . an analysis of the bottom panel of fig . [ fe-2 ] shows that both r - matrix data sets predict higher x and y lines , and a decreased w line , relative to the distorted - wave results . the net result of these differences is the increased @xmath2 and @xmath1 ratios displayed in fig . [ fe - gd ] . separate calculations ( not shown ) indicate that the increase in the x line is due to slightly higher r - matrix recombination rates rather than to sensitivity to the electron - impact excitation rates . this populating mechanism for the x line is consistent with the typical viewpoint in the literature ( e.g. * ? ? ? the y line , on the other hand , is sensitive to both electron - impact excitation and recombination rates at this high temperature ; for this case the recombination rates are dominant in determining the population of the excited state , but the excitation rate is non - negligible as y is an intercombination line . while this temperature ( @xmath88 k ) is above the peak of the dr hump ( see * ? ? ? 5 ) , it is still in a range where the resonances of the r - matrix cross section are important to the recombination rate . the high - temperature differences observed for the ni @xmath2 and @xmath1 ratios ( fig . [ ni - gd ] ) , for which only the recombination rates were changed among the various models , have a similar explanation . additionally , separate calculations ( not shown ) indicate that the differences in the w line are primarily due to differences in the electron - impact excitation data . the importance of excitation over recombination as a populating mechanism of the w line is expected since this transition is dipole allowed ( e.g. * ? ? ? the net effect of these differences is the increase in the r - matrix @xmath2 and @xmath1 ratios which is observed above the temperature of maximum abundance in fig . [ fe - gd ] . despite the subtle differences in the spectra presented above , we emphasise that the discrepancies in the important lines are well within the uncertainties ( 20 percent ) usually cited for r - matrix data . the disagreement in these spectra were among the largest seen in this study , which speaks to the excellent overall agreement between the rm and dw models . lastly , it should be noted that the line positions for the klm and higher satellite lines are a significant source of uncertainty in these calculations . while the accuracy of the line positions is estimated to be @xmath850.1 percent , a shift of that size could impact the spectra significantly by causing some of the strong klm lines , which blend with the w line in this present work , to move sufficiently far such that they should be considered with the bulk of the kll lines in the numerator of @xmath1 . this fact is underscored by the appearance of klm and higher lines @xmath857 ev blueward of the w line in fig . [ fe-1 ] , when they should instead converge upon the w line . if some of these higher lying satellite lines do in fact blend with the x line , the impact would be a corresponding increase in the @xmath1 ratio . new , more detailed calculations of the emission spectra of the he - like k@xmath0 complex of calcium , iron , and nickel have been carried out using atomic data from both distorted - wave and r - matrix calculations . spectra from these calculations are in excellent agreement , and demonstrate that satellite lines are important to both the spectra and the @xmath1 ratio across a wide temperature range that includes temperatures significantly above the temperature of maximum abundance for the he - like ionisation stage . a major conclusion of this work is the need to include satellite lines in the diagnosis of he - like k@xmath0 spectra of iron peak elements in low density , collisional ( coronal ) plasmas , even at temperatures well above the temperature of maximum abundance . when the satellite lines are appropriately taken into account , the @xmath1 ratio remains an excellent potential temperature diagnostic . another important application of the results presented herein , is in the well - known application of the @xmath2 or @xmath1 ratio to ascertain the ionization state of a plasma . as shown in figs . [ ca - gd][ni - gd ] , the @xmath1 ratio is far more sensitive to the ionization state at @xmath89 than the @xmath2 ratio , by as much as a factor of 100 . therefore , it is imperative to calculate the @xmath1 values as precisely as possible at temperatures where the dielectronic satellite intensities are rapidly varying . such conditions are known to occur in plasmas which are not in coronal equilibrium , as discussed by @xcite . furthermore , it should be noted that , while this work does not consider the effect of satellite lines on the density sensitive diagnostic ratio @xmath90 ( @xmath91 ) , the effect of these lines is significant enough that they would need to be taken into account under conditions where @xmath90 is used . this inclusion is warranted due to the manifestation of the satellites embedded within the k@xmath0 complex , and in many cases blended with the principal lines x , y and z. the excellent agreement between the spectra produced from r - matrix and distorted - wave data used in the models presented in this work bolsters confidence in both data sets . any disagreement between the two sets of spectra would have indicated an error in the fundamental atomic data because the ipir approach has been shown to give good agreement with close - coupling approaches when producing the fundamental rate coefficients ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the present work provides a more stringent test of this assumption by including those rate coefficients in a fully integrated spectral calculation that takes into account several ion stages and includes the coupling between all of the important atomic processes . the good agreement observed in this work reaffirms the fact that in highly charged systems , models based on data calculated from computationally less expensive distorted - wave methods can reproduce the results of models based on r - matrix data if the effect of resonances are taken into account as independent processes . this behaviour , however , is not expected to remain true for all conditions , especially when near neutral systems are prevalent . the results presented in this paper should be applicable to high - energy and high - resolution x - ray spectroscopy of laboratory and astrophysical plasmas . astrophysical observations of the k@xmath0 complex of high - z ions , particularly the 6.66.7 kev range of the fe k@xmath0 , were expected to be made by the high - resolution x - ray satellite suzaku , but could not be performed due to instrument failure . it is , however , expected that these calculated results would be valuable in future x - rays missions such as the recently planned joint esa - nasa international x - ray observatory . this work was partially conducted under the auspices of the united states department of energy at los alamos national laboratory . much of the development of gsm was also done at the ohio supercomputer center in columbus , ohio ( usa ) . the work by the osu group ( snn , akp ) was partially supported by a grant from the nasa astrophysical theory program .

new , more detailed calculations of the emission spectra of the he - like k@xmath0 complex of calcium , iron and nickel have been carried out using data from both distorted - wave and r - matrix calculations . the value of the @xmath1 ratio ( an extended definition of the @xmath2 ratio that accounts for the effect of resolved and unresolved satellite lines ) is significantly enhanced at temperatures below the temperature of he - like maximum abundance . furthermore it is shown that satellite lines are important contributors to the @xmath1 ratio such that @xmath3 at temperatures well above the temperature of maximum abundance . these new calculations demonstrate , with an improved treatment of the kln ( @xmath4 ) satellite lines , that k@xmath0 satellite lines need to be included in models of he - like spectra even at relatively high temperatures . the excellent agreement between spectra and line ratios calculated from r - matrix and distorted - wave data also confirms the validity of models based on distorted - wave data for highly charged systems , provided the effect of resonances are taken into account as independent processes . [ firstpage ] atomic data , atomic processes , line : formation , line : profiles , x - rays : general

complex networks have become a natural abstraction of the interactions between elements in complex systems @xcite . when the type of interaction is essentially identical between any two elements , the theory of complex networks provides with a wide set of tools and diagnostics that turn out to be very useful to gain insight in the system under study . however , there are particular cases where this classical approach may lead to misleading results , e.g. when the entities under study are related with each other using different types of relations in what is being called multilayer interconnected networks @xcite . representative examples are multimodal transportation networks @xcite where two geographic places may be connected by different transport modes , or social networks @xcite where users are connected using several platforms or different categorical layers . here , we focus our study on the transportation congestion problem in multiplex networks , where each node is univocally represented in each layer and so the interconnectivity pattern among layers becomes a one - to - one connection ( i.e. , each node in one layer is connected to the same node in the rest of the layers , thus allowing travelling elements to switch layer at all nodes ) . this representation is an excellent proxy of the structure of multimodal transportation systems in geographic areas @xcite . the particular topology of each layer is conveniently represented as a spatial network where nodes correspond to a certain coarse grain of the common geography at all layers @xcite . transportation dynamics on networks can be , in general , interpreted as the flow of elements from an origin node to a destination node . when the network is facing a number of simultaneous transportation processes , we find that many elements travel through the same node or link . this , in combination with the possible physical constraints of the nodes and links , can lead to network congestion , in which the amount of elements in transit on the network grows proportional to time @xcite . usually , to analyze the phenomenon , a discrete abstraction of the transportation dynamics in networks is used @xcite . multimodal transportation can also be mathematically abstracted as transportation dynamics on top of a multiplex structure . note that routings on the multilayer transportation system are substantially different with respect to routings on single layer transportation networks . in the multilayer case , each location of the system ( e.g. geographical location ) has different replicas that represent each entry point to the system using the different transportation media . thus , each element with the intention of traveling between locations @xmath0 and @xmath1 have the option to choose between the most appropriate media to start and end its traversal . we assume that elements traverse the network using shortest paths , so each element chooses the starting / ending media that minimizes the distance between the starting / ending locations . as we will show in this work , this `` selfish '' behavior provokes an unbalance in the load of the transportation layers inducing congestion , similarly to what is presented in the classical counterintuitive result of the braess paradox @xcite . note that in a multiplex network we can have two types of shortest paths : paths that only use a single layer ( intra - layer paths ) and paths that use more than one layer ( inter - layer paths ) . hereafter , we develop the analysis of transportation in multiplex networks , consisting of @xmath2 locations ( nodes per layer ) and @xmath3 layers , and quantify when this structure will induce congestion . to this aim , we describe , with a set of discrete time balance equations , ( one for each node at each layer ) , the increment of elements , @xmath4 , in the queue of each node @xmath0 on layer @xmath5 : @xmath6 where @xmath7 is the average number of elements injected at node @xmath0 in layer @xmath5 ( also called the injection rate , which can be assimilated to an external particle reservoir ) , @xmath8 is the average number of elements that arrive to node @xmath0 in layer @xmath5 from the adjacent links of that node ( ingoing rate ) , and @xmath9 $ ] corresponds to the average number of elements that finish their traversal in node @xmath0 in layer @xmath5 or that they are forwarded to other neighboring nodes . the control parameter is @xmath7 : small values of it correspond to low density of elements in the network and high values to high density of elements . a graphical explanation of the variables of the model is shown in fig . [ fig : modelexplanation ] . and @xmath10 of the node . ] before reaching congestion , the amount of elements in the queue of each node is constant in average , @xmath11 and consequently , @xmath12 , where @xmath13 is the maximum processing rate of the node . a node @xmath0 on layer @xmath5 becomes congested when it is requested to process more elements than its maximum processing rate , @xmath14 , and therefore , its onset of congestion is achieved when @xmath15 . we are interested on computing the maximum injection rate @xmath7 for which the network is congestion free . in the non - congested phase , as well as on the onset of congestion , the amount of ingoing elements to each node @xmath8 can be obtained in terms of the node s effective betweenness , see @xcite . our scenario is slightly different since we need to account for the effective betweenness of the multiplex . in addition to the intra - layer and inter - layer paths , our definition of the dynamics also accounts for the number of shortest paths that start ( @xmath16 ) and end ( @xmath17 ) at node @xmath0 on layer @xmath5 ( this can be computed using any classical shortest path algorithm ) . note that @xmath18 . these factors are essential to understand the unbalance of loads between layers in the multiplex network , and only depend on the distribution of shortest path in the full structure . in the following , we assume a constant injection rate , @xmath19 , being @xmath20 the common injection rate at all locations @xmath0 . in addition we also suppose , without loss of generality , that the maximum processing rate is the same for all nodes of the multiplex network , @xmath21 . these hypothesis simplify the analysis but are not crucial to develop it . to obtain the critical injection rate of the multiplex , we require expressions for @xmath7 and @xmath8 . the injection rate of node @xmath0 on layer @xmath5 can be obtained as the product of the amount of elements that enter the network using location @xmath0 , @xmath22 , and the fraction of multiplex shortest paths that start on node @xmath0 on layer @xmath5 , @xmath23 : @xmath24 the ingoing rate of each node , @xmath8 , depends on the fraction of shortest paths that pass through or end in it @xcite . thus , @xmath8 can be obtained as the number of generated elements overall the network at each time step , @xmath25 , times the fraction of them that arrive ( @xmath17 ) or traverse it ( @xmath26 , the topological betweenness ) : @xmath27 when the network is already congested , eq . ( [ multiplexsigmabeforethreshold ] ) does not generally holds since elements traversing congested paths stack in intermediate nodes resulting in a cascade effect not captured by the betweenness . therefore , our analysis only covers the onset of congestion and it can not be directly applied to the congested regime . an efficient algorithm to compute the betweenness on multiplex structures can be found in @xcite for shortest paths dynamics and in @xcite for random walk dynamics . the computation of @xmath16 and @xmath17 for shortest paths dynamics can be obtained modifying the previously cited algorithm to account for the amount of paths that reach the source and destination nodes . the onset of congestion of the multiplex is attained when a node @xmath0 in layer @xmath5 is required to process elements at its maximum processing rate , i.e. @xmath28 . therefore , the critical injection rate of the system , @xmath29 , becomes @xmath30 where @xmath31 and @xmath32 . in the following we call @xmath33 the interconnected betweenness . note that @xmath33 depends on intra - layer paths , inter - layer paths , and on the migration of shortest paths between layers ( more efficient layers contain a larger proportion of the starting and ending routes ) . we test the validity of eq . ( [ eq : criticalgeneratiorate ] ) against monte carlo simulations on top of erds - rnyi multiplex networks , see fig . [ fig : corr_criticgenrate ] . given by eq . ( [ eq : criticalgeneratiorate ] ) predicting the actual onset of congestion in experimental simulations on 500 random multiplex networks formed by two erds - rnyi networks ( of 500 nodes ) as layers . inset ( * a * ) shows the correlation between the experimentally obtained critical injection rate and the analytical approximation in eq . ( [ eq : approximatio_rho_c ] ) where @xmath34 is approximated by @xmath35 . @xmath36 is the coefficient of determination for linear fits.,scaledwidth=45.0% ] in the following , we investigate the role of the topology of the individual layers on the multiplex congestion . first of all , note that in the definition and computation of the multiplex betweenness ( see @xcite ) , the shortest paths ( possibly degenerated ) between all pair of multiplex locations , @xmath37 , are considered . the multiplex structure unbalances , in a highly non - linear way , the distribution of shortest paths among the layers . however , some approximations are possible to grasp the effect of the different contributions to the onset of congestion in multiplex structures . as stated before , an important parameter of traffic dynamics in multiplex networks is the fraction of inter - layer shortest paths , i.e. the fraction of shortest paths that contain , at least , one inter - layer edge . experiments with multiplex networks composed of two layers , each one being a different random erds - rnyi network , show that most of the shortest paths are fully contained within a layer , see fig . [ fig : percentpathusemultiplex ] . this effect becomes more evident as the degree of the layers increases . therefore , the fraction of shortest paths fully contained within layers , @xmath34 , is basically 1 , and the main factor influencing the traffic dynamics is the migration of shortest paths from the less efficient layer ( the one with larger shortest paths ) to the most efficient one . under this situation we can approximate the interconnected betweenness of node @xmath0 in layer @xmath5 , @xmath38 , in terms of the betweenness of node @xmath0 of layer @xmath5 , @xmath39 , when layer @xmath5 is considered as a single layer network : @xmath40 where @xmath41 is the fraction of shortest paths using only layer @xmath5 , satisfying @xmath42 . the effect of the product of @xmath43 is to precisely account for the fraction of all shortest paths that traverse only layer @xmath5 in the multiplex . note that the approximation in eq . ( [ eq : approx_betweennessmultiplex ] ) does not account for the betweenness contribution of the paths that use inter - layer edges . however , the high value @xmath44 , indicates that they are usually negligible , and we can even further approximate @xmath45 . of paths fully contained within layers . each multiplex network is formed by two erds - rnyi layers of 500 nodes each . we plot 100 random realizations for each pair of mean degrees @xmath46 and @xmath47.,scaledwidth=45.0% ] taking advantage of eq . ( [ eq : approx_betweennessmultiplex ] ) , the critical injection rate of the multiplex can be obtained by rescaling the critical injection rate of the individual layers : @xmath48 where @xmath49 is the critical injection rate of the most efficient layer @xmath50 . fractions @xmath51 and @xmath34 are genuine properties of the multiplex network structure that can be obtained by means of the multiplex extension of the brandes betweenness algorithm @xcite . figure [ fig : corr_criticgenrate ] * ( a ) * shows the accuracy of this approximation in the calculation of @xmath52 . the high accuracy obtained in the approximation evidences that the critical injection rate of the multiplex crucially depends on the migration of shortest paths between layers , which is captured in @xmath53 . as an example , consider a multiplex structure composed by two identical layers . in this case , there are no shortest paths using inter - layer edges since they would be longer than the ones fully included in one layer , thus @xmath54 . since paths in both layers are identical , there is a multiplex path degeneration : for each shortest path in layer 1 there is an equivalent shortest path in layer 2 . as a consequence , nodes on the paths only obtain @xmath55 of the betweenness contribution they would obtain if layers were separated , which results in @xmath56 . eventually , we see that for identical layers the multiplex betweenness is @xmath55 of the betweenness computed on any of the layers . [ cols= " < , < " , ] on the other side , consider a multiplex network in which most of the paths in layer @xmath35 have length @xmath57 and most of the paths in layer @xmath57 have length @xmath58 . again , there are very few shortest paths using inter - layer edges since their minimum length is @xmath58 ( i.e. one intra - layer edge , followed by a change of layer through an inter - layer link , and finally another intra - layer edge ) , therefore @xmath59 . moreover , most of the shortest paths make use of layer @xmath35 , where the lengths are shorter , so @xmath60 and @xmath61 . substitution in eq . [ eq : approx_betweennessmultiplex ] shows that the interconnected betweenness of the multiplex is equivalent to the betweenness of the most efficient layer , that in this case is layer @xmath35 . we can compute the congestion induced by a multiplex as the situation in which a multiplex network reaches congestion with less load than the worst of its layers when operating individually . in a multiplex with two layers @xmath35 and @xmath57 ( being @xmath57 the most efficient ) , this limiting situation is obtained when @xmath62 , and consequently : @xmath63 figure [ fig : probabilitymorelessresilient]*(a ) * shows the regions where the multiplex structure induces congestion for sets of erds - rnyi multiplex networks . in each experiment , two erds - rnyi networks with different mean degree are coupled to form a multiplex network . for each pair of mean degrees we have evaluated 100 random realizations of the multiplex network and for each realization we have computed the onset of congestion of the multiplex network and of the individual layers . we have then obtained the fraction of times that the multiplex network reaches congestion before both layers . the boundaries approximated by equation [ eq : congestioninducedbymultiplex_lambdamu ] determine accurately the regions where the multiplex induces congestion . as expected , the approximation using only @xmath64 works well except when both mean degrees are low since on these cases the amount of shortest paths using the multiplex structure is more relevant . surprisingly , for larger degrees ( in the diagonal ) the er networks generated present small fluctuations on the average degree that eventually make a node in one layer to have a maximum degree a little bit larger than in the other layer . this asymmetry , for such dense networks , is enough to provoke a load unbalance that is reflected in the simulations . we have used homogenous random networks multiplexes to demonstrate the use of the analytical approach , however the theory is general for any other multiplex network structure . to conclude this letter , we have also used a different type of topology , random geometric graphs , more akin to represent transportation networks @xcite . to this end , we propose a simple configuration of a random geometric multiplex . we assume each random geometric multiplex is composed of two types of transportation media : short range ( e.g. the bus network ) and long range ( e.g. the subway ) , see fig . [ fig : geometricmultiplex ] . our construction method allows to generate very extreme geometric multiplexes ; from configurations where the long range layer only contains some of the longer edges of the short range layer ( @xmath65 ) , to a long range layer that only contains edges larger than the ones in the short range layer ( @xmath66 ) . however , we usually obtain configurations where the long range layer have some degree of edge overlap with the short range layer . the test set where we have performed the experiments has been constructed by creating @xmath67 random geometric multiplex networks choosing uniformly at random the parameters of the model . figure [ fig : probabilitymorelessresilient]*(b ) * shows that eq . [ eq : congestioninducedbymultiplex_lambdamu ] accurately predicts the region where the multiplex structure induces congestion . in summary , we have analyzed the congestion phenomena on multiplex transportation networks . we have developed an standardized model of how elements traverse those networks and we have provided analytical expression for the onset of congestion . then , we have shown that the multiplex structure induces congestion and derived analytical expressions to determine the network parameters that raise this phenomena . all analytical expressions have been assessed on erds - rnyi and geometric multiplex networks , and showing a perfect agreement with the empirical results . the reason behind this phenomenology is the unbalance of shortest paths between layers . the flow follows the shortest path , increasing the load of the most efficient ( in terms of shortest paths ) layer , and eventually congesting it . theory and experiments developed in this paper are specially useful to understand transportation dynamics on multilayer networks and might help on the development of more efficient transportation networks and routing algorithms . ^2 $ ] ; these are our node locations . we then generate the first layer by adding edges between all locations @xmath0 and @xmath1 separated by a distance @xmath68 lower than a certain radius @xmath69 $ ] . the second layer is generated by adding edges between all node pairs with distance @xmath70 . the values of @xmath71 $ ] force minimum overlapping between both layers . the value of @xmath72 with @xmath73 $ ] ensures the range @xmath74 $ ] does not exceed the radius of first layer.,scaledwidth=35.0% ] this work has been supported by ministerio de economa y competitividad ( grant fis2012 - 38266 ) and european comission fet - proactive projects plexmath ( grant 317614 ) . a.a . also acknowledges partial financial support from the icrea academia and the james s. mcdonnell foundation . 26ifxundefined [ 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 * * , ( ) in link:\doibase 10.1109/asonam.2011.114 [ _ _ ] ( , ) pp . @noop * * , ( ) @noop * * ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * ( ) @noop * * , ( ) http://dblp.uni-trier.de/db/journals/transci/transci39.html#braessnw05 [ * * , ( ) ] in link:\doibase 10.1145/2615569.2615687 [ _ _ ] ( ) pp . @noop ( ) @noop * * ( )

multiplex networks are representations of multilayer interconnected complex networks where the nodes are the same at every layer . they turn out to be good abstractions of the intricate connectivity of multimodal transportation networks , among other types of complex systems . one of the most important critical phenomena arising in such networks is the emergence of congestion in transportation flows . here we prove analytically that the structure of multiplex networks can induce congestion for flows that otherwise will be decongested if the individual layers were not interconnected . we provide explicit equations for the onset of congestion and approximations that allow to compute this onset from individual descriptors of the individual layers . the observed cooperative phenomenon reminds the braess paradox in which adding extra capacity to a network when the moving entities selfishly choose their route can in some cases reduce overall performance . similarly , in the multiplex structure , the efficiency in transportation can unbalance the transportation loads resulting in unexpected congestion .

experiments at lep , slc and tevatron have provided a large number of high - precision data , which , being supplemented by detailed studies of higher - order corrections , allow to probe the standard model at the loop level and subsequently to predict the mass of the higgs boson . in this context , the leptonic effective weak - mixing angle , @xmath3 , plays the most crucial role . it can be defined through the effective vector and axial - vector couplings , @xmath4 and @xmath5 , of the @xmath6 boson to leptons ( @xmath7 ) at the @xmath6-boson pole , @xmath8 the effective weak - mixing angle can be related to the on - shell weinberg angle , @xmath9 , as @xmath10 where @xmath11 and @xmath12 . at tree level , @xmath13 and @xmath14 . the form factor @xmath15 incorporates the higher - order loop corrections . usually , the @xmath16-boson mass , @xmath17 , is not treated as an input parameter but it is calculated from the fermi constant , @xmath18 , which is precisely known from the muon lifetime . the relation between @xmath17 and @xmath18 can be cast in the form @xmath19 where the quantity @xmath20 @xcite contains all higher - order corrections . the presently most accurate calculation of the @xmath16-boson mass includes full two - loop and leading higher - order corrections @xcite . on the other hand , the quantity @xmath21 in eq . ( [ eq : sin ] ) incorporates all corrections to the form factors of the @xmath22 vertex . recently , the calculation of the two - loop electroweak corrections has been completed @xcite . the uncertainty on @xmath3 due to unknown higher orders has been estimated to be 0.000047 , which is substantially smaller than the error of the current experimental value @xmath23 @xcite , but still larger than the expected precision , @xmath24 , of a future high - luminosity linear collider running at the @xmath6-boson pole @xcite . the experimental value for @xmath3 is determined from six asymmetry measurements , @xmath25 , @xmath26 , @xmath27 , @xmath28 , @xmath29 , and @xmath30 . of those , the average leptonic and hadronic measurements differ by 3.2 standard deviations , which is one of the largest discrepancies within the standard model . the main impact stems from two measurements , the left - right asymmetry with a polarised electron beam at sld , @xmath31 , and the forward - backward asymmetry for bottom quarks at lep , @xmath28 . on the experimental side , the only possible source of this discrepancy are uncertainties in external input parameters , in particular parameters describing the production and decay of heavy - flavoured hadrons ; see section 5 of ref . @xcite for a discussion . however , the interpretation of the asymmetry measurements in terms of @xmath3 requires also some theoretical input . the leptonic asymmetries depend on lepton couplings only and can be translated straightforwardly into the leptonic effective weak - mixing angle , with small corrections due to @xmath32- and @xmath33-channel photon exchange . by contrast , the hadronic observables , @xmath29 , @xmath28 and @xmath30 , depend on the quark couplings , @xmath34 . these couplings are associated with a flavour - dependent hadronic effective weak - mixing angle , @xmath35 , @xmath36 the forward - backward pole asymmetry of a quark @xmath37 , @xmath38 , is related to the effective couplings , @xmath39 and @xmath40 , and the effective weak - mixing angle , @xmath35 , by vertex also has a scalar part , besides the vector and axial - vector parts . we checked explicitly that the contribution of this scalar form factor to @xmath28 is more than a factor 1000 smaller than the current experimental uncertainty and thus truly negligible . ] @xmath41 with @xmath42 at tree level , @xmath35 and @xmath3 are identical , but the relations between these quantities receive sizable radiative corrections that need to be included in the analysis . note that , due to the small electric charge of the bottom quark , @xmath43 , the parameter @xmath44 is close to 1 , and @xmath28 is only weakly sensitive to @xmath0 . therefore , it seems unlikely that the discrepancy between @xmath45 and @xmath28 could be explained by radiative corrections . nevertheless , the theoretical prediction for @xmath0 enters in the standard - model fits through several observables , so that a precise prediction of this quantity is important for a robust analysis . for all fermions except bottom quarks , the known radiative corrections to @xmath46 include at least two - loop fermionic electroweak contributions and some leading higher - order corrections ; see ref . @xcite for details . however , for the @xmath47 vertex only one - loop corrections , leading two - loop corrections for large values of the top - quark mass of @xmath48 , and two- and three - loop qcd corrections have been calculated @xcite and included in the zfitter program @xcite ( see also the new program gfitter @xcite ) , which is widely used for global standard - model fits . the remaining two - loop electroweak corrections beyond the @xmath49 contributions are still unknown , although they are expected to be larger than the @xmath48 term , based on experience from @xmath3 . as a result , the present treatment of higher - order electroweak corrections leads to inconsistencies , for example in @xmath28 , since the corrections to @xmath3 and @xmath50 include two - loop and leading three - loop corrections that are absent for @xmath0 and @xmath44 ( see recent discussion in ref . @xcite ) . in this paper , the part of the missing two - loop corrections to @xmath0 with closed fermion loops is presented . we begin by explaining the techniques employed for the calculation in the next section . in section [ results ] , numerical results for @xmath0 are given before the summary in section [ concl ] . we work in the standard model and adopt the on - shell renormalisation scheme , which relates the renormalised masses and couplings to physical observables . details on the renormalisation scheme and explicit expressions for the relevant counterterms can be found in refs . @xcite . for the loop integrations , we employ dimensional regularisation . the problem of @xmath51 matrices in two - loop vertex diagrams with fermion triangle sub - loops is treated in the same way as in refs . @xcite , by evaluating the finite non - anticommutative contribution from @xmath51 to the vertex diagrams in four dimensions . most aspects connected with the calculation of the effective weak - mixing angle for the @xmath52 vertex are the same as for the leptonic effective weak - mixing angle and are discussed in detail in ref . @xcite . the contributions for the two - loop renormalisation terms are identical to the case of @xmath3 , with the exception of the two - loop bottom - quark wave - function counterterm , which involves new self - energy diagrams with internal top - quark propagators ; the first terms of this quantity are given in ref . @xcite . for the two - loop @xmath47 vertex corrections , on the other hand , a number of new three - point diagrams need to be computed . in general , electroweak two - loop corrections can be divided into two groups , which are separately finite and gauge invariant : fermionic corrections ( with at least one closed fermion loop ) and bosonic corrections ( without any closed fermion loops ) . in this article , we focus on the fermionic diagrams as a first step . for the purpose of this calculation , all light - quark masses are neglected in the two - loop diagrams , including the bottom - quark mass . as a result , for many diagrams , known results from the @xmath3 calculation can be used @xcite . the loop integrals for diagrams with closed massless - fermion loops are given in analytical form , while large - mass expansions were employed for diagrams with top quarks in the loops . however , the two - loop corrections to @xmath0 include a new group of integrals that were not covered in previous calculations of @xmath3 , stemming from diagrams with internal @xmath16-boson and top - quark propagators ; see fig . [ diags ] . the computation of these diagrams will be discussed in detail in the following subsections . the two - loop diagrams are computed with several independent methods , so that cross checks can be performed . the first method , based on the observation that all new diagrams in fig . [ diags ] include internal top - quark propagators , uses asymptotic expansions for large top - quark mass . this method was already employed successfully for the calculation of @xmath3 @xcite . for references on the subject , we refer the reader to ref . @xcite . secondly , we develop a code for the evaluation of feynman diagrams with a semi - numerical method , based on the bernstein - tkachov ( bt ) method of ref . this method had already been used previously for one - loop problems @xcite . in a recent series of papers @xcite , it was extended to general two - loop vertices , and some applications to two - loop problems are already known : the leptonic effective weak - mixing angle was presented in ref . @xcite and corrections to the @xmath53 decay width in ref . @xcite . finally , we use another semi - numerical method based on dispersion relations @xcite , which was also used previously for @xmath3 @xcite . this method allows us to evaluate all self - energy diagrams , the vertex diagrams in figs . [ diags](a)(d ) , as well as the scalar integrals with the topology of figs . [ diags](e)(g ) . however , due to problems with the complex tensor structure , the complete diagrams in figs . [ diags](e)(g ) can not be checked with this technique . in the next subsections , we explain the applications of these methods for our purposes and present a comparison between them . we perform an expansion in a parameter @xmath54 , where @xmath55 for any two - loop problem , there are four regions to consider . let @xmath56 and @xmath57 represent the internal momenta in the loops and @xmath58 stand for any external momentum , while @xmath59 generically denotes all masses that are small compared to @xmath60 , @xmath61 . in our case , @xmath62 . then the four regions can be identified as follows : [ cols= " < , < " , ] following earlier publications on two - loop electroweak corrections , we express our results in terms of fitting formulas . the form factor @xmath63 , which contains the fermionic two - loop electroweak corrections to @xmath0 according to eq . , can be approximated as @xmath64 where @xmath65 is the one - loop result , and @xmath66 \delta_z & = \frac{m_z}{91.1876~\mathrm{gev } } -1 , & \delta_w & = \frac{m_w}{80.404~\mathrm{gev } } -1 . \end{aligned}\ ] ] fitting this formula to the exact result , we obtain @xmath67 this parametrisation reproduces the exact calculation with maximal and average deviations of @xmath68 and @xmath69 , respectively , as long as the input parameters stay within their @xmath70 ranges of the experimental errors quoted in table [ tab : input ] and the higgs - boson mass is in the range 10 gev @xmath71 1 tev . if the top - quark mass and the @xmath16-boson mass vary within 4@xmath72 ranges , the formula is still accurate to @xmath73 . we also present a simple parametrisation for the currently best prediction for @xmath0 , including all known corrections to @xmath74 and @xmath20 ( for the calculation of @xmath17 from @xmath18 see refs . @xcite ) . for @xmath74 , in addition to the one - loop and fermionic two - loop electroweak corrections , we include qcd corrections of @xmath75 @xcite and @xmath76 @xcite to the one - loop contribution , as well as universal corrections for large top - quark mass , of @xmath77 and @xmath78 @xcite . moreover , leading four - loop qcd correction to the @xmath79 parameter , which arise from top- and bottom - quark loops , are taken into account @xcite . we use the parametrisation @xmath80 with @xmath81 the best - fit numerical values for the coefficients are @xmath82 this parametrisation approximates the full result with maximal and average deviations of @xmath83 and @xmath84 , respectively , for 10 gev@xmath85 tev and the other input parameters in their @xmath70 ranges . in this paper , the calculation of the two - loop electroweak fermionic corrections to the effective weak - mixing angle for the @xmath52 vertex , @xmath0 , was presented . such an accurate theoretical prediction for @xmath0 is necessary for the interpretation of the bottom - quark asymmetry measurements at the @xmath6-boson pole . compared to the previously known corrections to @xmath0 , the new electroweak two - loop result turns out to be sizable , of order @xmath86 for a higgs - boson mass near 100 gev . the calculation was performed by using methods that had been used earlier for the computation of the leptonic effective weak - mixing angle , as well as a newly developed code based on the bt algorithm . the results of the different methods were checked against each other . although we did not perform a detailed analysis of the error from unknown high - order corrections , in particular the missing bosonic two - loop corrections and terms of order @xmath87 , we expect those to be of similar order as for the leptonic effective weak - mixing angle . the main difference between the leptonic and bottom - quark effective weak - mixing angles are the vertex diagrams with internal @xmath16-boson and top - quark propagators . while leading to numerical differences between @xmath3 and @xmath0 , these diagrams do not introduce special enhancement or suppression factors . therefore , we expect the theoretical uncertainty to our result for @xmath0 to be about @xmath88 , similar to ref . @xcite . the work of m.a . and b.a.k . was supported in part by the german research foundation ( dfg ) through grant no . kn 365/3 - 1 and through the collaborative research centre 676 _ particles , strings and the early universe the structure of matter and space time_. the work of m.c . was supported in part by the sofja kovalevskaja award of the alexander von humboldt foundation and by the tok program _ algotools _ ( mtkd - cd-2004 - 014319 ) . a.f . is grateful for warm hospitality at argonne national laboratory and the enrico fermi institute of the university of chicago , where part of his work on this project was performed . m. awramik , m. czakon , a. freitas , g. weiglein , phys . d 69 ( 2004 ) 053006 , arxiv : hep - ph/0311148 . m. awramik , m. czakon , a. freitas , g. weiglein , phys . lett . 93 ( 2004 ) 201805 , arxiv : hep - ph/0407317 ; + m. awramik , m. czakon , a. freitas , g. weiglein , in : proceedings of the international conference on linear colliders ( lcws 04 ) , paris , france , 1924 april 2004 , arxiv : hep - ph/0409142 ; + a. freitas , m. awramik , m. czakon , in : proceedings of the 2005 international linear collider workshop ( lcws 05 ) , stanford , california , 1822 march 2005 , p. 0610 , arxiv : hep - ph/0507159 . m. awramik , m. czakon , a. freitas , g. weiglein , nucl . b ( proc . suppl . ) 135 ( 2004 ) 119 , arxiv : hep - ph/0408207 . w. hollik , u. meier , s. uccirati , nucl . b 731 ( 2005 ) 213 , arxiv : hep - ph/0507158 ; + w. hollik , u. meier , s. uccirati , phys . b 632 ( 2006 ) 680 , arxiv : hep - ph/0509302 ; + w. hollik , u. meier , s. uccirati , nucl . b 765 ( 2007 ) 154 , arxiv : hep - ph/0610312 . m. awramik , m. czakon , a. freitas , phys . b 642 ( 2006 ) 563 , arxiv : hep - ph/0605339 ; + m. czakon , m. awramik , a. freitas , nucl . b ( proc . suppl . ) 157 ( 2006 ) 58 , arxiv : hep - ph/0602029 . m. awramik , m. czakon , a. freitas , jhep 0611 ( 2006 ) 048 , arxiv : hep - ph/0608099 . aleph , delphi , l3 and opal collaborations , s. schael , et al . , phys . 427 ( 2006 ) 257 , arxiv : hep - ex/0509008 . r. hawkings , k. mnig , eur . j. direct c 1 ( 1999 ) 8 , arxiv : hep - ex/9910022 . r. barbieri , m. beccaria , p. ciafaloni , g. curci , a. vicere , phys . b 288 ( 1992 ) 95 , arxiv : hep - ph/9205238 ; + r. barbieri , m. beccaria , p. ciafaloni , g. curci , a. vicere , phys . b 312 ( 1993 ) 511 , erratum ; + r. barbieri , m. beccaria , p. ciafaloni , g. curci , a. vicere , nucl . b 409 ( 1993 ) 105 ; + j. fleischer , o.v . tarasov , f. jegerlehner , phys . b 319 ( 1993 ) 249 ; + j. fleischer , o.v . tarasov , f. jegerlehner , phys . d 51 ( 1995 ) 3820 ; + a. denner , w. hollik , b. lampe , z. phys . c 60 ( 1993 ) 193 , arxiv : hep - ph/9305273 ; + j. fleischer , f. jegerlehner , m. tentyukov , o.l . veretin , phys . b 459 ( 1999 ) 625 , arxiv : hep - ph/9904256 . d. bardin , et al . , report no . cern th.6443/92 , arxiv : hep - ph/9412201 ; + d. bardin , m. bilenky , p. christova , m. jack , l. kalinovskaya , a. olchevski , s. riemann , t. riemann , comput . . commun . 133 ( 2001 ) 229 , arxiv : hep - ph/9908433 ; + a.b . arbuzov , m. awramik , m. czakon , a. freitas , m.w . grnewald , k. mnig , s. riemann , t. riemann , comput . 174 ( 2006 ) 728 , arxiv : hep - ph/0507146 . h. flcher , m. goebel , j. haller , a. hcker , k. mnig and j. stelzer , arxiv:0811.0009 [ hep - ph ] . a. freitas , k. mnig , eur . j. c 40 ( 2005 ) 493 , arxiv : hep - ph/0411304 . a. freitas , w. hollik , w. walter , g. weiglein , nucl . b 632 ( 2002 ) 189 , arxiv : hep - ph/0202131 ; + a. freitas , w. hollik , w. walter , g. weiglein , nucl . b 666 ( 2003 ) 305 , erratum . m. butenschn , f. fugel , b.a . kniehl , phys . lett . 98 ( 2007 ) 071602 , arxiv : hep - ph/0612184 ; + m. butenschn , f. fugel , b.a . kniehl , nucl . b 772 ( 2007 ) 25 , arxiv : hep - ph/0702215 . smirnov , evaluating feynman integrals , springer tracts mod . phys . 211 ( 2004 ) 1 ; + v.a . smirnov , applied asymptotic expansions in momenta and masses , springer , berlin , germany , 2002 . tkachov , nucl . instrum . meth . a 389 ( 1997 ) 309 , arxiv : hep - ph/9609429 . d.yu . bardin , l.v . kalinovskaya , f.v . tkachov , in : proceedings of the 15th international workshop on high - energy physics and quantum field theory ( qfthep 2000 ) , tver , russia , 1420 september 2000 , arxiv : hep - ph/0012209 ; + g. passarino , nucl . b 619 ( 2001 ) 257 , arxiv : hep - ph/0108252 ; + a. ferroglia , m. passera , g. passarino , s. uccirati , nucl . b 650 ( 2003 ) 162 , arxiv : hep - ph/0209219 . a. ferroglia , m. passera , g. passarino , s. uccirati , nucl . b 680 ( 2004 ) 199 , arxiv : hep - ph/0311186 ; + s. actis , a. ferroglia , g. passarino , m. passera , s. uccirati , nucl . b 703 ( 2004 ) 3 , arxiv : hep - ph/0402132 ; + g. passarino , s. uccirati , nucl . b 747 ( 2006 ) 113 , arxiv : hep - ph/0603121 . g. passarino , c. sturm , s. uccirati , phys . b 655 ( 2007 ) 298 , arxiv:0707.1401 [ hep - ph ] . s. bauberger , f.a . berends , m. bhm , m. buza , nucl . phys . b 434 ( 1995 ) 383 , arxiv : hep - ph/9409388 ; + b.a . kniehl , acta phys . b 27 ( 1996 ) 3631 , arxiv : hep - ph/9607255 . s. laporta , int . j. mod . a 15 ( 2000 ) 5087 , arxiv : hep - ph/0102033 . chetyrkin , f.v . tkachov , nucl . b 192 ( 1981 ) 159 . t. gehrmann , e. remiddi , nucl . b 580 ( 2000 ) 485 , arxiv : hep - ph/9912329 . m. czakon , ` diagen / idsolver ` ( unpublished ) . a. ghinculov , j.j . van der bij , nucl . phys . b 436 ( 1995 ) 30 , arxiv : hep - ph/9405418 . chetyrkin , j.h . khn , m. steinhauser , phys . lett . 75 ( 1995 ) 3394 , arxiv : hep - ph/9504413 . m. faisst , j.h . khn , t. seidensticker , o. veretin , nucl . b 665 ( 2003 ) 649 , arxiv : hep - ph/0302275 . vermaseren , report no . nikhef-00 - 032 , arxiv : math - ph/0010025 . t. hahn , comput . commun . 168 ( 2005 ) 78 , arxiv : hep - ph/0404043 . lep electroweak working group , d. abbaneo , et al . , + ` http://lepewwg.web.cern.ch/lepewwg/ ` . particle data group , s. eidelman , et al . b 592 ( 2004 ) 1 ; also 2005 partial update for edition 2006 , available on ` http://pdg.lbl.gov ` . bardin , a. leike , t. riemann , m. sachwitz , phys . b 206 ( 1988 ) 546 . a. freitas , w. hollik , w. walter , g. weiglein , phys . b 495 ( 2000 ) 338 , arxiv : hep - ph/0007091 ; + a. freitas , w. hollik , w. walter , g. weiglein , phys . b 570 ( 2003 ) 260 , erratum ; + m. awramik , m. czakon , phys . lett . 89 ( 2002 ) 241801 , arxiv : hep - ph/0208113 ; + m. awramik , m. czakon , a. onishchenko , o. veretin , phys . rev . d 68 ( 2003 ) 053004 , arxiv : hep - ph/0209084 ; + a. onishchenko , o. veretin , phys . b 551 ( 2003 ) 111 , arxiv : hep - ph/0209010 ; + m. awramik , m. czakon , phys . b 568 ( 2003 ) 48 , arxiv : hep - ph/0305248 . a. djouadi , c. verzegnassi , phys . b 195 ( 1987 ) 265 ; + a. djouadi , nuovo cim . a 100 ( 1988 ) 357 ; + b.a . kniehl , nucl . b 347 ( 1990 ) 86 ; + f. halzen , b.a . kniehl , nucl . b 353 ( 1991 ) 567 ; + b.a . kniehl , a. sirlin , nucl . b 371 ( 1992 ) 141 ; + b.a . kniehl , a. sirlin , phys . d 47 ( 1993 ) 883 ; + a. djouadi , p. gambino , phys . d 49 ( 1994 ) 3499 , arxiv : hep - ph/9309298 ; + a. djouadi , p. gambino , phys . d 53 ( 1996 ) 4111 , erratum . l. avdeev , j. fleischer , s. mikhailov , o. tarasov , phys . b 336 ( 1994 ) 560 , arxiv : hep - ph/9406363 ; + l. avdeev , j. fleischer , s. mikhailov , o. tarasov , phys . b 349 ( 1994 ) 597 , erratum ; + k.g . chetyrkin , j.h . khn , m. steinhauser , phys . b 351 ( 1995 ) 331 , arxiv : hep - ph/9502291 ; + y. schrder , m. steinhauser , phys . b 622 ( 2005 ) 124 , arxiv : hep - ph/0504055 . chetyrkin , m. faisst , j.h . khn , p. maierhofer , c. sturm , phys . 97 ( 2006 ) 102003 , arxiv : hep - ph/0605201 ; + r. boughezal , m. czakon , nucl . b 755 ( 2006 ) 221 , arxiv : hep - ph/0606232 .

we present the first calculation of the two - loop electroweak fermionic correction to the flavour - dependent effective weak - mixing angle for bottom quarks , @xmath0 . for the evaluation of the missing two - loop vertex diagrams , two methods are employed , one based on a semi - numerical bernstein - tkachov algorithm and the second on asymptotic expansions in the large top - quark mass . a third method based on dispersion relations is used for checking the basic loop integrals . we find that for small higgs - boson mass values , @xmath1 gev , the correction is sizable , of order @xmath2 . keywords : electroweak radiative corrections , effective weak - mixing angle , bernstein - tkachov algorithm pacs : 12.15.lk , 13.38.dg , 13.66.jn , 14.70.hp

it is well known that gravitational lensing is a powerful tool for directly probing the structure and distribution of dark matter in the universe ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) . by comparing the number of lenses found in a survey of remote sources ( e.g. , quasars , radio galaxies , or high redshift type ia supernova ) to theoretical predictions , we should be able to deduce the quantity of dark matter in the universe and how it is distributed ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * henceforth lo02 ; gladders et al . the joint observations of gravitational lensing , high redshift type ia supernova , cosmic microwave background ( cmb ) , and cluster abundances constrain the universe to be in all likelihood flat and accelerating , with the present mass density being composed of about 70% cosmological constant ( or dark energy ) , 26% dark matter , and 4% ordinary matter @xcite . however , the lensing cross - section ( and thus the lensing probability ) is found to be extremely sensitive to the inner density profile of lenses ( keeton & madau 2001 ; wyithe , turner , & spergel 2001 ; lo02 ) . for example , with fixed total mass , when the inner slope of the density profile , @xmath8 , changes from @xmath9 [ the nfw case @xcite ] to @xmath10 [ the singular isothermal sphere ( sis ) case @xcite ] while maintaining the same mass density in lenses , the integral lensing probability increases by more than two orders of magnitudes for the flat model of the universe ( lo02 ) . therefore , lensing also sensitively probes small scale structure . this complicates matters and renders it is hazardous to use observed lensing statistics to draw inferences with regard to cosmology before determining the sensitivity to other factors . in lo02 , we have shown that in order to explain the observed numbers of lenses found in the jvas / class survey , at least two populations of dark halos must exist in nature . one population , which corresponds to normal galaxies , has masses @xmath11 and a steep inner density profile ( @xmath12 , i.e. sis ) presumably determined by the distribution of baryonic material in the inner parts of galaxies ; the other one , which corresponds to groups or clusters of galaxies , has masses @xmath13 and a shallow inner density profile ( @xmath14 , i.e. similar to nfw ) . a similar conclusion has been obtained by @xcite for explaining the number of lenses found in the castles survey . these results are consistent with the theoretical studies on the cooling of massive gas clouds : there is a critical mass of halos @xmath15 below which cooling of the corresponding baryonic component will lead to concentration of the baryons to the inner parts of the mass profile @xcite . in this paper we investigate the lensing statistics produced by a compound population of halos . we assume that there are three populations of halos in the universe : population a : @xmath16 , @xmath17 ( sis ) ; + population b : @xmath18 , @xmath19 [ gnfw ( generalized nfw , * ? ? ? * ) ] ; + population c : @xmath20 , @xmath19 ( gnfw ) , where @xmath21 is the hubble constant in units of 100 km s@xmath22 mpc@xmath22 . population a corresponds to spiral and elliptical galaxies , whose centers are dominated by baryonic matter . population b corresponds to groups or clusters of galaxies , whose centers are dominated by dark matter . population c corresponds to dwarf galaxies or subgalactic objects , whose centers lack baryons due to feedback processes such as supernova explosions , stellar winds , and photoionizations @xcite , and so are also dominated by dark matter . we adopt an inner slope for the dark matter halos of @xmath19 consistent with the value @xmath23 found by @xcite and intermediate between the values advocated by @xcite of @xmath24 and @xcite of @xmath25 . we will calculate here the lensing probability of two measurable variables : image separation and time delay , examining in a subsequent paper the expected arc properties . recently , @xcite used lensing statistics to constrain the inner slope of lensing galaxies . using the schechter function @xcite , they constrained the inner slope of lensing galaxies to the range from @xmath26 to @xmath27 , at 95% confidence level ( cl ) . it is hard to predict how their result would change if the press - schechter function @xcite were used . our choice of @xmath17 for galaxies is supported by the following fact : stellar dynamics of elliptical galaxies , modeling of lensed systems , and flux ratios of multiple images all give an inner profile that is consistent with sis @xcite . @xcite have reported a remarkably flat inner slope in the lensing cluster ms2137 - 23 : @xmath28 at 99% cl . however , by measuring the average gravitational shear profile of six massive clusters of virial masses @xmath29 , @xcite have found that the data are well fitted by a mass density profile with @xmath30 for scdm model and @xmath31 for lcdm model , both at 68% cl . so , our choice of @xmath19 for population b looks reasonable . the inner density slopes for small mass halos are not well constrained . cdm simulations generally predict a cusped inner density , while other dark matter models , like warm dark matter @xcite , repulsive dark matter @xcite , and collisional dark matter @xcite , tend to predict flatter inner density ( see also ricotti 2002 ) . our choice of @xmath19 for population c should be a reasonable upper limit . in her recent paper , by requiring that the schechter luminosity function and the press - schechter mass function to give consistent predictions for the image separation below @xmath32 , @xcite has shown that the fraction of sis halos peaks around mass of @xmath33 and quickly drops for large and small mass halos . this is qualitatively consistent with the model that we adopt in this paper . the paper is organized as follows : in [ sec2 ] we write down the lensing cross - section produced by sis and gnfw halos . in [ sec3 ] we show how to calculate the lensing probability , assuming that halos are composed of the population defined above , whose mass function is given by the press - schechter function @xcite . in [ sec4 ] we present our results . in [ sec5 ] we summarize and discuss our results . issues related to image separation are presented in lo02 in detail , so here we focus on the time delay between multiple images produced by gravitational lensing . the density profile for an sis is @xcite @xmath34 where @xmath35 is the constant velocity dispersion . assuming that the angular - diameter distances from the observer to the lens and the source are respectively @xmath36 and @xmath37 , from the lens to the source is @xmath38 . then , the time delay between the two images of the remote source lensed by an sis halo is @xcite @xmath39 where @xmath40 @xmath41 is the redshift of the lens ( dark halo ) , @xmath42 is the distance from the source to the point where the line of sight through the lens center intersects the source plane , in units of @xmath43 . the cross - section for producing two images with a time delay @xmath44 is @xmath45\ , \vartheta(\delta t_1 - \delta t ) \ ; , \label{siga}\end{aligned}\ ] ] where @xmath46 is the step function . the density profile for a gnfw profile is ( zhao 1996 ; wyithe , turner , & spergel 2001 ; lo02 ) @xmath47 where @xmath48 , @xmath49 , and @xmath50 are constants . the case of @xmath51 corresponds to the nfw profile @xcite . the case of @xmath17 , @xmath52 but keeping @xmath53 constant , corresponds to the sis profile . in this paper , we take @xmath19 for populations b and c. in the lens plane , we denote the distance from the lens center to the point where the light ray of the source object intersects the lens plane by @xmath54 , in units of @xmath50 . in the source plane , we denote the distance from the source to the point where the line of sight through the lens center intersects the source plane by @xmath42 , in units of @xmath55 . then , the lensing equation is ( lo02 ) @xmath56^{\alpha -3 } \;,\end{aligned}\ ] ] where @xmath57 to a good approximation , the time delay between the two images produced by a gnfw halo is given by @xcite @xmath58 where @xmath59 is the positive root of @xmath60 , @xmath61 corresponds to the positive @xmath42 at @xmath62 . the cross - section for producing two images with a time delay @xmath63 is @xmath64\ , \vartheta(\delta t_2 - \delta t ) \;. \label{sigb}\end{aligned}\ ] ] the probability for a remote point source lensed by foreground dark halos is given by @xmath65 where @xmath66 is the redshift of the source , @xmath67 is the proper distance from the observer to a lens at redshift @xmath41 , @xmath68 is the proper number density of lens objects of masses between @xmath0 and @xmath69 , @xmath70 is the lensing cross - section of a dark halo of mass @xmath0 at redshift @xmath41 . when @xmath71 ( which is true in most cases for lensing statistics ) , we have @xmath72 . for both sis and gnfw profiles , the mass contained within radius @xmath73 diverges as @xmath74 . so , a cutoff in radius must be introduced . here , as is typically done in the literature , we define the mass of a dark halo to be the mass within a sphere of radius @xmath75 , where @xmath76 is the radius within which the average mass density is @xmath77 times the critical mass density of the universe at the redshift of the halo . as in lo02 , we consider three kinds of cosmological models : lcdm , ocdm , and scdm . we assume that the number density of dark halos is distributed in mass according to the press - schechter function @xcite . we compute the cdm power spectrum using the fitting formula given by @xcite , where , to be consistent with the recent observations of _ wmap _ @xcite , we assume the hubble constant @xmath78 and the primordial spectrum index @xmath79 . for ocdm and scdm , we determine the value of @xmath80 by the cluster abundances constraint @xcite @xmath81 where @xmath82 . for lcdm , we take @xmath83 and @xmath80 to be consistent with the observations of _ wmap _ @xcite : @xmath84 ( then @xmath85 ) , @xmath86 . a new cluster abundances constraint has recently been obtained by @xcite with the sdss data . the best - fit cluster normalization is given by @xmath87 ( for @xmath88 ) for the flat model of the universe with a hubble constant @xmath89 . @xcite found that the best - fit parameters of the observed mass function are @xmath90 and @xmath91 . recent calibration of the cluster data based on x - ray observations @xcite are closer to the @xcite result . so , for comparison , we will also present some results for a flat lcdm model with the bahcall et al . normalization to show the sensitivity of results to normalization . for the case of image separation , the cross - section @xmath92 can be found in lo02 ( eqs . [ 37 ] for sis and [ 48 ] for gnfw ) . for the case of time delay , the cross - section is given by equation ( [ siga ] ) for sis halos , and equation ( [ sigb ] ) for gnfw halos . we normalize the gnfw profile so that the concentration parameter @xmath93 satisfies @xcite @xmath94 throughout the paper we fix @xmath95 , in consistence with the simulations @xcite . for the model of compound halo population considered in this paper , the integration over mass @xmath0 is divided into three parts : @xmath96 for gnfw with @xmath19 , @xmath97 for sis , and @xmath98 for gnfw with @xmath19 ; where @xmath99 , @xmath100 . with the formalism described above , we are ready to calculate the lensing probability for images separation and time delay . the models to be calculated are listed in table [ tab1 ] . as explained in the previous section , we take three different normalizations for lcdm models : in most of calculations we choose parameters to be consistent with _ wmap _ @xcite , but , for comparison , we will also present some results corresponding to the normalization of @xcite . for ocdm and scdm models , we adopt equation ( [ s8 ] ) for normalization . throughout the paper we take @xmath78 and @xmath79 . for image separation , we have calculated the differential lensing probability @xmath101 where @xmath102 is given by equation ( [ ip ] ) with @xmath103 . we show the results for different cosmological models in figure [ fig1 ] , separately for the three different components in the whole population : population a ( galaxies , the highest island ) , population b ( groups and clusters of galaxies , the second high island ) , and population c ( dwarf galaxies and subgalactic objects , the lowest island ) . the source object is assumed to be at @xmath104 . from the figure we see that , population a ( galaxies ) contributes most to the total number of lenses , due to its steep inner density slope ( @xmath105 ) ; population b contributes less ; population c contributes least , due to its small mass and shallow inner density slope ( @xmath19 ) . consistent with the results in lo02 , the lensing probability produced by the @xmath106 gnfw halos is smaller than the lensing probability produced by sis halos by two orders of magnitudes in the overlap regions . ( the results here are slightly different from those in lo02 due to the fact that in this paper we use a different normalization in the concentration parameter , i.e. eq . [ [ c1 ] ] . ) in figure [ fig2 ] , we show the lcdm ( @xmath84 , @xmath107 ) results corresponding to different redshift of the source object : from @xmath108 to @xmath109 . we see that the lensing probability increases quickly with the source redshift , increasing by an order of magnitude between @xmath110 and @xmath111 ( cf . wambsganss , bode , & ostriker 2003 ) . however , the rate of increase in the lensing probability decreases with the source redshift , this is because that the proper distance from the source object to the observer approaches a finite limit as @xmath112 ( due to the existence of a horizon in an expanding universe ) . we also see that , as the source redshift increases , the splitting angle corresponding to the peak probability of each island shifts toward larger values . in figure [ fig3 ] , we show the corresponding integral lensing probability @xmath113 to compare the predictions with observations , the effect of magnification bias must be considered ( turner , ostriker , & gott 1984 ; schneider , ehlers , & falco 1992 ; lo02 ; oguri et al . 2002 ) . when the source objects have a flux distribution @xmath114 ( @xmath115 ) and the probability density for magnification is @xmath116 , the magnification bias is given by ( lo02 ) @xmath117 where @xmath118 is the minimum of the total amplification . for sis lenses we have @xmath119 . for gnfw lenses , @xmath118 can be approximated by @xmath120 where @xmath121 . equation ( [ amnfw ] ) is an improvement to the equation ( 68 ) of lo02 . the magnification bias calculated with equations ( [ bias ] ) and ( [ amnfw ] ) agrees with that calculated with the more complicated formula of @xcite with errors @xmath122 for @xmath123 . for gnfw lenses with @xmath19 , we show the average magnification bias @xmath124 ( defined by the ratio of the biased lensing probability to that without bias ) as a function of image separation in figure [ fig4a ] ( as an improvement to the fig . 10 of lo02 ) for the jvas / class survey @xcite , where we have assumed @xmath125 @xcite and @xmath126 @xcite . the magnification bias for gnfw lenses depends on cosmological models , decreases with increasing image separation , and is bigger than the magnification bias for sis lenses ( which is a constant @xmath127 ) by about @xmath128 order of magnitude on average ( for @xmath129 ) . in figure [ fig4 ] , we compare our predictions ( including magnification bias ) for the compound model with observations from the jvas / class survey . the data are updated compared to @xcite . the new data contain @xmath130 lenses found in a sample of @xmath131 of radio sources which form a statistical sample @xcite . considering error bars , both lcdm and ocdm models with both normalizations are marginally consistent with the jvas / class observational data . @xcite . ] comparing lcdm2 with lcdm3 , we find that even for the same cluster normalization there is significant discriminatory power available from lensing statistics ( if data is available ) in breaking the degeneracy on the @xmath132 plane . this is consistent with our previous results ( lo02 ) . the three different lcdm models do not differ significantly in their predictions at small splittings but for splittings above 10 arcseconds the bahcall et al normalization , lcdm3 , predicts few lenses by more than a factor of five . for time delay , we have calculated the differential lensing probability @xmath133 where @xmath134 is given by equation ( [ ip ] ) with @xmath135 . we show the results in figure [ fig5 ] , for the same models in [ sec4.1 ] . we see that , the distribution of lensing probability over time delay is very similar to the distribution over image separation ( compare fig . [ fig2 ] to fig . [ fig1 ] ) . again , the contribution to lensing events is overwhelmingly dominated by population a due to its steep inner density slope . population c contributes the least . we have also calculated the lensing probability for time delay corresponding to different source redshift : from @xmath108 to @xmath109 . the results for the lcdm model ( @xmath84 , @xmath136 ) are shown in figure [ fig6 ] for the differential lensing probability @xmath137 , and figure [ fig7 ] for the integral lensing probability @xmath138 from these figures we see that , like in the case for image separation , the lensing probability sensitively depends on @xmath66 for small @xmath66 . for large @xmath66 , the lensing probability becomes less sensitive to the source redshift , due to the fact that @xmath139 decreases with increasing @xmath66 . we can calculate the joint lensing probability @xmath140 by using the joint cross - section @xmath141 where @xmath142 for sis and @xmath143 for gnfw . the cross - section @xmath144 is given by equation ( 37 ) of lo02 for sis , and equation ( 48 ) of lo02 for gnfw . then , we can calculate the conditional lensing probability @xmath145 defined by @xmath146 which gives the distribution of lensing events over time delay for a given image separation . knowing @xmath145 , we can calculate the median time delay @xmath147 as a function of @xmath148 , where @xmath147 is defined by @xmath149 the prediction for @xmath147 as a function of @xmath148 is not sensitive to the magnification bias since it is determined by the ratio of two probabilities . so , the correlation between @xmath150 and @xmath148 provides a test of lensing models independent of the determination of magnification bias . the results of @xmath151 for the lcdm ( @xmath152 , @xmath136 ; indeed the results are insensitive to the cosmological parameters ) model are shown in figure [ fig8 ] , where the source object is again assumed to be at @xmath153 . in figure [ fig8 ] we also show the quadrant deviations ( dashed lines ) , which are defined by equation ( [ med ] ) with the @xmath154 on the right - hand side being replaced by @xmath155 and @xmath156 , respectively . the observational data , taken from @xcite , fit the lcdm model well . comparison of figure [ fig8 ] with @xcite s figure 6 indicates that our compound model fits the observations better . the single population model predicts a single ( almost ) straight line in the @xmath157 space . for the compound model , a `` step '' is produced at the point where the mass density profile changes . the `` step '' that we see in figure [ fig8 ] corresponds to the transition from population a ( galaxies ) to population b ( galaxy groups / clusters ) . as an extension of our previous work ( lo02 ) , we computed the lensing probability produced by a compound population of dark halos . we have calculated the lensing probability for both image separation and time delay . the calculations confirm our previous results ( lo02 ) that the lensing probability produced by gnfw halos with @xmath158 is lower than that produced by sis halos with same masses by orders of magnitudes , where @xmath159 is the inner slope of the halo mass density . so , for the compound population of halos , both the number of lenses with large image separation ( @xmath160 ) and the number of lenses with small image separation ( @xmath161 ) are greatly suppressed . the same conclusion holds also for the number of lenses with large time delay ( @xmath162 ) and the number of lenses with small time delay ( @xmath163 ) . ( see figs . [ fig1 ] and [ fig5 ] . this conclusion holds even when the effect of magnification bias is considered , see figs . [ fig4a ] and [ fig4 ] . ) we have also tested the dependence of the lensing probability on the redshift of the source object ( figs . [ fig2 ] , [ fig3 ] , [ fig6 ] , and [ fig7 ] ) . the results show that , the lensing probability is quite sensitive to the change in the redshift of the source object . the number of lenses significantly increases as the source redshift increases . however , the rate of the increase decreases as the source redshift becomes large , which is caused by the fact that the proper cosmological distance approaches a finite limit when @xmath164 . another interesting result is that , the peak of the lensing probability for each population moves toward large image separation or time delay , as the source redshift increases . we see that population c ( dwarf halos ) in an lcdm model has a unique signature in the time domain , c.f . figures [ fig5 ] and [ fig6 ] . time delays of less than @xmath165 seconds and greater than 0.1 second are predicted and should be found in gamma - ray burst sources which are at cosmological distances and have the requisite temporal substructure . variants of cdm , such as warm dark matter @xcite , repulsive dark matter @xcite , or collisional dark matter @xcite would not produce this feature . however , current surveys do not go deep enough to provide a sufficiently large sample to test the prediction . when more observational data on gamma - ray burst time delay and small splitting angles become available , our calculations can be used to distinguish different dark matter models @xcite . we have compared the distribution of the number of lenses over image separation predicted by our model with the updated jvas / class observational data , with the new _ wmap _ cosmological parameters ( fig . [ fig4 ] ) . since the jvas / class survey is limited to image separation @xmath166 @xcite , we can not test our predictions for small image separations . however , in the range that is probed by jvas / class , we see that both the lcdm and ocdm models fit the observation reasonably well and current data do not allow us to distinguish between the two proposed normalizations for the lcdm spectrum , even though these produce predictions that differ by a factor of roughly @xmath167 . an explicit search for lenses with image separation between @xmath168 and @xmath169 has found no lenses @xcite , which rules out the sis model for image separation in this range ( lo02 ) . this together with our figure [ fig4 ] supports our model of compound population of halos . for separations greater than @xmath170 the differently normalized lcdm models produce significantly different results , thus producing an additional lever to break the degeneracies in the wmap results ( cf . bridle et al . 2003 ) we have also calculated the distribution of the mean time delay vs image separation for the lcdm model ( fig . [ fig8 ] ) . we see that , the compound model fits observations quite well , better than the model of single population of halos @xcite . the compound model predicts a unique feature in the @xmath157 plane : there is a `` step '' corresponding to the transition in mass density profile . this can be better tested when more observation data are available . a controlled survey of lenses with double the sample size of class , perhaps obtainable via sdss @xcite , should allow one to better distinguish between lcdm variants and perhaps between lcdm models and those based on quintessence @xcite rather than a cosmological constant . we thank b. paczyski for many helpful discussions , and the anonymous referee whose comments helped to improve our results . lxl s research was supported by nasa through chandra postdoctoral fellowship grant number pf1 - 20018 awarded by the chandra x - ray center , which is operated by the smithsonian astrophysical observatory for nasa under contract nas8 - 39073 . jpo s research was supported by the nsf grants asc-9740300 ( subaward 766 ) and ast-9803137 . lllllll lcdm & @xmath171 & @xmath172 & @xmath173 & @xmath174 & @xmath175 & _ wmap _ + lcdm2 & @xmath176 & @xmath177 & @xmath178 & @xmath179 & @xmath175 & @xmath180 + lcdm3 & @xmath181 & @xmath179 & @xmath179 & @xmath179 & @xmath175 & @xmath180 + ocdm & @xmath181 & @xmath182 & @xmath183 & @xmath179 & @xmath175 & @xmath184 + scdm & @xmath185 & @xmath182 & @xmath186 & @xmath179 & @xmath175 & @xmath184

based on observed rotation curves of galaxies and theoretical simulations of dark matter halos , there are reasons for believing that at least three different types of dark matter halos exist in the universe classified by their masses @xmath0 and the inner slope of mass density @xmath1 : population a ( galaxies ) : @xmath2 , @xmath3 ; population b ( cluster halos ) : @xmath4 , @xmath5 ; and population c ( dwarf halos ) : @xmath6 , @xmath5 . in this paper we calculate the lensing probability produced by such a compound population of dark halos , for both image separation and time delay , assuming that the mass function of halos is given by the press - schechter function and the universe is described by an lcdm , ocdm , or scdm model . the lcdm model is normalized to the _ wmap _ observations , ocdm and scdm models are normalized to the abundance of rich clusters . we compare the predictions of the different cosmological models with observational data and show that , both lcdm and ocdm models are marginally consistent with the current available data , but the scdm model is ruled out . the fit of the compound model to the observed correlation between splitting angle and time delay is excellent but the fit to the number vs splitting angle relation is only adequate using the small number of sources in the objective jvas / class survey . a larger survey of the same type would have great power in discriminating among cosmological models . furthermore , population c in an lcdm model has a unique signature in the time domain , an additional peak at @xmath7 seconds potentially observable in grbs , which makes it distinguishable from variants of cdm scenarios , such as warm dark matter , repulsive dark matter , or collisional dark matter . for image separations greater than 10 arcseconds the differently normalized lcdm models predict significantly different lensing probabilities affording an additional lever to break the degeneracies in the cmb determination of cosmological parameters .

marginally outer trapped surfaces are natural candidates for quasi - local black hole boundaries in general relativity . they are analogues of minimal surfaces in riemannian geometry , and in particular there is a notion of stability for marginally outer trapped surfaces , closely related to stability for minimal surfaces , which allows one to prove curvature bounds analogous to those which are known for minimal surfaces . it is natural to consider outermost marginally outer trapped surfaces , which enclose every weakly outer trapped surface . for these , we have area bounds , as well as a replacement for the strong maximum principle , which sheds light on the process of black hole coalescence . let @xmath0 be a cauchy hypersurface in a 3 + 1 dimensional lorentzian spacetime @xmath1 . let @xmath2 be a spacelike surface in @xmath3 with null normals @xmath4 , see figure [ fig : sigma ] . we set @xmath5 then @xmath6 is the second fundamental form of @xmath2 in @xmath3 , @xmath7 is the restriction of @xmath8 to @xmath2 , and @xmath9 are the null second fundamental forms associated to @xmath4 . taking traces yields @xmath10 then @xmath11 is the mean curvature of @xmath2 in @xmath3 , @xmath12 is the trace of @xmath7 on @xmath2 and @xmath13 are the * null expansions * of @xmath2 . we declare @xmath14 to be the outer null normal . @xmath2 is a * marginally outer trapped surface * ( mots ) if @xmath15 . recall that @xmath16 is the logarithmic variation of area along @xmath14 , @xmath17 if @xmath2 is a mots then outgoing null rays are marginally collapsing . we call @xmath2 ( weakly ) outer trapped if ( @xmath18 ) @xmath19 . the null energy condition ( nec ) holds if @xmath20 for any null vector @xmath21 , where @xmath22 is the einstein tensor . the usual definition of trapped surface is @xmath23 , @xmath24 . by the singularity theorems of hawking and penrose , a maximal globally hyperbolic spacetime satisfying suitable energy conditions , eg . , nec , and which contains a trapped surface , is causally incomplete . we note that also the presence of an outer trapped surface implies incompletness . in particular , if nec holds in @xmath1 and @xmath1 contains a cauchy surface @xmath3 with an outer trapped surface which separates and has noncompact exterior , then @xmath1 is null geodesically incomplete , cf thus motss may be viewed as black hole boundaries . the null expansion @xmath25 is elliptic when viewed as a functional of @xmath26 , since , as is well known , the mean curvature @xmath11 has this property , and @xmath12 may be viewed as a lower order term . for variations along @xmath14 , we have @xmath27 so @xmath28 is _ not _ an elliptic functional with respect to null variations . however , variations within @xmath3 , of the form @xmath29 define an elliptic operator , see section [ sec : stability - op ] below . it is well known that if there are surfaces @xmath30 with @xmath31 > 0 $ ] and @xmath32 < 0 $ ] , which form barriers for the problem of minimizing area , then there is a minimal ( @xmath33 ) surface between them . suppose we have an analogue for motss of existence in the presence of barriers , in this case surfaces @xmath30 with the inner barrier satisfying @xmath34 < 0 $ ] , while the outer barrier satisfies @xmath35 > 0 $ ] . then , in view of ( [ eq : ray ] ) , motss should _ persist _ if nec holds . this can in fact be proved using the results of @xcite , given an outer barrier , cf . @xcite , see section [ sec : motspersist ] . in particular , if @xmath36 is an asymptotically flat initial data set , then there is an outer barrier in @xmath3 . we shall , throughout the rest of this note , assume the presence of an outer barrier . due to the persistence of motss , we expect that motss are generically in a * marginally outer trapped tube * ( mott ) , i.e. a hypersurface of @xmath1 , foliated by motss . this has been proved for outermost motss , modulo a genericity condition , cf . theorem [ thm : ams05amms ] and @xcite . in this case , the mott is weakly spacelike if nec holds . if in addition @xmath24 on the mots , the mott is a dynamical horizon , cf . @xcite . . then @xmath38\ ] ] where @xmath39 . the operator @xmath40 is the analogue of the minimal surface stability operator . we note the following facts which hold for @xmath40 . the operator @xmath40 is 2:nd order elliptic and non - self adjoint in general . there is a unique principal eigenvalue @xmath41 , with positive eigenfunction @xmath42 . if @xmath2 is a locally outermost mots then @xmath43 further , if @xmath43 then there is @xmath44 such that @xmath45 , i.e. , if @xmath43 the maximum principle holds . @xmath2 is * stable * if @xmath43 . in particular , if @xmath2 is locally outermost then @xmath2 is stable . the proof of the following theorem makes use of the definition of @xmath40 , the implicit function theorem , as well as the above mentioned version of the maximum principle . the following theorem was proved in @xcite . [ thm : ams05amms ] suppose @xmath2 is stable ( @xmath43 ) . if @xmath46 , assume in addition is not identically zero may be viewed as a genericity condition ] @xmath47 is not identically zero . then @xmath48 a mott @xmath49 containing @xmath2 . @xmath49 is weakly spacelike if nec holds . theorem [ thm : ams05amms ] is a local result . in general , the outermost mots can jump . this may happen for example through the formation of a new mots outside the existing ones , a process that can be caused by the coalescence of black holes , cf . section [ sec : coalescence ] . as a mots is created , the mott bifurcates in general , see @xcite , see also figure [ fig : mott3 ] . [ thm : am05 ] let @xmath2 be a stable mots . then @xmath50 the proof of theorem [ thm : am05 ] applies several techniques used in @xcite , including the simons identity , a kato inequality , and the hoffmann - spruck sobolev inequality . the moser iteration used in @xcite to achieve the @xmath51 estimate for @xmath52 is replaced by a stampacchia iteration . further , the symmetrized stability estimate of @xcite is used . by applying the local area bound of pogorelov , it is possible to avoid a dependence on the area @xmath28 in the curvature estimate . due to the use of this result , which in turn relies upon the gauss - bonnet theorem , the above form of the curvature estimate applies only to the case of a 2-dimensional surface in a 3 + 1 dimensional spacetime . consider @xmath53 , with metric @xmath54 , see figure [ fig : jang ] . define @xmath55 by pullback of @xmath8 . let @xmath56 be the graph of @xmath57 . on @xmath56 we have induced mean curvature @xmath58 and @xmath59 . jang s equation is @xmath60 : = { \mathcal h}- { \mathcal p}= 0\ ] ] this is the analogue of the equation @xmath15 by translation invariance , @xmath61 = j[f + t]$ ] , we have that the stability operator for @xmath56 has @xmath62 , with @xmath63 . here @xmath40 is the analogue of the minimal surface stability operator for @xmath56 . by the work in @xcite , we have local curvature bounds for @xmath56 , which yields compactness . this allows one to prove existence of solutions to jang s equation using a capillarity deformation together with leray - schauder theory . one considers the deformed equation @xmath64 as @xmath65 goes from 0 to 1 , we have a limit @xmath66 . letting @xmath67 , we have by compactness , convergence of a subsequence of @xmath66 to a solution @xmath57 of jang s equation . the solution has blowups in general . as observed in @xcite , blowups project to motss , cf . figure [ fig : jangblowmots ] . therefore jang s equation can be used to prove existence of motss . it was proved in @xcite that the blowup surfaces are _ stable _ motss . [ thm : motsexist ] suppose @xmath3 is compact with barrier boundaries @xmath68 , such that @xmath69 < 0 , \quad \theta^+[\partial^+ m ] > 0\ ] ] then @xmath3 contains a mots @xmath2 . theorem [ thm : motsexist ] provides the analogue of the barrier argument for existence of minimal surfaces . the proof considers a sequence of dirichlet problems for jang s equation , which forces a blowup solution . we solve @xmath60 = 0 , \quad f \bigg{|}_{\partial^{\pm } m } = \mp z\ ] ] see figure [ fig : jangdirichlet ] . if we let @xmath70 , then the solution converges to solution with blowups . in order to prove boundary gradient estimates necessary to apply leray - schauder theory , one makes use of a deformation of the cauchy surface , see figure [ fig : jangbend ] , to get @xmath71 at @xmath72 . the limiting solution must blow up somewhere , which implies the existence of a mots . we have a foliation by barriers near @xmath72 . using this fact , and the maximum principle , one can show the motss constructed are in the _ undeformed _ region of @xmath3 . by deforming the data inside @xmath73 , we can allow @xmath74 \leq 0 $ ] . we remark that @xcite has studied the plateau problem for motss using perron s method . by rauchaudhuri , @xmath75 if nec holds . [ thm : amms ] let @xmath1 be a spacetime which satisfies nec . let @xmath76 , be a cauchy foliation of @xmath1 , and assume we have outer barriers . if @xmath77 contains a mots , then each @xmath78 , @xmath79 contains a mots . based on theorem [ thm : amms ] , it seems natural to view the collection of outermost mots in @xmath78 as the black hole boundary in @xmath1 . if it is smooth , this collection is a mott . for further regularity and continuation results for motts , cf . [ thm : areabound ] suppose @xmath3 has an outer barrier . there is a constant latexmath:[$c = c(|{\operatorname{riem}}|_{c^0 } , for a bounding mots @xmath2 in @xmath3 , either @xmath81 or there is a mots @xmath82 _ outside _ @xmath2 . the idea of proof of theorem [ thm : areabound ] is the following . if @xmath28 is very large , then due to curvature bounds and the bounded @xmath83 , @xmath2 must nearly meet itself from the outside , which implies that the outer injectivity radius @xmath84 must be small . in this situation we can use surgery and heat flow to show the existence of a mots outside @xmath2 . . ] . ] in a location where @xmath2 nearly meets itself on the outside , we glue in a neck with @xmath23 , see figure [ fig : volumefigs ] . the resulting surface is then deformed using the @xmath16 heat flow @xmath85 this gives a family @xmath86 , @xmath87 . the maximum principle can be used to show that for @xmath88 , @xmath89 is outside @xmath2 , with @xmath90 < 0 $ ] . thus , @xmath86 is an inner barrier , which means that we can apply the existence result theorem [ thm : motsexist ] . it follows there is a mots @xmath91 _ outside _ @xmath2 . each time the above argument is applied it uses at least @xmath92 of the volume outside @xmath2 , and hence after finitely many steps , one has a @xmath93 outside @xmath2 with outer injectivity radius @xmath94 . the surface @xmath93 has the claimed area bound . to estimate the area , we use the estimates on curvature and @xmath95 to estimate the volume of a tube around @xmath93 from below , using the divergence theorem , in terms of @xmath96 . this tube must have volume bounded by @xmath83 , which leads to an estimate for @xmath96 . this area bound , together with the curvature bound from theorem [ thm : am05 ] , gives compactness for the family of outermost motss in a sequence of cauchy data sets with suitable uniformity properties , cf . @xcite . it is a direct consequence of the gluing and heat flow construction used in the proof of the area bound , that if @xmath97 are locally outermost motss which are sufficiently close , then there is a mots @xmath2 surrounding them . this may be interpreted as stating that black holes must coalesce once they are sufficiently close . the phenomenon described here is seen in numerical simulations . as the mots @xmath2 is formed in an evolution , it has principal eigenvalue @xmath46 and generically there is a mott which bifurcates into existence when @xmath2 is formed . see @xcite for details . the above result gives a _ `` maximum principle for mots''_. note that the usual maximum principle _ does not apply _ for motss which meet on the _ outside_. let @xmath36 be an af data set . the trapped region is @xmath98 if @xmath48 @xmath99 , with @xmath100 weakly outer trapped , then @xmath101 has smooth boundary @xmath102 , with @xmath103 = 0 $ ] . in particular , @xmath104 is the _ unique outermost mots in @xmath3_. for the proof , replace @xmath101 by @xmath105 \leq 0 , \quad \text { and } i^+(\partial \omega ) \geq \delta _ * \}\ ] ] for the collection of subsets defining @xmath106 we have _ compactness _ by the area bound for surfaces with @xmath95 bounded from below . this , together with a gluing construction to smooth corners gives that @xmath107 is a mots . to complete the proof we have to show that @xmath108 . to see this , suppose there is a weakly outer trapped surface @xmath109 . we can argue that this means @xmath110 . smoothing gives a barrier , see figure [ fig : smoothing2 ] and hence there is a mots outside . this can be taken to be in @xmath106 . hence @xmath111 , which completes the proof . by @xcite , the outermost mots is a union of finitely many @xmath112 , assuming nec . bray and khuri have proposed generalized apparent horizons ( gah ) , satisfying the condition @xmath113 as well as a generalized jang s equation motivated by the gah condition , as part of an approach to the general penrose inequality . eichmair @xcite proved existence of outermost gah . these are area outer minimizing . however , it is not clear how they are related to black holes . large families of gah conditions can be treated using the techniques discussed here . the known conditions for existence of motss in a cauchy data set @xcite involve nonvacuum data . a better understanding of conditions for the existence of motss in vacuum , due to concentration of curvature in terms of , say , curvature radii , conformal spectral gap , etc . is needed . i thank the mittag - leffler - institute , djursholm , sweden for hospitality and support . this work was supported in part by the nsf , under contract no . dms 0407732 and dms 0707306 with the university of miami .

i will discuss some recent results on marginally outer trapped surfaces , apparent horizons and the trapped region . a couple of applications of the results developed for marginally outer trapped surfaces to coalescence of black holes and to the characterization of the trapped region are given . address = albert einstein institute + am mhlenberg 1 + d-14467 golm + germany

the notion of asymptotic dimension of a metric space was introduced by gromov in @xcite . it is a large scale analog of topological dimension and it is invariant by quasi - isometries . this notion has proved relevant in the context of novikov s higher signature conjecture . yu @xcite has shown that groups of finite asymptotic dimension satisfy novikov s conjecture . dranishnikov ( @xcite ) has investigated further asymptotic dimension generalizing several theorems from topological to asymptotic dimension . in this paper we are concerned with the relationship between asymptotic dimension of a gromov - hyperbolic space ( see @xcite ) and the topological dimension of its boundary . gromov in @xcite , sec . @xmath3 sketches an argument that shows that complete simply connected manifolds @xmath4 with pinched negative curvature have asymptotic dimension equal to their dimension . he observes that the same argument shows that @xmath5 for @xmath6 a hyperbolic group and asks whether such considerations lead further to the inequality @xmath7 . bonk and schramm ( @xcite ) have shown that if @xmath0 is a gromov - hyperbolic space of bounded growth then @xmath0 embeds quasi - isometrically to the hyperbolic @xmath8-space @xmath9@xmath10 for some @xmath8 . it follows that @xmath11 ( see also @xcite for a proof of this ) . if @xmath12 is any metric space one can define ( @xcite , @xcite ) a hyperbolic space @xmath13 with @xmath14 . if @xmath0 is a visual hyperbolic space then @xmath0 is quasi - isometric to @xmath15 ( i.e. the boundary determines the space ) . so it is natural to ask whether @xmath16 for visual hyperbolic spaces in general . besides the argument sketched in @xcite , sec . @xmath3 makes sense in this context too . in this paper we give an example of a visual hyperbolic space @xmath0 such that @xmath17 and @xmath18 . so the inequality @xmath19 does nt hold for this space . we remark finally that gromov s question for hyperbolic group was settled in the affirmative recently by buyalo and lebedeva @xcite . * metric spaces*. let ( x , d ) be a metric space . the _ diameter _ of a set b is denoted by diam(b ) . a _ path _ in x is a map @xmath20 where i is an interval in @xmath21 . a path @xmath22 joins two points x and y in x if i [ a , b ] and @xmath22(a ) = x , @xmath22(b ) = y . the path @xmath22 is called an infinite ray starting from @xmath23 if i=[0,@xmath24 ) and @xmath25 . a geodesic , a geodesic ray or a geodesic segment in x is an isometry @xmath26 where i is @xmath21 or @xmath27 or a closed segment in @xmath21 . we use the term geodesic , geodesic ray etc for the images of @xmath22 without discrimination . on a path connected space x given two points x , y we define the path metric to be @xmath28 where the infimum is taken over all paths @xmath29 that connect @xmath30 and @xmath31 ( of course @xmath32 might be infinite ) . it is easy to see that inside a ball b(x , n ) of the hyperbolic plane or the euclidian plane the path metric and the usual metric coincide . a metric space @xmath33 is called _ geodesic metric space _ if @xmath34 ( the path metric is equal to the metric ) . * hyperbolic spaces*. let ( x , d ) be a metric space . given three points x , y , z in x we define the _ gromov product _ of x and y with respect to the basepoint w to be : @xmath35 a space is said to be _ @xmath36- hyperbolic _ if for all x , y , z , w in x we have : @xmath37 a sequence of points @xmath38 in x is said to converge at infinity if : @xmath39 two sequences @xmath38 and @xmath40 are equivalent if : @xmath41 this is an equivalence relation which does not depend on the choice of w ( easy to see ) . the boundary @xmath42 of x is defined as the set of equivalence classes of sequences converging at infinity . two sequences are close if @xmath43 is big . this defines a topology on the boundary . the boundary of every proper hyperbolic space is a compact metric space . if @xmath0 is a geodesic hyperbolic metric space and @xmath44 then @xmath42 can be defined as the set of geodesic rays from @xmath23 where we define to rays to be equivalent if they are contained in a finite hausdorf neighborhood of each other . we equip this with the compact open topology . a metric @xmath45 on the boundary @xmath42 of x is said to be _ visual _ if there are @xmath46 and @xmath47 such that @xmath48 for every z , w in @xmath42 . the boundary of a hyperbolic space always admits a visual metric ( see @xcite ) . a hyperbolic space x is called _ visual _ if for some @xmath49 there exists a @xmath50 such that for every @xmath51 there exists a geodesic ray @xmath52 from @xmath23 in @xmath42 such that @xmath53 ( see more on @xcite ) . it is easy to see that if @xmath0 is visual with respect to a base point @xmath23 then it is visual with respect to any other base point . * topological dimension*. a covering @xmath54 has _ @xmath8 if no more than @xmath55 sets of the covering have a non empty intersection . the _ mesh _ of the covering is the largest of the diameters of the @xmath56 . we will use in this paper the following definition of topological dimension for compact metric spaces which is equivalent to the other known definitions : a compact metric space has _ dimension @xmath57 _ if and only if it has coverings of arbitrarily small mesh and order @xmath57 . ( see @xcite ) * asymptotic dimension*. a metric space y is said to be d - disconnected or that it has dimension 0 on the d - scale if @xmath58 such that : @xmath59 , dist(@xmath60 ) @xmath61 @xmath62 where dist(@xmath60 ) = @xmath63 \{dist(a , b ) a@xmath64 , b @xmath65 _ ( asymptotic dimension 1)_. we say that a space x has asymptotic dimension n if n is the minimal number such that for every @xmath66 we have : @xmath67 for k = 1,2 , ... n and all @xmath68 are d - disconnected . we then write asdim = n we say that a covering @xmath54 has _ d - multiplicity _ , k if and only if every d - ball in x meets no more than k sets @xmath56 of the covering.a covering has _ n if no more than n + 1 sets of the covering have one a non empty intersection . a covering @xmath69 is _ d - bounded _ if diam @xmath70 _ ( asymptotic dimension 2)_. we say that a space @xmath0 has asdim = n if n is the minimal number such that @xmath71 there exists a covering of x of uniformly d - bounded sets @xmath56 such that d - multiplicity of the covering @xmath72 . the two definitions are equivalent . ( see @xcite ) * the hyperbolic plane*. the hyperbolic plane @xmath73@xmath74 is a visual hyperbolic space of bounded geometry . it is easy to see that @xmath75@xmath76 ( see @xcite ) . we will use the standard model of the hyperbolic plane given by the interior of a disk in @xmath77@xmath74 . let @xmath78@xmath74 be the hyperbolic plane and let @xmath79 be geodesic rays starting from a point @xmath23 and extending to infinity such that the angle between @xmath80 is @xmath81 . let @xmath82 be the sector defined by the rays @xmath80 . in other words @xmath82 is the convex closure of @xmath80 . since geodesics diverge in @xmath78@xmath74 there is an @xmath83 such that the ball of radius @xmath8 and center @xmath30 , @xmath84 is contained in @xmath82 . let @xmath85 be such that @xmath86 . let @xmath87 let s call @xmath88 the upper arc of @xmath89 , i.e. @xmath90 we subdivide @xmath88 into small pieces of length between @xmath91 and 1 marking the vertices . then we consider the geodesic rays starting from @xmath23 to every vertex we defined and we extend them to infinity . so we arrive at the `` comb '' space which is the union of all the @xmath89 together with these rays and looks like this : 10.0 cm 8.0 cm * @xmath2 . for every @xmath8 we have that @xmath88 is bounded . that means that we define a finite number of vertices on every @xmath88 so we add a finite number of geodesic rays . so , all the infinite geodesic rays are countable . so @xmath92 is countable . now a countable metric space has dimension 0 ( see @xcite page 18 ) . so dim(@xmath92)=0 * x is a hyperbolic space with the `` path '' metric . that is true since every pair of points of @xmath0 can be joined by a path of finite length . also let @xmath93 be a closed curve of x then @xmath93 is a closed curve in @xmath73@xmath74 and @xmath94 . but since @xmath73@xmath74 is hyperbolic we have the isoperimetric inequality @xmath95 so @xmath96 which means that x is hyperbolic.(see @xcite , @xcite ) * @xmath97 . that is because @xmath0 contains arbitrarily large balls @xmath98@xmath74 for every @xmath99 . * @xmath0 is a visual hyperbolic space with @xmath100 since for every @xmath30 in @xmath0 there exists a geodesic from @xmath23 to @xmath30 . let s call that @xmath101 . if @xmath101 can be extended to infinity then we have nothing to prove . let @xmath101 be finite , then @xmath30 must belong to a sector @xmath89 . we extend @xmath101 until it meets @xmath88 at a point @xmath102 . then by the construction of @xmath0 there exists an infinite geodesic @xmath52 corresponding to the vertex on @xmath88 @xmath103 such that @xmath104 is less than 1 . then obviously @xmath105 is less than 1 . 99 m. gromov , _ hyperbolic groups _ , essays in group theory ( s. m. gersten , ed . ) , msri publ . 8 , springer - verlag , 1987 pp . asymptotic invariants of infinite groups _ , geometric group theory , ( g.niblo , m.roller , eds . ) , lms lecture notes , vol . 182 , cambridge univ . press ( 1993 ) m.bonk and o.schramm,_embeddings of gromov hyperbolic spaces _ , gafa geom.funct.anal , vol 10(2000 ) , 266 - 306 . s.buyalo , n.lebedeva _ capacity dimension of locally self similar spaces _ , preprint , august 2005 . w.hurewitz and h.wallman , _ dimension theory _ , princeton university press ( 1969 ) . a.dranishnikov_asymptotic topology _ , russian math.surveys 55(2000 ) , no 6 , 71 - 116 . g.yu,_the novicov conjecture for groups with finite asymptotic dimension _ , ann . of math . 147(1998 ) , no 2 , 325 - 335 . j.roe , _ lectures on coarse geometry _ ams university lecture series , 2003 b.h.bowditch _ a short proof that a subquadratic isoperimetric inequality implies a linear one _ , michigan math j.42(1995 )

we give an example of a visual gromov - hyperbolic metric space @xmath0 with @xmath1 and @xmath2 .

gilbert s mechanical model is sketched in fig . 1 . a rigid cylindrical stick of length @xmath13 , with one end fixed at the origin , is pointing in a direction described by the angles @xmath14 and @xmath15 . the magnetization is aligned along the effective magnetic field @xmath7 at equilibrium . due to the application of a vertical force oriented along the @xmath16 axis , the stick is precessing around the vertical axis at angular velocity @xmath17 . the magnetic energy is @xmath18 where @xmath19 is the effective field and @xmath20 is the magnetization ( @xmath21 is the unit vector defined in fig.1 ) . furthermore , the stick is spinning around its own symmetry axis at angular velocity @xmath22 . this motion corresponds to the rotation of the electric carrier of ampere s dipole ( see below ) . the phase space of this rigid rotator is defined by the angles @xmath23 and the three components of the associated angular momentum @xmath1 . the relation between the angular momentum and the angular velocity @xmath24 is @xmath25 , where @xmath26 is the inertia tensor . in the rotating frame , or body - fixed frame @xmath27 , the inertial tensor is reduced to the principal moments of inertia @xmath28 . the symmetry of revolution of the spinning stick imposes furthermore that @xmath29 : @xmath30 in the fixed body frame , the angular velocity reads ( see fig . 1 ) : @xmath31 [ h ! ] [ cols="^ " , ] the kinetic equation is obtained from the angular velocity : for any vector @xmath0 of constant modulus carried with the rotating body , we have : @xmath32 let us start with gilbert s hypothesis of vanishing inertia @xcite : @xmath33 so that @xmath34 . however , we have @xmath35 . since @xmath36 , the conservation of angular momentum @xmath1 imposes @xmath37 constant ( this is also valid in the case of damping @xcite ) . without loss of generality , we can define the modulus of the vector @xmath0 with the help of the constant @xmath38 , such that @xmath39 where @xmath38 defines the well - known gyromagnetic ratio . the effective magnetic field is defined by the canonical relation @xmath40 where @xmath41 is the gradient defined on the configuration space @xmath42 ( which is the surface of the sphere of radius @xmath13 ) . the torque exerted on the system is defined by the vectorial product @xmath43 . by convention , we defined the direction @xmath8 along the effective field @xmath44 . the third newton s law @xmath45 gives then the kinetic equation of the magnetization : @xmath46 according to the gyromagnetic relation eq.([gyromag ] ) , we have @xmath47 and equation eq.([llg ] ) is nothing but the well - known equation of the precession of the magnetization without damping : @xmath48 . furthermore , since @xmath49 , the kinetic equation reads @xmath50 . inserting the precession angular velocity @xmath51 , we have : @xmath52 which is the definition of the larmor angular velocity , as expected for a precessing magnetic moment . the geometric phase is the phase difference acquired over the course of a precession loop . the precession time @xmath53 ( i.e. the slow characteristic time of our problem ) is the time at which the axis @xmath21 is rotating one cycle around the axis @xmath8 , i.e. such that @xmath54 . according to eq.([larmor ] ) , the precession time is given by : @xmath55 we can now give the expression of the number @xmath56 ( the subscript @xmath57 stands for the non - inertial approximation ) of rotation around the @xmath21 axis ( spinning rotation ) during the time of a precession of the same axis around @xmath8 . according to the relation @xmath58 , we have : @xmath59 where we used eq.([precession_time ] ) , eq.([gyromag ] ) , and the expression of the larmor angular velocity @xmath60 . anticipating over the next section , we introduce the `` slowness parameter '' @xmath61 defined as the dimensionless angular momentum @xmath62 scaled with the angular momentum @xmath63 , i.e. the ratio of the slow over the fast angular momentum , or equivalently of the slow over the fast time - scale : @xmath64 the last term in the right - hand side of eq.([slowg ] ) defines the fast characteristic time @xmath65 of the motion . the expression of @xmath66 now reads : @xmath67 the first term in the right hand side can be defined as the dynamical angle , while the second term @xmath68 can be defined as the geometric phase ( see however the discussion in reference @xcite ) . note that the factor @xmath69 also defines a time ratio @xmath70 , where @xmath71 is another possible fast characteristic time of the movement . this parameter will be discussed below . the expression eq.([deltapsig ] ) is completed in section iv below , in the case of inertia , with an expansion as a series of power of @xmath72 . from the viewpoint of the geometric phase , the gilbert s magnetic dipole is defined by the two magnetic monopoles @xmath73 that radiate from the center of a sphere of radius @xmath74 through both north ( + ) and south ( - ) hemispheres . the parameter @xmath74 is defined by ampere s magnetic dipole @xmath75 that is generated by the electric carrier of charge @xmath5 and mass @xmath4 rotating inside the loop of radius @xmath74 . the phase @xmath66 then allows to link the mechanical definition of gilbert s magnetic dipole to ampere s magnetic dipole . if we define the _ radial field _ @xmath76 by a potential vector @xmath77 , the circulation of @xmath78 around a closed loop of radius @xmath74 defines a phase @xcite @xmath79 which is the geometric phase calculated above . eq.([deltapsi2 ] ) and eq.([deltapsi1 ] ) gives the expression of @xmath80 : @xmath81 on the other hand , in the framework of the ampere s model of the `` molecular currents '' , a microscopic magnetic moment is defined by the * bohr magneton * @xmath82 generated by an electron of mass @xmath4 and charge @xmath5 moving in a loop of bohr radius @xmath74 . the gyromagnetic ratio is @xmath83 and the moment of inertia associated to the loop of radius @xmath74 is @xmath84 . furthermore , the flux @xmath85 of the external magnetic field ( by convention along @xmath86 ) @xmath44 through the microscopic hemisphere of radius @xmath74 is also quantified , with the well - known quantized flux : @xmath87 equation ( [ monopolemag ] ) then reads : @xmath88 this expression defines the classical counterpart of the magnetic monopole @xcite . note that the corresponding geometric phase eq.([deltapsig ] ) reduces to : @xmath89 . the scalar gyromagnetic relation eq.([gyromag ] ) used above in the framework of the mechanical ( or gilbert s ) model of the magnetic dipole coincides with the usual vectorial definition @xmath2 of the gyromagnetic relation if the inertial effects are neglected @xmath10 . if we take into account inertial effects , @xmath90 , the gyromagnetic relation @xmath2 is no longer valid in this form . the generalized equation is obtained , by cross- multiplication of eq.([kinetic0 ] ) with the vector @xmath20 . @xmath91 or : @xmath92 newton s law @xmath93 becomes , with the constant @xmath94 : @xmath95 where the characteristic time @xmath96 has already been introduced in eq.([slowg ] ) . equation ( [ illg ] ) generalizes eq.([llg ] ) with the inertial term ( @xmath97 ) . this equation is the adiabatic limit ( i.e. without damping ) of the inertial llg presented in previous studies @xcite . it is convenient to rewrite eq.([illg ] ) , with the dimensionless time @xmath98 and the slowness parameter @xmath61 ( both defined in eq.([slowg ] ) ) . the equation of motion eq.([illg ] ) takes the following vectorial form : @xmath99 eqs . ( [ illg2 ] ) becomes @xmath100 where @xmath101 . this equation is the dynamical equation of the magnetization generalized to inertial effects ( in the absence of damping ) . these equations allow the adiabatic movement to be studied below in terms of the geometric phase . the generalized equation including gilbert damping has been studied in previous reports @xcite . the number @xmath102 of rotation around the @xmath21 axis performed by the magnetization vector during the ( dimensionless ) time @xmath103 of one precession is : @xmath104 where @xmath105 is the dimensionless angular velocity @xmath106 . due to the conservation of the angular momentum component @xmath37 , @xmath107 is constant which implies @xmath108 the hannay angle @xmath109 is @xmath110 which is the solid angle swept by the axis in one precession cycle . following ref.@xcite we seek for the slow manifold , _ i.e. _ the set of initial conditions in the phase space for which the particular solution of the equations of motion eqs.([illg3 ] ) corresponds to pure precession , which means precession in the absence of nutation . it therefore corresponds to @xmath111 , from which inserted in eq.([illg3]a ) gives @xmath112 the dynamics of pure precession therefore give two corresponding precessional velocities , a slow one @xmath113 and a fast one @xmath114 , which are given by @xmath115 the square root in this equation shows that the pure precession requires @xmath116 . therefore , pure precession without nutation is possible for @xmath117 for any inclination angle @xmath14 , whereas for @xmath118 , pure precession is only possible for inclination angles such that @xmath116 . we now consider the slow precession velocity @xmath113 given by eq.([pureprecessions ] ) . for such slow pure precession it is possible to derive exact results from eq.([psi1 ] ) . in this case @xmath119 and @xmath14 are constant , and since @xmath113 is negative whatever the sign of @xmath120 , the precession time reads @xmath121 . combined with @xmath122 , eq.([psi1 ] ) gives @xmath123 using from eq.([gpureprecession ] ) @xmath124 and using the slow precession velocity from eq.([pureprecessions ] ) @xmath125 eq.([psi3 ] ) gives @xmath126 this expression generalizes eq.([deltapsig ] ) of section iii to the inertial regime for the pure precession . this is of course the same expression as that obtained for the spinning top in ref.@xcite . in this framework , the first term @xmath127 of the expansion was the dynamical phase . the question that was discussed in ref . @xcite , was about the nature of the second term @xmath128 . there was an ambiguity about associating it to the dynamical phase or to the geometric phase . it appears below that , in the framework of the `` bohr magneton '' approach used in section iii - d for the magnetic monopole , the two first terms in the right hand side of eq.([psi4 ] ) are identical . indeed , according to the ii - d , we have @xmath129 and eq.([psi4 ] ) reads : @xmath130 the geometric phase @xmath131 is a function of the precession angle @xmath14 and the slowness parameter @xmath61 . note that if we remove the dynamical angle @xmath132 , the developpement is a function of a single parameter @xmath133 only . the generalization of the magnetic monopole eq.([monopolemag ] ) is @xmath134 so that @xmath135 \\ & = & \frac{2\cos\theta}{r^2 } \left ( 2 + \left ( \frac{cos \theta}{g^2 } \right ) - \left(\frac{cos \theta}{g^2 } \right)^2 + 2 \left ( \frac{cos \theta}{g^2 } \right)^3 - 5 \left ( \frac{cos \theta}{g^2 } \right)^4 + ... \right ) \nonumber \label{monopolegene2}\end{aligned}\ ] ] this equation gives the influence of the inertia ( i.e. the fast magnetic degrees of freedom ) on the magnetic monopole , in the case of the pure precession . magnetization dynamics have been investigated beyond the usual assumption of the total separation of time scales between slow and fast magnetic degrees of freedom , for the adiabatic limit . we have exploited the analogy with the spinning top by pushing the mechanical model of the magnetic dipole beyond gilbert s assumption . fast degrees of freedom are introduced with the angular momentum @xmath1 and its time variation ( with non - zero first and second principal moment of inertia @xmath136 ) . the problem is investigated from the viewpoint of the geometric phase which allows the magnetic monopole to be defined naturally . the effect of inertia is then taken into account , and an analytical expression is obtained in the case of the _ pure precession _ , for which the nutation vanishes . in the case of pure precession with precession angle @xmath14 , the calculation of the geometric phase shows that , beyond a dynamical phase of the form @xmath137 , the hannay angle is a simple function of the parameter @xmath138 , where @xmath139 is the slowness parameter ( i.e. the ratio of the slow characteristic time of the precession over the fast characteristic time ) . the magnetic monopole ( defined as the radial magnetic field produced from a punctual center ) , is derived directly from the geometric phase . in the usual case without inertia ( @xmath140 ) , the bohr magneton approach gives a very simple expression of the magnetic monopole as a function of the precession angle @xmath141 . in the case of pure precession , the correction due to the action of the fast degrees of freedom is given as a simple expression @xmath142 $ ] . note that in an experimental context , the magnetic monopole @xmath80 is constant because it is related to a given material , and the precession angle @xmath14 depends the parameter @xmath61 . this result suggests that the pure precession - i.e. the slow manifold for the dynamics of the magnetization @xcite - should not be a purely formal concept , but could correspond to the actual motion of the magnetization for the ultrafast precession of the magnetization , that would correspond to the minimum power dissipated by the system ( in comparison with the motion that includes nutation oscillations superimposed to the precession ) . this point should however still be clarified in further studies . j .- e . w is grateful to michael v. berry for helpful comments . + p. bruno , _ berry phase effects in magnetism _ , in `` magnetisme goes nano '' ( matter and material * 26 * ) , edited by s. blgel , t. brckel , c. m. schneider , forschungszentrums jlich 2005 . http://hdl.handle.net/2128/560 . h. kurebayashi , jairo sinova , d. fang , a. c. irvine , t. d. skinner , j. wunderlich , v. novk , r. p. campion , b. l. gallagher , e. k. vehstedt , l. p. zrbo , k. vborn , a. j. ferguson and t. jungwirth , _ an antidamping spinorbit torque originating from the berry curvature _ , nature nanotechnology * 9 * , 211 ( 2014 ) . t. l. gilbert , _ a phenomenological theory of damping in ferromagnetic materials _ , ieee trans . mag . * 40 * , 3443 ( 2004 ) . the discussion related to the assumption @xmath10 is confined in footnotes 7 and 8 . o. v. pylypovskyi , v. p. kravchuk , d. d. sheka , d. makarov , o. g. schmidt , y. gaididei , _ coupling of chiralities in spin and physical spaces : the mbius ring as a case study_. phys . lett . * 114 * , 197204 ( 2015 ) . j. miltat , g. alburquerque , a. thiaville , _ an introduction to micromagnetics in the dynamics regime _ , in _ spin dynamics in confined magnetic structures i _ , edited by b. hillebrands , k. ounadjela ( springer , berlin , 2002 ) . c. aron , d. g. barci , l. f. cugliandolo , z. g. arenas and g. s. lozano , _ magnetization dynamics : path - integral formalism for stochastic landau - lifshitz - gilbert equation _ , j. stat . mech . * 2014 * p09008 ( 2014 ) .

the landau - lifshitz - gilbert ( llg ) equation that describes the dynamics of a macroscopic magnetic moment finds its limit of validity at very short times . the reason for this limit is well understood in terms of separation of the characteristic time scales between slow degrees of freedom ( the magnetization ) and fast degrees of freedom . the fast degrees of freedom are introduced as the variation of the angular momentum responsible for the inertia . in order to study the effect of the fast degrees of freedom on the precession , we calculate the geometric phase of the magnetization ( i.e. the hannay angle ) and the corresponding magnetic monopole . in the case of the pure precession ( the slow manifold ) , a simple expression of the magnetic monopole is given as a function of the slowness parameter , i.e. as a function of the ratio of the slow over the fast characteristic times . recently , important efforts have been devoted to both the reformulation of well known effects and to the description of new phenomena by means of the geometric phase ( the quantum berry phase @xcite or the classical hannay angle @xcite ) , in particular in relation to spin systems @xcite . the geometric phase is indeed an efficient tool that allows the essential physics to be extracted from a complex system , in which gauge invariance plays a fundamental role ( e.g. in terms of `` curl forces '' @xcite or `` equilibrium currents '' @xcite ) . an important application can be found for electronic transport in ferromagnets , typically for the anomalous hall effect @xcite , or for the recent developments about electronic devices that exploit spin - orbit interactions @xcite . the geometric phase appears to be also a necessary tool for the description of the transport of magnetic moments or spins @xcite , or for the description of magnetic excitations traveling throughout chiral structures @xcite . in the above mentioned cases , the magnetic configuration is not always at equilibrium . instead , a transport effect occurs also inside the magnetic or spin configuration space , at each point of the real space . the corresponding magnetization dynamics are described by the well - known landau - lifshitz - gilbert equation ( llg ) @xcite . if one consider both the transport throughout the usual configuration space and inside the magnetization space , the set of possible magnetic excitations is extraordinarily rich and complex @xcite . even if one consider only the case of uniform magnetization ( no space variable ) , the llg equation already describes a wide variety of effects , including ferromagnetic resonance and rotational brownian motion in a field of force @xcite . furthermore , recent investigations suggest that , at the ultra - fast regime , the llg equation should be generalized with considering inertial terms @xcite . the goal of the present work is to investigate the inertial regime for the uniform magnetization with the help of the geometric phase . in this context , we focus our attention to the connection between three fundamental concepts ; the _ geometric phase _ of the magnetization , the _ magnetic monopole _ , and the _ inertial regime _ of the magnetization . the three concepts are coupled because the dynamics of a magnetic dipole are composed of both fast and slow dynamics , and the geometric phase is an efficient tool for the study of the separation of time - scales between slow and fast degrees of freedom @xcite . the influence of the fast variables on the slow motion is treated in perturbation expansions @xcite in which the ratio of small and fast time scales define a slowness parameter , and the successive terms are interpreted as reaction forces of the fast variables on the slow motion @xcite + the magnetization @xmath0 of a uniformly magnetized body is usually defined as a magnetic dipole . the description of the dynamics of a classical magnetic dipole is however still problematic today @xcite . ampere s magnetic dipole is defined by an electric charge that is moving _ at high speed _ about a microscopic `` loop '' , typically an atomic orbital . this simple model allows the gyromagnetic relation to be derived : the magnetization @xmath0 of the magnetic dipole then follows the angular momentum @xmath1 of the electric carrier , with the relation @xmath2 where @xmath3 is the gyromagnetic ratio ( @xmath4 is the mass and @xmath5 is the electric charge of the electric carrier , and @xmath6 is the land factor ) . if a static magnetic field @xmath7 ( oriented along @xmath8 ) is applied , the magnetization precesses at the larmor angular velocity @xmath9 around the axis defined by @xmath8 . in other terms , a _ slow motion _ ( precession ) is added to the _ fast motion _ ( moving electric carrier ) that defines the magnetic dipole . in the absence of dissipation the dynamics of the dipole are reduced to a simple precessional motion . however , this reduction is valid only if the velocity of the electric charge is much higher than the precession velocity , i.e. if the typical time - scales are well separated . indeed , if the larmor angular velocity is high enough and becomes of the same order as the angular velocity of the electrical carrier moving in the loop , the amperian magnetic dipole @xmath0 is no longer defined by a simple expression ( the exact trajectory of the punctual electric carrier should be taken into account instead of averaging over the loop ) @xcite . however there is an other way to define a magnetic dipole , namely the _ gilbert s dipole _ ( according to d. j. griffiths , the gilbert dipole is a double monopole @xcite ) . in our non - relativistic context , the gilbert magnetic dipole is defined by its dynamical properties , based on the mechanical analogy with the spinning top @xcite . this mechanical approach allowed t. h. gilbert to derive the well - known landau - lifshitz - gilbert equation ( llg ) , providing that the first two principal moments of inertia vanish @xmath10 , but not the third one @xmath11 @xcite . this ad - hoc assumption is related to the electrodynamic limitation of the amperian magnetic dipole mentioned above . in this context , fast degrees of freedom have been taken into account as inertial variables ( so that @xmath12 ) by enlarging the configuration space to the corresponding phase space , i.e. including the angular momentum . the corresponding generalized llg equation then contains a supplementary term proportional to the second time - derivative of the magnetization @xcite . + in the present work , we show that the hannay angle and the corresponding magnetic monopole are able to describe , in the adiabatic limit , the transition from the usual precession to more complex dynamics containing the inertial effects . the analysis follows the method recently proposed by m. v. berry and p. shukla in ref.@xcite for the study of the spinning top . within this approach , the dynamics of the magnetization are interpreted as the reaction of the fast dynamics on the slow . a simple analytical result is obtained by reducing the phase space to the slow manifold . the paper is composed as follows . section 1 below is devoted to the mechanical definition of the adiabatic gilbert dipole without taking into account the fast degrees of freedom . section 2 describes the adiabatic kinetic equation . the geometric phase is presented in section _ 2.3 _ , and the corresponding magnetic monopole is described in section _ 2.4_. section 3 studies the effect of the fast degrees of freedom . in particular , the calculation of the adiabatic dynamics of the magnetization that includes inertia is presented in section _ 3.1 _ , and the calculation of the geometric phase with inertia is given in section _ 3.2_. the case of the pure precession is studied in section _ 3.3 _ , and the corresponding magnetic monopole is given in section _ 3.4_. the conclusion is proposed in section _ 4_.

quasars ( qsos ) serve as tools , in conjunction with studies of the intergalactic medium , for probing conditions in the early universe . these studies rely on the fact that the spectra are , to the lowest order , rather uniform ( e.g. , the construction and application of qso composite spectra ) . we know , however , that the spectra do exhibit differences : the spectral slopes , as well as the line profiles , differ among quasars . in fact , even in a single spectrum , the widths of the emission lines can be vastly different . although these differences may provide insights for understanding the physical environments in the vicinity of quasars ( by constructing inflow or outflow models for different kinds of elements in the surroundings ) , they present substantial challenges when modeling broad and narrow line regions ( blrs and nlrs ) . a quantitative understanding of the variation in quasar spectra is therefore a necessary and important study . in the pioneering work by francis ( 1992 ) , the authors applied a principal components analysis ( pca ) to 232 quasar spectra ( i.e. , spectral pca , in which the concerned variables are the observed flux densities in the wavelength bins of a spectrum ) from the large bright quasar survey ( lbqs ; hewett et al . 1996 ) and found that the mean spectrum plus the first two principal components in the rest - wavelength range @xmath8 describe the majority of the variation seen in the uv - optical spectra of quasars . in this spectral region , the quasars are shown to have a variety of spectral slopes and equivalent widths , ranging from broad , low - equivalent - width lines to narrow , high - equivalent width lines , with other spectral properties also varying along this trend . furthermore , boroson and green ( 1992 ) identified several important parameters in describing quasars and carried out a pca on 87 quasars from the bright quasar survey ( bqs ; schmidt & green 1983 ) in this parameter space ( i.e. , parameter pca , in which the variables are the physical quantities of interest ) , from which an anti - correlation was found between ( optical , around the h@xmath10 spectral region ) and @xmath11@xmath7 $ ] . ( this correlation is widely quoted as `` eigenvector-1 '' ) . more recently , shang et al . ( 2003 ) considered a wider rest - wavelength range covering ly@xmath6 to h@xmath6 , and constructed eigenspectra from 22 optically selected quasars from the bqs . their results agreed with boroson and green s eigenvector-1 , and supported the speculated anti - correlation between ( optical ) and ( uv ) . the conclusions of these studies , however , are drawn from small ranges of redshifts ( @xmath12 ; @xmath13 ; @xmath14 ) respectively . the sdss spectroscopic survey has the advantage of a large number of quasars , and most importantly , a large redshift range . it provides a unique opportunity for investigating how quasars differ from one another , and whether they form a continuous sequence ( francis et al . 1992 ) . in this paper , we apply the karhunen - love ( kl ) transform to study this problem in the 16,707 quasars from the sdss . the primary goals of this paper are to 1 ) obtain physical interpretations of the eigenspectra , 2 ) determine the effects of redshift and luminosity on the spectra of quasars , and 3 ) study the correlations between broad emission lines and uv - optical continua . with this data set , in which @xmath15 % of quasars were discovered by the sdss , our analysis is the most extensive of its kind to date . we discuss the sdss quasar sample used in this work in [ section : data ] , followed by a review of the kl transform and the gap - correcting procedures in [ section : kl ] . the set of quasar eigenspectra for the whole sample covering @xmath16 in rest - wavelength are presented in [ section : global ] . we quantitatively detect the redshift and luminosity effects through a commonality analysis of the eigenspectra sets constructed from quasar subsamples in [ section : similar ] . the quasar eigenspectra in several subsamples of different redshifts and luminosities are shown in [ section : zbin ] , and we make a comparison between the kl - reconstructed spectra using either sets of eigenspectra ( i.e. , the subsamples versus the global case ) . in [ section : crossbin ] , we perform a kl transform on cross - redshift and -luminosity bins , from which evolutionary ( [ section : evolution ] ) and luminosity effects ( [ section : baldwin ] ) are found in the quasar spectra . in [ section : class ] , we discuss the possible classification of quasar spectra by invoking the eigencoefficients in these subsamples . correlations among the broad emission lines and the local eigenspectra are presented in [ section : linecorr ] , including the well - known `` eigenvector-1 '' . [ section : conclusion ] summarizes and concludes the present work . the sample we use is an early version of the first data release ( dr1 ; abazajian et al . 2003 ) quasar catalog @xcite from the sloan digital sky survey ( sdss ; york et al . 2000 ) , which contains 16,707 quasar spectra and was created on the 9th of july , 2003 . the official dr1 quasar catalog includes slightly more objects ( 16,713 ) and was created on the 28th of august , 2003 . all spectra in our sample are cataloged in the official dr1 quasar catalog except one : ( i.e. , there are 7 dr1 qsos not included in our sample ) . the sdss operates a ccd camera @xcite on a 2.5 m telescope located at apache point observatory , new mexico . images in five broad optical bands ( with filters @xmath17 and @xmath18 ; fukugita et al . 1996 ) are being obtained over @xmath19 deg@xmath20 of the high galactic latitude sky . the astrometric calibration is described in pier et al . the photometric system is described in smith et al . ( 2002 ) while the photometric monitoring is described in hogg et al . the details of the target selection , the spectroscopic reduction and the catalog format are discussed by schneider et al . ( 2003 ) and references therein . about @xmath21 % of the quasar candidates in our sample are chosen based on their locations in the multi - dimensional sdss color - space @xcite , while @xmath22 % are targeted solely by the serendipity module . the remaining qsos are primarily targeted as first sources , rosat sources , stars or galaxies . all quasars in the dr1 catalog have absolute magnitudes ( @xmath23 ) brighter than @xmath24 , where @xmath23 are calculated using cosmological parameters @xmath25 km s@xmath26 mpc@xmath26 , @xmath27 and @xmath28 ; and that the uv - optical spectra can be approximated by a power - law @xmath29 with the frequency index @xmath30 @xcite . the absolute magnitudes in five bands are corrected for galactic extinction using the dust maps of schlegel , finkbeiner & davis ( 1998 ) . quasar targets are assigned to the 3 diameter fibers for spectroscopic observations ( the tiling process ; blanton et al . ( 2003 ) ) . spectroscopic observations are discussed in detail by york et al . ( 2000 ) ; castander et al . ( 2001 ) ; stoughton et al . ( 2002 ) and schneider et al . the sdss spectroscopic pipeline , among other procedures , removes skylines and atmospheric absorption bands , and calibrates the wavelengths and the fluxes . the signal - to - noise ratios generally meet the requirement of @xmath31 of 15 per spectroscopic pixel @xcite . the resultant spectra cover @xmath32 in the observed frame with a spectral resolution of @xmath33 . at least one prominent line in each spectrum in the dr1 quasar catalog is of full - width - at - half - maximum ( fwhm ) @xmath34 km s@xmath26 . type ii quasars and bl lacs are not included in the dr1 quasar catalog . all of the 16,707 quasars are included in our present analysis , including quasars with broad absorption lines ( balqsos ) . to perform the kl transforms , the spectra are shifted to their restframes , and linearly rebinned to a spectral resolution @xmath35 , with @xmath36 being the lowest redshift of the whole sample ( [ section : global ] ) or of the subsamples of different @xmath37-bins ( defined in [ section : zbin ] ) . skylines and bad pixels due to artifacts are removed and fixed with the gap - correction procedure discussed in [ section : kl ] . unless otherwise specified , in this paper we present every quasar spectrum as flux densities in the observed frame and wavelengths in the restframe for the convenience of visual inspection . following the convention of the sdss , wavelengths are expressed in vacuum values . the karhunen - love transform ( or principal component analysis , pca ) is a powerful technique used in classification and dimensional reduction of massive data sets . in astronomy , its applications in studies of multi - variate distributions have been discussed in detail ( efstathiou & fall 1984 ; murtagh & heck 1987 ) . the basic idea in applying the kl transforms in studying the spectral energy distributions is to derive from them a lower dimensional set of _ eigenspectra _ @xcite , from which the essential physical properties are represented and hence a compression of data can be achieved . each spectrum can be thought of as an axis in a multi - dimensional hyperspace , @xmath38 , which denotes the flux density per unit wavelength at the @xmath39-th wavelength in the @xmath0-th quasar spectrum . for the moment , we assume that there are no gaps in each spectrum ; we will discuss the ways we deal with missing data later . from the set of spectra we construct the correlation matrix @xmath40 where the summation is from @xmath41 to the total number of spectra , @xmath42 , and @xmath43 is the normalized @xmath0-th spectrum , defined for a given @xmath0 as @xmath44 the eigenspectra are obtained by finding a matrix , @xmath45 , such that @xmath46 where @xmath47 is the diagonal matrix containing the eigenvalues of the correlation matrix . @xmath45 is thus a matrix whose @xmath0-th column consists of the @xmath0-th eigenspectrum @xmath48 . we solve this eigenvalue problem by using singular value decomposition . the observed spectra are projected onto the eigenspectra to obtain the eigencoefficients . in these projections , every wavelength bin in each spectrum is weighted by the error associated with that particular wavelength bin , @xmath49 , such that the weights are given by @xmath50 . the observed spectra can be decomposed , with no error , as follows @xmath51 where @xmath52 is the total number of eigenspectra , and @xmath53 are the expansion coefficients ( or the _ eigencoefficients _ ) of the @xmath0-th order . it is straightforward to see that , if the number of spectra is greater than the number of wavelength bins , @xmath52 equals the total number of wavelength bins in the spectrum . an assumption that the spectra are without any gaps was made previously . in reality , however , there are several reasons for gaps to exist : different rest - wavelength coverage , the removal of skylines , bad pixels on the ccd chips all leave gaps at different restframe wavelengths for each spectrum . all can contribute to incomplete spectra . the idea behind the gap - correction process is to reconstruct the missing regions in the spectrum using its principal components . the first application of this method to analyze galaxy spectra is due to connolly & szalay ( 1999 ) , which expands on a formalism developed by everson & sirovich ( 1994 ) for dealing with two - dimensional images . initially , we fix the missing data by some means , for example , linear - interpolation . a set of eigenspectra are then constructed from the gap - repaired quasar spectra . afterward , the gaps in the original spectra are corrected with the linear combination of the kl eigenspectra . the whole process is iterated until the set of eigenspectra converges . from our previous work on the sdss galaxies @xcite , the eigenspectra set converges both as a function of iteration steps in the gap - repairing process and the number of input spectra . to measure the commonality between two sets of eigenspectra ( i.e. , how alike they are ) , two subspaces @xmath54 and @xmath55 are formed respectively for the two sets . the sum of the projection operators of each subspace is calculated as follows @xmath56 where @xmath57 are the basis vectors which span the space @xmath54 ( see , for example , merzbacher 1970 ) . a basis vector is an eigenspectrum if @xmath54 is considered to be a set of eigenspectra . if the two subspaces are in common , we have @xmath58 where @xmath59 is the trace of the products of the projection operators , and @xmath60 is the ( common ) dimension of both subspaces . the two subspaces are disjoint if the trace quantity is zero , which hence serves as a quantitative measure for the similarity between two arbitrary subspaces of the same dimensionality . models of accretion on black holes and scenarios for the formations of -blends often predict relationships between the uv and optical quasar spectral properties ( for example , the strong anti - correlation between the `` small bump '' and the optical blends was suggested by netzer and wills 1983 ) . using our sample with 16,707 quasar spectra , we construct a set of eigenspectra covering 900 to 8000 in the restframe . for each quasar spectrum , the spectral regions without the sdss spectroscopic data are approximated by the linear combinations of the calculated eigenspectra by the gap - correction procedure described in [ section : kl ] . a quantitative assessment of this procedure on quasar spectra is discussed in detail in appendix [ appendix : gapcorr ] . to determine the number of iterations needed for this gap - correcting procedure , we calculate the commonality between the two subspaces spanned by the eigenspectra in one iteration step and those in the next step . for the subspace spanned by the first two modes , the convergence rate is fast and it requires about three iterations at most to converge . including higher - order components , in this case the first 100 modes , the subspace takes about 10 iteration steps to converge . in this work , all eigenspectra are corrected for the missing pixels with 10 iteration steps . the gaps in each spectrum are corrected for using the first 100 eigenspectra during the iteration . the partial sums of weights ( i.e. , accumulative weights , where the weights are the eigenvalues of the correlation matrix ) in different orders of the global eigenspectra are shown in table [ tab : weights_all ] . the first eigenspectrum accounts for about 0.56 of the total sample variance and the first 10 modes account for @xmath61 . to account for @xmath62 of the total sample variance , about @xmath63 modes are required . the first four eigenspectra are shown in figure [ fig : global_eigenspec ] , and their physical attributes will be discussed below . the first eigenspectrum ( the average spectrum of the data set ) reveals the dominant broad emission lines that exist in the range of @xmath64 . these , presumably doppler - broadened lines , are common to most quasar spectra . as can be seen in figure [ fig : compare_1steigspec_edrcomposite ] , this eigenspectrum exhibits a high degree of similarity with the median composite spectrum @xcite constructed using over 2200 sdss quasars , but with lesser noise at the blue and red ends , probably due to the larger sample used in this analysis . the 2nd eigenspectrum shows a striking similarity in the optical region ( @xmath65 ) with the 1st galaxy eigenspectrum ( i.e. , mean spectrum ) from the sdss galaxies ( of @xmath66 galaxy spectra ; yip et al . figure [ fig : compare_qso_gal ] shows a comparison between the two . besides the presence of the ca@xmath67 and ca@xmath67 lines and the balmer absorption lines as reported previously in the composite quasar spectrum , the triplet ( which appears to be composed of two lines because of the limited resolution , i.e. , @xmath685169+@xmath685174 , and @xmath685185 ) is also seen in this mode . the presence of the balmer absorption lines ( see the inset of figure [ fig : compare_qso_gal ] ) implies the presence of young to intermediate stellar populations near the nuclei ( because of the sdss 3 spectroscopic fiber ) . the main differences between the quasar 2nd eigenspectrum and the galaxy mean spectrum lie in the balmer lines h@xmath6 and h@xmath10 , which are , as expected , doppler - broadened for the qso spectra . the quasar eigenspectrum also has a redder continuum , meaning that _ if _ this eigen - component represents all contributions from the host - galaxies , the galaxies would be of earlier spectral type than the average spectral type in the sdss main galaxy sample . our ability to detect significant host - galaxy features in this eigenspectrum triggers an important application , that is , the removal of the host - galaxy contributions from the quasar spectra . the properties the host - galaxies of quasars have recently attracted interest ( e.g. , bahcall et al . 1997 , mclure et al . 1999 , mclure et al . 2000 , nolan et al . 2001 , hamann et al . 2003 ) , mainly because of their obvious relationship with the quasars they harbor and the probable co - evolution that happens between them . therefore , the evolution of massive galaxies , which are believed to be at one time active quasar hosts ( see hamann & ferland 1999 ) , can also be probed . on the other hand , narrow emission lines in active galactic nuclei ( agns ) have been considered less useful than broad emission lines as diagnostic tools , because agns with prominent narrow lines have low luminosities ( see , for example , the discussion in chapter 10 of krolik 1999 ) , in which case contributions from the host galaxies may affect both the continuum and the lines , obscuring their true appearances . hence , the removal of host - galaxy components can potentially fix the narrow emission lines and reveal their true physical nature . preliminary results ( vanden berk et al . 2004 , in prep . ) show that it is possible to remove the galaxy continuum in the lower - redshift quasars in the sdss sample . related issues such as the effects on the broad and narrow emission lines from such a removal procedure are beyond the scope of this paper and are currently being studied . the second mode also shows slight anti - correlations between major broad emission lines which exist in @xmath69 smaller and larger than @xmath70 ( see figure [ fig : global_eigenspec ] ) . the change of the continuum slope , with a zero - crossing ( i.e. , a node ) at around 3990 , dominates this global eigenspectrum . the optical continuum appears to be galaxy - like , but not as much as the 2nd global eigenspectrum . for example , in this component the [ ] @xmath683728 is missing , and the nebular lines are generally weaker . the node at @xmath71 is in partial agreement with the 2nd principal component of 18 low - redshift ( @xmath72 ; balqsos _ excluded _ ) quasar spectra @xcite , which showed the uv - optical continuum variation ( except the node is at @xmath73 ) . this particular wavelength ( 4000 ) marks the modulation of the slope between the uv and the optical regions . one related effect is the `` ultra - violet excess '' , describing the abrupt rise of quasar flux densities from about 4000 to 3500 . this observed excess flux was suggested to be due to the balmer continuum @xcite , as there seem to be no other mechanisms which can explain this wavelength coincidence . in malkan & sargent s work , an exact wavelength for this onset was not clear . the node at @xmath71 can serve the purpose of defining that wavelength . other possible physical reasons for the modulations between the uv and optical continua are the intrinsic change in the quasar continuum ( e.g. , due to intrinsic dust - reddening ) and the stellar light from the host galaxy . there is also a second node located in ly@xmath6 showing an anti - correlation between the continua blueward and redward of the ly@xmath6 . since the number of quasars with spectroscopic measurements in the vicinity of ly@xmath6 is much smaller than those with measurements in the uv - optical regions that are redward of ly@xmath6 , the significance of this anti - correlation is less than that of the uv - optical continuum variation in this eigenspectrum . this mode shows the correlations of broad emission lines , namely , ly@xmath6 , , + @xmath7 $ ] , @xmath7 $ ] , , @xmath11@xmath7\lambda$]5008 and also the balmer emission lines h@xmath6 , h@xmath10 , h@xmath74 , h@xmath75 and h@xmath76 . these are in partial agreement with the 3rd eigenspectrum of shang et al . , in which emission lines @xmath7 $ ] , , h@xmath6 , h@xmath10 are found to be involved . it seems natural that these balmer lines are correlated , as presumably they are formed coherently by some photo - ionization processes . however , it is not known why they appear in this low - order mode . the fact that @xmath7 $ ] and h@xmath10 vary similarly was seen previously @xcite , and it was suggested that h@xmath10 and @xmath7 $ ] may arise from the same optically - thick disk . by construction , subsequent higher - order eigenspectra show more nodes , causing small modulations of the continuum slope . they also show broad absorption line features . since quasars with bals are not the dominating populations in our sample ( there are 224 broad absorption line quasars in the 3814 quasars from the sdss edr quasar catalog , reichard et al . 2003 ) , their signatures preferentially show up at higher orders in this global set of eigenspectra . the bal components are not confined to only one particular mode , but span a number of orders . to investigate the effects of balqsos on the global eigenspectra , our approach is to perform the kl transform on our original sample ( including balqsos ) and on the same sample but with the balqsos excluded , and make a comparison between them . there are 682 balqsos ( with balnicity index @xmath77 ) found in our sample according to the balqso catalog for the sdss spectra by trump et al . ( private communication ) . figure [ fig : weight_global_bal_balexcluded ] compares the weights at different orders between the balqso - included and the balqso - excluded global eigenspectra . since the balqso - included global eigenspectra contain information describing both the non - balqsos and the balqsos , the weight of each mode is larger than that of the balqso - excluded eigenspectra . that is , the balqso - excluded eigenspectra set is more compact . the magnitude of this offset , however , is small and is apparent only after the 5-th order , which is consistent with the fact that the balqsos form a minority population ( about @xmath78 % ) . this difference is seen to extend to higher orders , implying that the features describing the balqsos span a number of higher - order eigenspectra and are not confined to only one particular mode . a comparison of the 6th global eigenspectrum between the balqso included and excluded samples is shown in figure [ fig : comp_all_allcutbal_6thmode ] . absorption features ( in this case , in + @xmath7 $ ] and ) are found in the first set of eigenspectra but are missing in the latter . we have to note that the discrepancies in the spectral features of these two sets of eigenspectra attributed to the weight differences are not only confined to the existence or non - existence of bal absorption troughs as shown here , as the difference in the normalizations between the two can in general also yield different eigenspectra sets . we will leave the discussion of the reconstruction of the balqso spectra using eigenspectra till [ section : bal ] . to study the possible evolution and luminosity effects in the quasar spectra , our first step is to investigate whether the set of eigenspectra of a given order derived from quasar spectra in different redshift and luminosity ranges differ . the trace quantity mentioned in [ section : kl ] is adopted for these quantitative comparisons . as a null measure , two subsamples are chosen with approximately the same redshift and luminosity distributions , such that any differences in the two sets of eigenspectra would be due to noise and the intrinsic variability of the quasars . we fix the rest - wavelengths of this study to be @xmath79 , and require a full rest - wavelength coverage of the input quasars ; redshifts are limited to 0.9 to 1.1 . one subsample contains 472 objects ( subsample 1 ) and the other subsample , 236 objects ( subsample 2 ) . subsample 2 is , by construction , a subset of the original 472 objects . the reason behind this construction is to ensure a high commonality of the two sets of resultant eigenspectra . they both have luminosities from @xmath80 to @xmath81 , and the actual distributions of redshifts and luminosities are similar . the line on the top in figure [ fig : common_subsample ] shows the commonality of these two subsamples as we increase the number of eigenspectra forming the subspace . as higher orders of eigenspectra are included in the subspaces , the commonality drops , meaning that the two subspaces become more disjoint . as mentioned above , this disjoint behavior is mainly due to the noise and the intrinsic variability among quasars , both are unlikely to be completely eliminated . at about 20 modes and higher , the commonality levels off , which implies that the eigenspectra mainly contain noise . with this null measure in place , the differences of our test subsamples are further relaxed to include luminosity effects alone ( subsamples 1 and 3 , see table [ tab : subsample ] ) , redshift effects alone ( subsamples 3 and 4 ) , and lastly , both effects combined ( subsamples 1 and 4 ) . the commonalities of these subsamples are overlaid in figure [ fig : common_subsample ] . the first modes constructed in all these subsamples , including the null measure , are always very similar to each other ( more than 99 % similar ) . this shows that a single mean spectrum can be constructed across the whole redshift coverage , which was presumed to be true in many previous constructions of quasar composite spectra . the validity of construction of the mean spectrum in a given sample may seem trivial , but it is not if we take into account the possibility that the quasar population may evolve at different cosmic epochs . similar to the null measure , as higher orders are included in the subspaces , the eigenspectra subspaces become more disjoint . in addition , the commonalities in these condition - relaxed cases actually drop _ below _ the null measure for orders of modes higher than @xmath3 . therefore , the eigenspectra of the same order but derived from quasars of different redshifts and luminosities describe different spectral features . in addition , our results show that both luminosity and evolution effects have detectable influences on the resultant sets of eigenspectra , very much to the same degree ( in terms of commonality ) . in the case of the combined effects , the commonality drops to the lowest value among all cases , as expected . the actual redshift and luminosity effects found in the quasar spectra will be presented in sections [ section : evolution ] and [ section : baldwin ] . we learn from this analysis that there does not exist a unique set of kl eigenspectra across the whole redshift range , with the number of modes equal or smaller than approximately 10 . the implications are twofold . on one hand , the classification of quasar spectra , in the context of the eigenspectra approach , has to be redshift and luminosity dependent . in other words , the _ weights _ of different modes are in general different when quasars of different redshifts and luminosities are projected onto the same set of eigenspectra . so , eigenspectra derived from quasars of a particular redshift and luminosity range in general do not _ predict _ quasar spectra of other redshifts and luminosities . on the other hand , the existence of the redshift and luminosity effects in our sample can be probed quantitatively by analyzing the eigenspectra subspaces . kl transforms are performed on subsamples with different redshift and luminosity ranges , that allow us to explicitly discriminate the possible luminosity effects on the spectra from any evolution effects , and vice versa . the constructions of these bins are based on requiring that the maximum gap fraction among the quasars , that is , the wavelength region without the sdss data , is smaller than 50 % of the the total spectral region we use when applying the kl transforms . the total spectral region , by construction , is approximately equal to the largest common rest - wavelengths of all the quasars in that particular bin . we find that constraining the gap fraction to be a maximum of 50 % improves the accuracy of the gap - correcting procedure for most quasars ( see appendix [ appendix : gapcorr ] for further explanation ) . as a result , five divisions are made in the whole redshift range @xmath82 ( where the quasars of redshifts larger than 5.13 are discarded to satisfy the constraint of 50 % minimum wavelength - coverage in all related luminosity bins ) , and four in the whole luminosity range @xmath83 . these correspond to * zbin 1 * to * 5 * and the @xmath23 bins * a * to * d * for the redshift and luminosity subsamples respectively . in the following , we denote each subsample in a given luminosity and redshift range , for example , the bin * a4*. such divisions are by no means unique and can be constructed according to one s own purposes , but we find that important issues such as the correlation between continua and emission lines remain unchanged as we construct bins with slightly different coverages in redshift , in luminosity and in the total rest - wavelength range . the actual rest - wavelength range and the number of spectra in each bin are shown in table [ tab : cuts ] , which also lists the fractions of qsos in each bin that are targeted either in the quasar color - space @xcite or solely by the serendipity module . while the majority of the quasars from most of the bins are targeted by using the multi - dimensional color - space , in which the derived eigenspectra are expected to be dominated the intrinsic quasar properties , there is one bin ( * c4 * ) in which most quasars are targeted by the serendipity module . in principle , the eigenspectra in the latter case will represent the properties of the serendipitous objects and lack a well motivated color distribution . in general for all @xmath37-bins , the first 10 modes or less are required to account for more than 92 % of the variances of the corresponding spectra sets ( table [ tab : weights ] ) . in the iterated calculation of the @xmath37-binned eigenspectra , the first 50 modes are used in the gap correction . the first 4 orders of eigenspectra of each ( @xmath84)-bin are shown in figures [ fig : zbin1_eigenspec ] @xmath85 [ fig : zbin5_eigenspec ] , arranged in 5 different redshift ranges . in each figure , eigenspectra of different luminosities are plotted along with the ones which are constructed by combining all luminosities ( shown in black curves ) . by visual inspection , the eigenspectra in different orders show diverse properties for each @xmath37-bin . in the following , properties associated with different orders are extracted by considering _ all _ @xmath37-bins generally . eigenspectra which are distinct from the average population will be discussed separately . as in the global case , the lowest - order eigenspectra are simply the mean of the quasars in the given subsamples . for every redshift bin , the first eigenspectrum shows approximately a power - law shape ( either a single or broken power - law ) , with prominent broad emission lines . different luminosity bins show differences in the overall spectral slopes to various degrees . in every redshift range , the spectra of higher - luminosity quasars are bluer than their lower luminosity counterparts . for example , * c1 * ( figure [ fig : zbin1_eigenspec ] ; @xmath87 ) shows a harder spectral slope blueward of @xmath71 than that of * d1 * ( @xmath88 ) . however , for the higher redshift ( @xmath89 ) quasars , e.g. , in * zbin 4 * ( figure [ fig : zbin4_eigenspec ] ) and * 5 * ( figure [ fig : zbin5_eigenspec ] ) , the difference in spectral slope seems to be confined mainly to changes in the flux densities blueward of ly@xmath90 . the 2nd mode in every @xmath86-bin has one node at a particular wavelength . this implies that the linear - combination of the first 2 modes changes the spectral slope . this is similar to the galaxy spectral classification by the kl approach @xcite , in which the first two eigenspectra give the spectral shape . for the lowest redshift bin ( * zbin 1 * ; figure [ fig : zbin1_eigenspec ] ) , the node of the second eigenspectrum occurs at about 3850 for the lower luminosity qsos ( * d1 * ) , but at @xmath91 for the higher luminosity ones ( * c1 * ) . possible physical reasons underlying the modulation of the uv - optical slopes were discussed previously in [ section : globalslope ] . interestingly , the luminosity averaged 2nd eigenspectrum ( black curve ) in this redshift range also shows galactic features ( as found for the 2nd global eigenspectrum ) . the continuum redward of @xmath71 is very similar to that in galaxies of earlier - type . absorption lines ca@xmath67 and ca@xmath67 , and the balmer absorption lines h 9 , h 10 , h 11 and h 12 are seen in the lower - luminosity bin * d * ( and are not present in the higher - luminosity bin * c * , hence a luminosity dependent effect is implied ) . in addition to the finer - modulation of the continuum slope provided by the 3rd eigenspectrum compared with the 2nd mode , in the redshift range @xmath92 ( * zbin 2 * ; figure [ fig : zbin2_eigenspec ] ) , averaging over all luminosities , this mode shows a strong anti - correlation between the quasi - continuum in the ( uv ) regions around ( the `` small bump '' , with its estimated location indicated in the 3rd eigenspectrum in figure [ fig : zbin2_eigenspec ] ) and the continuum in the vicinity of h@xmath10 . around the h@xmath10 emission , the continuum is blended with the optical blends , the h@xmath75 , h@xmath74 and [ ] lines . the wavelength bounds are found to be @xmath93 for the ultraviolet blends and @xmath94 4050 upward ( to @xmath94 6000 , which is the maximum wavelength of this redshift bin ) for the optical continuum around h@xmath10 . this appears to support the calculations that strong optical emissions require a high optical depth in the resonance transitions of the ( uv ) @xcite , hence a decrease in the strength of the latter . the actual wavelengths of the nodes bounding the ( uv ) region are shown in figure [ fig : zbin2_eigenspec ] . for brighter quasars ( * b2 * ) , the small bump is smaller ( @xmath95 ) than that found in fainter qsos . to examine the intrinsic broad absorption line features in the @xmath37-binned eigenspectra , we study the reconstructed spectra using different numbers of eigenspectra . figure [ fig : recon_bal_b3_277_51908_437 ] shows one of the edr bal quasars ( reichard et al . 2003 ) found in the bin * b3 * , and its reconstructed - spectra using different numbers of eigenspectra . this hibal ( defined as having high - ionization broad absorption troughs such as ) quasar is chosen for its relatively large absorption trough in for visual clarity . the findings in the following are nonetheless general . the first few modes ( @xmath96 for this spectrum ) are found to fit mainly the continuum , excluding the bal troughs . with the addition of higher - order modes the intrinsic absorption features ( in this case , in the emission lines and ) are gradually recovered . some intrinsic absorption features are found to require @xmath97 modes for accurate description , as was found in the global eigenspectra ( [ section : globalhighorder ] ) . we should note that in the reconstructions using different numbers of modes ; the _ same _ normalization constant is adopted ( meaning the eigencoefficients are normalized to @xmath98 ) . clearly , a different normalization constant in the case of reconstructions using fewer modes ( e.g. , figure [ fig : recon_bal_b3_277_51908_437]a ) will further improve the fitting in the least - squares sense . while the fact that a large number of modes are required to reconstruct the absorption troughs probably suggests a non - compact set of kl eigenspectra ( referring to those defined in this work ) for classifying bal quasars , the appropriate truncation of the expansion at some order of eigenspectra in the reconstruction process will likely lead to an _ un - absorbed _ continuum , invaluable to many applications . the proof of the validity of such a truncation will require detailed future analyses . one method is to construct a set of eigenspectra using only the known bal quasars in the sample and to make comparisons between that and our current sets of eigenspectra . by comparing the different orders of both sets of eigenspectra we may be able to recover the bal physics . we expect that this separate set of balqso - eigenspectra will likely reduce the number of modes in the reconstruction , which is desirable from the point of view of classification . reconstructions of a typical non - bal quasar spectrum are shown in figure [ fig : recon_czbin3_spec500 ] , using from ( a ) 2 to ( d ) 20 orders of eigenspectra . this particular quasar is in the @xmath86-bin * c3*. the bottom curve in each sub - figure shows the residuals from the original spectrum . the first 10 modes are sufficient for a good reconstruction . the reconstructions of the same quasar spectrum but using the global set of eigenspectra are shown in figure [ fig : recon_all_spec2017 ] , from ( a ) 2 modes to ( f ) 100 modes . to obtain the same kind of accuracy , more eigenspectra are needed in the global case ; in this case about 50 modes . this is not surprising as the global eigenspectra must account for the intrinsic variations in the quasar spectra as well as any redshift or luminosity evolutions . there are , therefore , two major factors we should consider when adopting a global set of quasar eigenspectra for kl - reconstruction and classification of quasar ( instead of redshift and luminosity dependent sets ) . first , we need to understand and interpret about @xmath99 global eigenspectra . this is significantly larger than found for galaxies ( 2 modes are needed to assign a type to a galaxy spectrum according to connolly et al . this is a manifestation of the larger variations in the quasar spectra . second , the `` extrapolated '' spectral region , @xmath100 , in figure [ fig : recon_all_spec2017 ] ( which is the rest - wavelength region without spectral data ) show an unphysical reconstruction even when 100 modes are used , although this number of modes can accurately reconstruct the spectral region with data . this agrees with the commonality analysis in [ section : similar ] , that there are evolutionary and luminosity effects in the qsos in our sample . as such , eigenspectra derived in a particular redshift and luminosity range are in general not identical to those derived in another range . the accuracy of the extrapolation in the no - data region using the kl - eigenspectra remains an open question for the @xmath37-bins . it will be an interesting follow - up project to confront the repaired spectral region with observational data , which ideally cover the rest - wavelength regions where the sdss does not . for example , uv spectroscopic observations using the hubble space telescope . to study evolution in quasar spectra with the eigenspectra , we must ensure that the eigencoefficients reflect the same physics independent of redshift . we know however that the eigenspectra change as a function of redshift ( see [ section : similar ] ) . to overcome this difficulty , and knowing that the overlap spectral region between the two sets of eigenspectra in any pair of adjacent redshift bins is larger than the common wavelength region ( @xmath101 ) for the full redshift interval , we study the differential evolution ( in redshift ) of the quasars by projecting the observed spectra at higher redshift onto the eigenspectra from the adjacent bin of lower redshift . in this way , the eigencoefficients can be compared directly from one redshift bin to the next . without the loss of generality , we project the observed quasar spectra in the higher redshift bin ( or dimmer quasars for the cross - luminosity projection ) onto the eigenspectra which are derived in the adjacent lower - redshift one ( or brighter quasars for the cross - luminosity projection ) . for example , @xmath102 ( i.e. , the spectra in the @xmath37-bin * b3 * ) are projected onto @xmath103e@xmath104 ( the set of eigenspectra from the @xmath37-bin * b2 * ) , and similarly for the different luminosity bins but the same redshift bin . from that , we can derive the relationship between the eigencoefficients and redshift ( or luminosity ) . the most obvious evolutionary feature is the small bump present in the spectra at around @xmath105 to @xmath106 . this feature is mainly composed of blended emissions ( @xmath107 , wills et al . 1985 ) and the balmer continuum ( @xmath108 ) . when we project quasar spectra of redshifts @xmath109 ( i.e. , @xmath110 ) onto eigenspectra constructed from quasars of redshifts @xmath111 ( i.e. , \{@xmath112(*c2 * ) } ) , the coefficients from the second eigenspectrum show a clear trend with redshift , as shown in figure [ fig : extrac32_a2_a1_redshift ] . in this figure , only those quasars with @xmath113 are chosen ( 900 objects ) , as such the redshift trend does not primarily depend on the absolute luminosities of the quasars . to understand this relation _ observed _ spectra are selected along the regression line in figure [ fig : extrac32_a2_a1_redshift ] ( with the locations marked by the crosses ) and are shown in figure [ fig : extrac32_a2_a1_realspec.new ] . the two dotted lines mark the bandpass where the cross - redshift projection is performed . the small bump is found to be present and is prominent in the lower - redshift quasars , whereas it is small and may be absent in the higher - redshift ones . the spectra marked by the arrows in figure [ fig : extrac32_a2_a1_realspec.new ] lie relatively close to the regression line . an example of the range of evolution in the small bump as a function of redshift is shown by the remaining 3 spectra which deviate from the regression line . the observed evolution is present independent of which of the spectra we consider . the mean spectra ( figure [ fig : extrac32_a2_a1_compositespec ] ) as a function of redshift , constructed using a bin width in redshift ( @xmath114 ) of 0.2 , show a similar behavior . each mean spectrum is calculated by averaging the valid flux densities of all objects in each wavelength bin . the regression of the eigencoefficient - ratios with redshift ( with outliers of @xmath115 removed from the calculation ) is @xmath116 where the subscript @xmath117 denotes that the eigenspectra are from @xmath118 . the correlation coefficient ( @xmath119 ) is calculated to be 0.1206 with a two - tailed p - value of 0.00027 ( the probability that we would see such a correlation at random under the null hypothesis of @xmath120 ) , as such the correlation is considered to be extremely significant by conventional statistical criteria . this redshift dependency can be explained by either the evolution of chemical abundances in the quasar environment @xcite , or an intrinsic change in the continuum itself ( which , of course , could also be due to the change in abundances through indirect photo - ionization processes ) . green , forster & kuraszkiewicz ( 2001 ) found in the lbqs that the primary correlations of the strengths of emission lines are probably with redshift ; an evolutionary effect is therefore implied . kuhn et al . ( 2001 ) also supported the evolution of the small bump region @xmath121 from high - redshift ( @xmath122 ) to lower - redshifts ( @xmath123 ) by comparing two qso subsamples with evolved luminosities . as the second mode in the @xmath37-binned eigenspectra describes the change in the spectral slope of the sample , the above findings support the idea that the balmer continuum , as a part of the small bump , changes with redshift . to further understand this effect , the 3rd eigenspectrum in * c2 * is taken into consideration , which presumably describes the iron lines ( see [ section : feanticorr ] ) . we find that the third eigencoefficient - ratio @xmath124 also shows a slight redshift dependency ( not shown ) with the regression relation ( with outliers of @xmath125 removed from the calculation , resulting in 901 objects ) @xmath126 and the correlation coefficient is calculated to be 0.0030 with a two - tailed p - value of 0.93 , which is considered to be not statistically significant . while the strength of this effect shown by the two ratios are of similar magnitude ( 0.0820 versus 0.0478 ) , the difference in their correlation coefficients implies that the sample variation is much greater in the ratio @xmath127 than @xmath128 . the non - trivial value of the regression slope in the case of @xmath127 agrees with the change in shape of the observed line profiles in the small bump regions seen in the local wavelength level ( smaller in width than what is expected in the continuum change ) with redshift . in conclusion , this implies that there exists the possibility of an evolution in iron abundances but with a larger sample variation compared with that for the continuum change . to our knowledge , our current analysis is the first one without invoking assumptions of the continuum level or a particular fitting procedure of the blends that finds an evolution of the small bump ; directly from the kl eigencoefficients . because of the large sample size , the conclusion of this work that the small bump evolves is drawn from spectrum - to - spectrum variation independent of the luminosity effect , in contrast to the previous composite spectrum approaches @xcite , in which the authors found that the composite spectra in two subsamples with mean redshifts @xmath129 and @xmath130 , and that from the large bright quasar survey of lower redshifts ( @xmath131 ) are similar in the vicinity of and hence did not suggest the existence of a redshift effect . the variation of the small bump with redshift is further confirmed with the study of composite quasar spectra of the dr1 data set ( vanden berk et al . , in preparation ) . at this point we make no attempt to quantitatively define and deblend the optical lines and the balmer continuum , as that would be beyond the scope of this paper . it is a well - known and unsolved problem to identify the true shape of total flux densities due to the emission lines . this difficulty arises because there are too many lines to model and they form a quasi continuum . luminosity effects on broad emission lines can also be probed in a similar way to the cross - redshift projection . one prominent luminosity effect is found by projecting @xmath132 onto @xmath133 . these samples have the same redshift range but different luminosities ( for * d1 * , @xmath134 and for * c1 * , @xmath135 ) . figure [ fig : extradc1_a2_a1_redshift ] shows the eigencoefficient @xmath136 as a function of absolute luminosity , with redshifts fixed at @xmath137 ( 235 quasars ) . the ratio of the first 2 eigencoefficients decreases with increasing quasar luminosity . the regression line ( with outliers of @xmath115 removed from the calculation ) is @xmath138 with a correlation coefficient of 0.2305 with an extremely significant two - tailed p - value of 0.0003 . along this luminosity trend , the equivalent widths of emission lines such as h@xmath10 and @xmath11@xmath7 $ ] lines are found to decrease typically , as a function of increasing absolute magnitude @xmath23 ( as shown in the spectra in figure [ fig : extradc1_a2_a1_realspec.2]a ) . this is the baldwin ( 1977 ) effect . we note that the host - galaxy may come into play in this case ( at low redshifts and low luminosities ) . the geometric composite spectra of different luminosities within the range from @xmath2 to @xmath81 are shown in figure [ fig : extradc1_a2_a1_realspec.2]b , in which a spectral index of @xmath30 for the continua is assumed . the baldwin effect for the emission lines is also present . in the highest redshift bins , the baldwin effect can be found in the first and the second eigenspectra . figure [ fig : zbin4_eigenspec ] shows that the addition ( with positive eigencoefficients ) of the first two eigenspectra _ enhances _ the flux density around 1450 and reduces the equivalent width of . ly@xmath6 and other major bels are also shown to be anti - correlated with the continuum flux . hence , the baldwin effect is not limited to the emission line , and is also observed in many broad emission lines ( see , for example , a summary in sulentic et al . the linear - combination of the first and third modes in this redshift range also shows a similar modulation between the flux density around 1450 and the line equivalent width . this effect is , however , not general for all luminosities , with the third eigenspectrum in * c4 * showing only a small value in the 1450 flux density . the baldwin effect can also be seen by comparing the first eigenspectra constructed for different luminosity bins . figure [ fig : baldwin_civ ] shows the first eigenspectra derived in different luminosities in the second highest redshift bin ( i.e. , the @xmath37-bins * a4 * , * b4 * and * c4 * , with @xmath139 ) and the highest one ( * a5 * and * b5 , * with @xmath140 ) . the eigenspectra are normalized to unity at @xmath141 . the continua for wavelengths approximately greater than 1700 in figure [ fig : baldwin_civ]a are not perfectly normalized ( which is difficult to define in the first place ) , but a more careful normalization would only lead to an increase in the degree of the baldwin effect in the emission lines @xmath7 $ ] and . the ly@xmath6 and lines demonstrate the most profound baldwin effect . other broad emission lines such as @xmath681640 , ] and also exhibit this effect . for the controversial line , an `` anti - baldwin '' correlation is found at redshifts @xmath142 , such that flux densities are smaller for lower - luminosity quasars . at the highest redshifts in this study ( @xmath143 , figure [ fig : baldwin_civ]b ) , however , a normal baldwin effect of is found . the redshift dependency in the baldwin effect for may explain the contradictory results found in previous studies ( a detection of baldwin effect of in tytler & fan 1992 ; and non - detections in steidel & sargent 1991 ; osmer et al . 1994 ; and laor et al . while most studies have shown little evidence of the baldwin effect in the blended emission lines + @xmath7 $ ] , our results support the existence of an effect ( though at a much weaker level than that of ly@xmath6 and ) . this is in agreement with two previous works ( laor et al . ( 1995 ) which used 14 hst qsos , and green , forster & kuraszkiewicz ( 2001 ) which used about 400 qsos from the lbqs ) . in the optical region , at least @xmath684687 was reported to show the baldwin effect @xcite . to further verify that the luminosity dependency of the eigencoefficients implies a baldwin effect , we also study the eigencoefficients corresponding to the baldwin effect seen in figure [ fig : baldwin_civ ] . we find that when @xmath144 are projected onto @xmath145 the luminosity dependency is also seen in the eigencoefficients , with @xmath146 ( @xmath147 , and an insignificant two - tailed p - value of 0.14 ) and @xmath148 ( @xmath149 , and a very significant two - tailed p - value of 0.0043 ) , both for objects with redshifts within @xmath150 ( 161 objects in the case of @xmath151 and 166 in that of @xmath124 ) . figure [ fig : bzbin3_a1_a5 ] shows plots of the first five eigencoefficients of the @xmath37-bin * b3 * , where the properties are typical for all @xmath86-bins . the eigencoefficients are normalized as : @xmath98 . the plot of @xmath153 versus @xmath154 shows a continuous progression in the ratio of these coefficients which is similar to that found in the kl spectral classification of galaxies @xcite , in which the points fall onto a major `` sequence '' of increasing spectral slopes . as higher orders are considered , for example @xmath155 vs @xmath156 ( figure [ fig : bzbin3_a1_a5]d ) , no significant correlations are observed . observed quasar spectra are inspected along this trend of @xmath153 versus @xmath154 ( figure [ fig : bzbin3_pickspec_a1_a2 ] ) . the top of each sub - figure shows the values of @xmath152 . along the sequence with decreasing @xmath153 values , the quasar continua are progressively bluer . the relatively red continua in figures [ fig : bzbin3_pickspec_a1_a2]a to [ fig : bzbin3_pickspec_a1_a2]c may be due intrinsic dust obscuration @xcite . the quasar in figure [ fig : bzbin3_pickspec_a1_a2]c is probably a high - ionization balqso ( hibal ) according to the supplementary sdss edr bal quasar catalog @xcite . we do , however , emphasize that the appearance of this balqso ( or any balqso in general ) in this particular sequence of quasar in the @xmath153 versus @xmath154 plane does not imply two modes are enough to achieve an accurate classification for a general balqso ( for the reasons described in [ section : bal ] ) . the steepness of the spectral slope of this particular balqso is the major reason which causes such values of @xmath154 and @xmath153 eigencoefficients . on the variations of the emission lines along these major @xmath86 sequences , we can appreciate some of the difficulties in obtaining a _ simple _ classification concerning _ all _ emission lines by inspecting the examples listed in table [ tab : fwhmzbin ] . the addition of the 2nd eigenspectrum to the 1st , weighted with ( signed ) medians of the eigencoefficients for all objects in a given sample , broadens some emission lines while making others narrower ; a similar effect is seen for the addition of the 3rd eigenspectrum to the 1st , but in two _ different _ sets of lines . this shows the large intrinsic variations in the emission line - widths of the qsos . one of the utilities of the kl transform is to study the linear correlations among the input parameters , in this case , the pixelized flux densities in a spectrum . due to possible uncertainties in any continuum fitting procedure in quasar spectra and the fact that no quasar spectrum in our sample completely covers the rest wavelength range @xmath16 , correlations among the broad emission lines are first determined locally around the lines of interest by studying the first two eigenspectra in a smaller restricted wavelength range using the wavelength - selected qso spectra . this process is then repeated from 900 to 8000 . each local wavelength region is chosen to be @xmath157 wide in the restframe . empirically , we find that at these spectral widths the correlations among broad emission lines can be isolated in the first two eigenspectra without interference by the continuum information ( except in the vicinity of doublet , for which the adjacent strong emission lines are located well beyond the ( uv ) region , which can be as broad as @xmath158 @xmath79 ) , _ in contrast _ to the property of the @xmath37-bins in which the 2nd eigenspectra generally describe the variations in the spectral slopes . the actual procedures to determine the correlations among the strengths of the major emission lines are as follows : @xmath159 in each bin , the eigencoefficients of all objects are computed , and the distribution of the first two eigencoefficients , @xmath153 versus @xmath154 , are divided into several ( @xmath3 ) sections within @xmath160 of the @xmath153 distribution . in each section the mean eigencoefficients , @xmath161 and @xmath162 , are calculated ( discarding outliers @xmath163 ) . @xmath164 along this trend of mean eigencoefficients , synthetic spectra are constructed by the linear - combination of the first two eigenspectra using the weights defined by the mean eigencoefficients . @xmath165 the equivalent widths of emission lines in the synthetic spectra are calculated along the trend of mean eigencoefficients , so that the correlations among the strengths of the broad emission lines can be deduced . linear regression and linear correlation coefficients are calculated from the ew - sequence of a particular emission line relative to that of another line , which is fixed to be the emission line with the shortest wavelength of each local bin . the equivalent widths are calculated by direct summation over the continuum - normalized flux densities within appropriate wavelength windows . from such procedures , the correlations found are ensemble - averaged properties of redshifts and luminosities over the corresponding range , and are physical . table [ tab : linedata ] shows the rest - wavelength bounds , the redshift range , the number of quasar spectra in each bin , and regression and correlation coefficients for each major emission line . the range of the possible restframe equivalent widths ( ew@xmath166 ) along @xmath152 is listed in decreasing @xmath153 values . since the redshifts are chosen such that each quasar spectrum has a full coverage in the corresponding wavelength region , the gap - correcting procedure is implemented to correct only for skylines and bad pixels . the ew@xmath166 of the emission lines vary at different magnitudes along the @xmath152 sequence ; some change by nearly a factor of two ( e.g. , ly@xmath6 , ) , while some show smaller changes ( e.g. , + @xmath7 $ ] , @xmath681906 ) . within a single local bin , the rest equivalent widths of some emission lines increase while others decrease along the trend @xmath152 with decreasing @xmath153 values . these results are the testimonies to the fact that quasar emission lines are diverse in their properties . we also note that some pairs of emission lines change their correlations as a function of redshift ( i.e. , different local bins ) . for example , is correlated with + ( opt82 ) in the local bin of @xmath167 but anti - correlated in that of @xmath168 . another example is the @xmath11@xmath7$]@xmath169 and @xmath11@xmath7$]@xmath170 pair . hence if correlations are interpreted between the emission lines from one local bin with those from an adjacent bin , caution has to be exercised . the uncertainty in the continuum estimation ( e.g. , the iron contamination in the continuum in the vicinity of ) prevents us from drawing an exact physical interpretation of this phenomenon . two examples of the locally - constructed eigenspectra are shown in figures [ fig : lyalpha_eigenspec ] and [ fig : hbeta_eigenspec ] . in figure [ fig : lyalpha_eigenspec ] , the eigenspectra are constructed using wavelength - selected qso spectra in the rest - wavelengths @xmath8 ( with @xmath171 ) , so that both ly@xmath6 and are covered . excellent agreement is shown between our eigenspectra and those selected from the large bright quasar survey in the @xmath172 range @xcite . the second eigenspectrum ( corresponding to the first principal component in francis et al . ) shows the line - core components of emission lines . in contrast , the 3rd mode ( corresponding to their 2nd principal component ) shows the continuum slope , with the node located at around 1450 . besides , the addition ( with positive eigencoefficient ) of the 3rd eigenspectrum to the 1st one enhances the fluxes at shorter wavelengths while _ increases _ the blueshift . this supports the finding of a previous study @xcite that blueshift is greater in bluer sdss qsos . at longer wavelengths , the sdss quasars with redshifts @xmath173 show the anti - correlation between ( optical ) and [ ] ( figure [ fig : hbeta_eigenspec ] ) , in agreement with the eigenvector-1 ( boroson & green 1992 ) . the first two eigenspectra in figure [ fig : hbeta_eigenspec ] demonstrate that both the h@xmath10 and the nearby [ ] forbidden lines are anti - correlated with the ( optical ) emission lines , which are the blended lines blueward of h@xmath10 and redward of [ ] . in the 3rd local eigenspectrum , the balmer emission lines are prominent , which was noted previously in the pca work by shang et al . in addition , we find a correlation between the continuum and the balmer lines in this local 3rd eigenspectrum , so that their strengths are stronger in bluer quasars . to date , it is generally believed that the anti - correlation between ( optical ) and [ ] is not driven by the observed orientation of the quasar . one of the arguments by boroson & green was that the [ ] @xmath685008 luminosity is an isotropic property . subsequent studies of radio - loud agns have put doubt on the isotropy of the [ ] emissions . recent work by kuraszkiewicz et al . ( 2000 ) , however , showed a significant correlation between eigenvector-1 and the evidently orientation - independent [ ] emission in a radio - quiet subset of the optically selected palomar bqs sample , which implies that external orientation probably does not drive the eigenvector-1 . an interesting future project to address this problem is to relate the quasar eigenspectra in the sdss to their radio properties . enlargements of the first two locally constructed eigenspectra focusing on major broad emission lines are illustrated in figure [ fig : linecore ] . except for the almost perfectly symmetric and zero velocity of the line centers of the 1st and 2nd eigenspectra exhibited by [ ] @xmath685008 , most broad emission lines do show asymmetric and/or blueshifted profiles . these demonstrate the variation of broad line profiles of quasars and the generally blueshifted broad emission lines relative to the forbidden narrow emission lines . the forbidden lines in the narrow line regions of a qso are always adopted in calculating the systemic host - galaxy redshift , so the clouds associated with blueshifted bels probably have additional velocities relative to the host . this line - shift behavior was found in many other studies ( see references in vanden berk et al . 2001 ) . the behavior of the shift led richards et al . ( 2002b ) to suggest that orientation ( whether external or internal ) may be the cause of the effect . it is also obvious from figure [ fig : linecore ] that the 2nd eigenspectra are generally narrower ( except for , in which the conclusion is complicated by the presence of the surrounding lines ) than their 1st eigenspectra counterparts . the line - widths of the sample - averaged kl - reconstructed spectra using only the first eigenspectrum or the first two eigenspectra are listed in table [ tab : fwhmlocal ] . the addition of the first two modes , weighted by the medians of the eigencoefficients , causes the widths of 76 % of the emission lines ( with fwhm @xmath174 km s@xmath26 ) to be narrower than those reconstructed from the first mode only . hence , most broad emission lines can be mathematically decomposed into broad , high - velocity components and narrow , low - velocity components . appearing in the second local eigenspectra , the line - widths are thus the most important variations of the quasar broad emission lines . the line - core components were reported by francis et al . ( 1992 ) for and ly@xmath6 ; and shang et al . ( 2003 ) for some major broad emission lines . one nice illustration of the line - core component of the 2nd mode is the splitting of h@xmath74 and its adjacent [ ] in figure [ fig : hbeta_eigenspec ] , for they are blended in the 1st mode . similar properties may be expected in the 2nd @xmath37-binned eigenspectra . table [ tab : fwhmzbin ] lists the average fwhm of different linear combinations using the first 3 eigenspectra in constructing some major broad emission lines . comparatively , for most emission lines the second @xmath37-binned eigenspectra do not show as narrow line components as the second eigenspectra , in which the widths of 61 % of the emission lines with fwhm @xmath174 km s@xmath26 become narrower by adding the 2nd eigenspectrum to the 1st one . this effect is mainly due to the difference in the numbers of quasars , and more importantly , the inclusion of a wider spectral region causes the ordering of the weights of different physical properties to re - arrange . in this case , the spectral slope variations are more important than those of the line - cores . while the 3rd @xmath86-binned eigenspectra ( weighted by medians of the eigencoefficients of the sample ) also do not represent prominent changes in the emission line - cores , except for ly@xmath6 and ( the fwhm of appears to be larger because the line - core 3rd mode is pointing downward in * zbin 4 * ) , on average the quasar populations with _ negative _ 3rd eigencoefficients do show narrower widths for 77 % of the emission lines . similarly , the 2nd global eigenspectrum does not carry dominant emission line - core components , which are found to be represented more prominently by the 3rd mode ( table [ tab : fwhmglobal ] ) . the narrower emission features in the 2nd local eigenspectrum compared with the 1st one , and the fact that almost every broad emission line is pointing towards positive flux values in both of these two modes , imply that there is an anti - correlation between fwhms and the equivalent widths of broad emission lines . in fact , as suggested by francis et al . ( 1992 ) , this may form a basis for the classification of quasar spectra in @xmath175 , by arranging them accordingly into a sequence varying from narrow , large - equivalent - width to broad , low - equivalent - width emission lines . from the locally constructed eigenspectra , such an anti - correlation is not generally true for every broad emission line as we find that there exists at least one exception : a positive correlation between the fwhm and the ew of in the local bin of the redshift range @xmath176 . an assumption in these measurements is that the continuum underneath can be approximated by a linear - interpolation across the window @xmath177 . one complication , however , is the contamination due to the many emission lines in the vicinity of , so the true continuum may be obscured . the positive fwhm - ew correlations appear to exist in some other weaker emission lines as well , but the weak strengths of those lines do not permit us to draw definitive conclusions under the current spectral resolution . in conclusion , the fwhm - ew relation can help us to classify most broad emission lines individually , but _ this relation can not be used in a general sense , nor does it represent the most important sample variation _ , if the surrounding continua are included to the extent of the rest - wavelength ranges of the @xmath86-binned spectra . nonetheless , most broad emission lines can be viewed mathematically as the combinations of broad and narrower components . a future study will focus on finding the best physical parameters for classifying the spectra in the wide spectral region , which will be the subject of a second paper . one possible approach is to study the distributions of the eigencoefficients and their relations with other spectral properties ( e.g. , francis et al . 1992 ; boroson & green 1992 ) . the shapes of the continua and the correlations among the broad emission lines of the second locally constructed eigenspectra are all identified in either the 3rd or the 4th @xmath37-binned eigenspectra . we do expect , and it is indeed found to be true , that the local properties of the spectra can be found in the latter , though the ordering may be different . the identifications are marked in figures [ fig : zbin1_eigenspec ] @xmath85 [ fig : zbin5_eigenspec ] by the redshift ranges of the local eigenspectra , with reference to the luminosity averaged * zbin * eigenspectra . the correlations of broad emission lines are generally found in higher - order @xmath37-binned eigenspectra compared with the orders representing the spectral slopes . we perform kl transforms and gap - corrections on 16,707 sdss quasar spectra . in rest - wavelengths @xmath178 , the 1st eigenspectrum ( i.e. , the mean spectrum ) shows agreement with the sdss composite quasar spectrum @xcite , with an abrupt change in the spectral slope around 4000 . the 2nd eigenspectrum carries the host - galaxy contributions to the quasar spectra , hence the removal of this mode can probably prevent the obscuration of the real physics of galactic nuclei by the stellar components . whether this eigenspectrum is the only one containing galaxy information requires further study . the 3rd eigenspectrum shows the modulation between the uv and the optical spectral slope , in agreement with the 2nd principal component of shang et al . the 4th eigenspectrum shows the correlations between balmer emission lines . locally around various broad emission lines , the eigenspectra from the wavelength - selected quasars qualitatively agree with those from the large bright quasar survey , the properties in the eigenvector-1 @xcite , and the anti - correlations between the fwhms and the equivalent widths of ly@xmath6 and @xcite . the anti - correlation between the fwhm and the equivalent width is found in most broad emission lines with few exceptions ( e.g. , is discrepant ) . from the commonality analysis of the subspaces spanned by the eigenspectra in different redshifts and luminosities , the spectral classification of quasars is shown to be redshift and luminosity dependent . therefore , we can either use of order 10 @xmath37-binned eigenspectra , or of order @xmath179 global eigenspectra to represent most ( on average 95 % ) quasars in the sample . we find that the first two modes can describe the spectral slopes of the quasars in all @xmath37-bins under study , which is the most significant sample variance of the current qso catalog . the simplest classification scheme can be achieved based on the first two eigencoefficients , so that a physical sequence can be formed upon the linear - combinations of the first two eigenspectra . the diversity in quasar spectral properties , and the inevitable different restframe wavelength coverages due to the nature of the survey , increase the sparseness of the data . hence , higher - order modes enter into the construction of the broad emission lines with the eigenspectra , in contrast to the galaxy spectral classification , in which most emission lines vary monotonically with the spectral slope @xcite . this result is also a manifestation of the high uniformity of galaxy spectra compared with quasar spectra . we find that bal features do not only appear in one particular order of eigenspectrum but span a number of orders , mainly higher - orders . this may indicate substantial challenges to the classification of bal quasars by the current sets of eigenspectra in terms of arriving at a compact description . a separate kl - analysis of the bal quasars is desirable for studying the classification problem . nonetheless , the appropriate truncation of the number of eigenspectra in reconstructing a quasar spectrum can in principle lead to an un - absorbed continuum . we find evolution of the small bump by the cross - redshift kl transforms , in agreement with the quasars from the large bright quasar survey @xcite and in other independent work @xcite . the baldwin effect is detected in the cross - luminosity kl transforms , as well as from the mean qso spectra derived for different luminosities . one implication of these redshift and luminosity effects is that they have to be accounted for in the spectral classification of quasars , consistent with our finding from the commonality analysis . the high quality of the data allows us to obtain quasar eigenspectra which are generic enough to study spectral properties . despite the presence of diverse quasar properties such as different continuum slopes and shapes , and various emission line features known for several decades , our analysis shows that there are unambiguous correlations among various broad emission lines and with continua in different windows . a second paper is being prepared to address the classifications of the dr1 quasars in greater detail . one interesting direction is to relate the current eigenspectra approach to the radio properties of the quasars , so that further discriminations of intrinsic and extrinsic properties can be achieved , for example , the orientation effects on the observed spectra ( e.g. , richards et al . another application currently being addressed is the removal of host - galaxy components from the sdss quasar spectra . in addition , the cross - projections can also be applied to study future larger samples of quasars ( e.g. , @xmath94 100,000 at the completion of the sdss ) for possibly new evolution and luminosity effects . we thank david turnshek for the discussion of the baldwin effect . we thank ravi sheth for various comments and discussions . we thank the referee zhaohui shang for the helpful comments . cwy and ajc acknowledge partial support from an nsf career award ast99 84924 and a nasa ltsa nag58546 . ajc acknowledges support from an nsf itr awards ast-0312498 and aci-0121671 . funding for the creation and distribution of the sdss archive has been provided by the alfred p. sloan foundation , the participating institutions , the national aeronautics and space administration , the national science foundation , the u.s . department of energy , the japanese monbukagakusho , and the max planck society . the sdss web site is http://www.sdss.org/. the sdss is managed by the astrophysical research consortium ( arc ) for the participating institutions . the participating institutions are the university of chicago , fermilab , the institute for advanced study , the japan participation group , the johns hopkins university , los alamos national laboratory , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , university of pittsburgh , princeton university , the united states naval observatory , and the university of washington . the construction of the @xmath37-bins in this work ( [ section : zbin ] ) is performed by constraining the gap fraction to be smaller than 50 % for each spectrum to improve the accuracy of spectral reconstructions using eigenspectra . here we discuss in detail how this value is arrived at . we artificially mask out ( i.e. assign a zero weight ) to given spectral intervals and study how well we can reconstruct these `` gappy '' regions from the eigenspectra @xcite . the comparison of the kl - reconstructed spectrum with the original unmasked spectrum gives a direct assessment to the accuracy of the gap - correction procedure . we perform this test for the @xmath37-binned quasar spectra from this work . to simulate the effects of un - observed spectral regions due to different rest - wavelength coverage for quasars at different redshifts ( the principal reason for gaps in the quasar spectra in our sample ) , each spectrum in all @xmath37-bins is artificially masked at the short- and the long - wavelength ends . the masked spectra are then projected onto the appropriate eigenspectra and the reconstructed spectra are calculated using the first 50 modes . the fractional change in the flux density per wavelength bin ( weighted by @xmath180 ) , @xmath181 , between the observed spectrum @xmath182 and the reconstructed spectrum @xmath183 , averaged over all quasar spectra in each bin , are shown in figure [ fig : reconspecerrvsgap]a as a function of the spectral gap fraction . the gap fraction is calculated relative to the full restframe wavelength range , a variable for each quasar spectrum . the reconstruction from @xmath184 modes has an intrinsic error of approximately @xmath185 % ( due to the noise present in each spectrum , and the existence of @xmath186 % bad pixels on average for each spectrum ) , which is estimated by reconstructing the spectra with no artificial gaps . as expected , the difference between the unmasked observed spectrum and the reconstructed spectrum increases gradually with gap fraction . averaging over all @xmath37-bins ( figure [ fig : reconspecerrvsgap]b ) , at a spectral gap fraction of @xmath187 % the mean error in the 50-mode reconstruction is @xmath188 % , which is @xmath189 % above the noise - dominated average reconstruction error in the flux . while a smaller gap fraction is in principle more desirable , 50 % is chosen to be the upper bound to compromise the fewer @xmath37-bins . in the construction of the global eigenspectra set covering the rest - wavelength range @xmath16 , there are 89 % of the qsos ( table [ tab : gapglobal ] ) having spectral gap fractions larger than 50 % . from figure [ fig : reconspecerrvsgap]b , we find that a gap fraction larger than @xmath190 % gives substantial reconstruction errors ( @xmath191 % ) , implying @xmath192 % of the qsos used in defining the global eigenspectra may be poorly constrained when correcting for the missing data . we stress that in defining the global eigenspectra from the sdss this is strictly the best estimation that can be made at present , as no sdss spectroscopic observations are available in the gap regions at the red and the blue ends of the spectrum . the impact of this gap correction is , as expected , wavelength dependent . wavelengths shortward of @xmath193 are very well constrained even with the global eigenspectra with less than 1 % of qsos having gap corrections in excess of 76 % ( table [ tab : gapglobal ] ) . determining the impact of the gaps and the use of additional spectroscopic observations to complement the sdss data will be addressed in a future paper . we also find that quasar broad emission lines can be reconstructed locally using the @xmath37-binned eigenspectra with errors that are typically small relative to the noise level . for example , if @xmath7 $ ] is masked ( over the region of influence @xmath194 ) , averaging over all qsos in the bins * b3 * and * c3 * , the 50-mode reconstruction error described above is 10.4 % ; and for ( over the region of influence @xmath195 ) , 11.3 % . for the case in which at least one broad emission line is masked and with a substantial total gap fraction ( in our case , @xmath7 $ ] ; and a mean spectral gap fraction of @xmath196 % ) , the average reconstruction error per pixel is found to be @xmath197 % when averaging over the bins * b3 * and * c3*. figure [ fig : recon_maskciii_50percentgap ] shows the observed and the reconstructed spectra of an object with a reconstruction error approximately equal to the average value . while the reconstructed continuum has a small difference from the observed continuum , the emission line @xmath7 $ ] is reconstructed well , extremely well if considering the fact that the whole region of influence is within the masked region . the actual quality of the reconstruction depends on the individual spectrum and position and size of the gaps . cccc 0.4 & 16,420 ( 0.98 ) & 15,313 ( 0.92 ) & 10,423 ( 0.62 ) + 0.5 & 15,050 ( 0.90 ) & 13,561 ( 0.81 ) & 6,421 ( 0.38 ) + 0.6 & 12,696 ( 0.76 ) & 10,275 ( 0.62 ) & 1,682 ( 0.10 ) + 0.7 & 7,424 ( 0.44 ) & 3,519 ( 0.21 ) & 423 ( 0.025 ) + 0.75 & 2,920 ( 0.17 ) & 1,131 ( 0.068 ) & 100 ( 0.0060 ) + 0.8 & 873 ( 0.052 ) & 416 ( 0.025 ) & 0 ( 0.00 ) + 0.9 & 0 ( 0.00 ) & 0 ( 0.00 ) & 0 ( 0.00 ) + [ tab : gapglobal ]

we study 16,707 quasar spectra from the sloan digital sky survey ( sdss ) ( an early version of the first data release ; dr1 ) using the karhunen - love ( kl ) transform ( or principal component analysis , pca ) . the redshifts of these quasars range from 0.08 to 5.41 , the @xmath0-band absolute magnitudes from @xmath1 to @xmath2 , and the resulting restframe wavelengths from 900 to 8000 . the quasar eigenspectra of the full catalog reveal the following : 1st order the mean spectrum ; 2nd order a host - galaxy component ; 3rd order the uv - optical continuum slope ; 4th order the correlations of balmer emission lines . these four eigenspectra account for 82 % of the total sample variance . broad absorption features are found not to be confined in one particular order but to span a number of higher orders . we find that the spectral classification of quasars is redshift and luminosity dependent , as such there does not exist a compact set ( i.e. , less than @xmath3 modes ) of eigenspectra ( covering 900 to 8000 ) which can describe most variations ( i.e. , greater than @xmath4 % ) of the entire catalog . we therefore construct several sets of eigenspectra in different redshift and luminosity bins . from these eigenspectra we find that quasar spectra can be classified ( by the first two eigenspectra ) into a sequence that is defined by a simple progression in the steepness of the slope of the continuum . we also find a dependence on redshift and luminosity in the eigencoefficients . the dominant redshift effect is a result of the evolution of the blended emission ( optical ) and the balmer continuum ( the `` small bump '' , @xmath5 ) . a luminosity dependence is also present in the eigencoefficients and is related to the baldwin effect the decrease of the equivalent width of an emission line with luminosity , which is detected in ly@xmath6 , + @xmath7 $ ] , , , @xmath7 $ ] and , while the effect in seems to be redshift dependent . if we restrict ourselves to the rest - wavelength regions @xmath8 and @xmath9 , the eigenspectra constructed from the wavelength - selected sdss spectra are found to agree with the principal components by francis et al . ( 1992 ) and the well - known `` eigenvector-1 '' @xcite respectively . ascii formatted tables of the eigenspectra are available .

the search for new superconducting materials and the opportunity to discover further evidence of non - bcs mechanisms of electron pairing attracted attention of researchers to iron pnictides @xcite and chalcogenides @xcite . among these materials the systems from the `` 11 '' group , namely fe@xmath7se @xcite , fe@xmath7te@xmath8se@xmath9 @xcite , and fe@xmath7te@xmath8s@xmath9 @xcite , have the simplest crystallographic structure with iron atoms arranged in characteristic planes ( figure [ struct ] ) . these fe(1 ) atoms , tetrahedrally coordinated by chalcogen atoms , form layers separated by van der waals gaps . in consequence the `` 11 '' systems can be regarded as quasi two - dimensional . nevertheless , this structure features an intrinsic disorder due to both excess iron in partially occupied fe(2 ) positions @xcite and substituted atoms , which are displaced with respect to the te crystallographic positions . it is known that doped s atoms have a _ z _ coordinate considerably different from that of te @xcite , as displayed in figure [ struct ] ( b ) . ( color online ) ( a ) crystallographic structure of fe@xmath7te with marked atomic positions , tetrahedral coordination of fe(1 ) atoms , van der waals gap ( vdw ) , elementary unit cell and orientation of crystallographic axes . the atomic positions of fe(1 ) and te are completely filled , while fe(2 ) positions are only partially occupied . ( b ) elementary unit cell of sulphur doped fe@xmath7te@xmath8s@xmath9 with coordination of fe(1 ) atoms . s substitutes te but is displaced along c crystallographic axis with respect to te position @xcite . both te and s positions are shown.,width=316 ] the superconducting critical temperature for `` 11 '' chalcogenides is relatively low under ambient pressure and reaches barely 14 k @xcite for fe@xmath7te@xmath8se@xmath9 , 13 k for fese @xcite and 10 k for fe@xmath7te@xmath8s@xmath9 @xcite while fe@xmath10te remains a non - superconducting antiferromagnet @xcite . moreover , the superconducting fraction of untreated fete@xmath11s@xmath12 as determined from magnetic susceptibility is close to 20 % @xcite . the direct connection between the iron overstoichiometry , magnetism and superconductivity can be exposed by topotactic deintercalation using iodine @xcite or other oxidation processes like annealing in oxygen @xcite . samples with the lowest content of excess iron have the highest sc fraction reaching 100% . the promising fact is that under high pressure the transition to superconductivity reaches t=37 k for fese @xcite . while the mechanism of electron pairing in the fe - based superconductors is still under debate , the electronic band structure can impose certain conditions on possible scenarios @xcite . therefore , the fermi surface ( fs ) and the electronic band structure of the discussed systems have been extensively studied by means of angle resolved photoemission spectroscopy ( arpes ) , quantum oscillations and density functional theory ( dft ) calculations @xcite . in particular , the previous arpes studies on `` 11 '' chalcogenides covered both non - superconducting fe@xmath7te @xcite and superconducting fe@xmath7te@xmath13se@xmath9 @xcite but corresponding results for fese or fe@xmath7te@xmath13s@xmath9 are absent in the literature so far . while the published data for fe@xmath7te@xmath13se@xmath9 are relatively consistent , studies of fe@xmath7te present two aspects : on the one hand clearly visible band topography @xcite , on the other hand intrinsically broad spectra in a paramagnetic state with emergence of quasiparticle peaks in the spin density wave ( sdw ) state @xcite . the latter scenario is confirmed by a more recent study of fe@xmath14te and becomes understood in terms of polaron formation @xcite . the fermi surface of superconducting fe@xmath7te@xmath13se@xmath9 chalcogenides consists of hole pockets located around the @xmath3(z ) point and electron pockets in the region of the m(a ) point , which is typical of both iron pnictides and chalcogenides . however , the newer a@xmath15fe@xmath16se@xmath17 systems ( a = k , cs , rb , tl , etc . ) are exceptional in that respect as they exhibit electron pockets at @xmath3(z)point @xcite . the current paper presents the band structure and dominant orbital characters obtained by arpes for fe@xmath0te@xmath1s@xmath2 superconductor . the data are compared to theoretical calculations . a flat band close to the chemical potential ( @xmath4 ) is found in the region of the @xmath3 point . the resulting high density of states at @xmath4 should be an important factor for the emergence of superconductivity in the sulphur doped `` 11 '' compounds . single crystals with targeted stoichiometry fe@xmath10te@xmath18s@xmath19 were grown in nist by similar techniques as reported earlier @xcite . stoichiometric quantities of the elements were sealed in evacuated quartz tubes and heated at 775 @xmath20c for 48 h with intermediate step at 450 @xmath20c . after regrinding the product was reheated at 825 @xmath20c for 12 h and slowly cooled to room temperature . x - ray diffraction performed at 290 k indicated single crystals with a composition of fe@xmath21te@xmath22s@xmath23 as obtained from the rietveld refinement to x - ray data . the determined crystal structure as shown in figure [ struct ] is consistent with the previous studies @xcite and remains tetragonal to the lowest temperature t=35 k reached in the experiment . the composition of the single crystals was also determined using a jeol jxa 8900 microprobe in wavelength dispersive mode ( wds ) from 10 flat points spread over the surface . the average composition was found to be fe@xmath24te@xmath25s@xmath26 and will further be used in the text as more reliable than the estimate from the diffraction data . the single crystals exhibited onset of the superconducting transition at t=9 k in magnetic susceptibility and electrical resistivity . however , according to the magnetic susceptibility studies the meissner phase at t=2k covered 23 % of the volume . the arpes experiments were carried out at the ape beamline @xcite of the elettra synchrotron using scienta ses2002 electron spectrometer . the crystals were cleaved at a pressure of @xmath27 mbar and studied with linearly or circularly polarized radiation . the energy and wave vector ( k ) resolution were 20 mev and 0.01 @xmath28 respectively . low energy electron diffraction was used to check the surface quality . fermi edge determination was performed regularly on evaporated gold . band structure calculations were carried out with the akaikkr software @xcite based on the korringa - kohn - rostoker ( kkr ) green s function method with coherent potential approximation ( cpa ) . this method is able to model the effect of disorder in alloys @xcite and should treat properly random occupancies of fe(2 ) and te / s atomic positions . cpa is considered as the most relevant approach for disordered `` 11 '' systems @xcite . a von barth and hedin type exchange - correlation potential @xcite was applied . the width of the energy contour for the integration of the green s function was 1.9 ry and the added imaginary component of energy was 0.002 ry . the bloch spectral function was calculated for 255 k - points in the irreducible brillouin zone ( ibz ) . other calculations were performed for stoichiometric fete by means of the linearized augmented plane wave with local orbitals ( lapw+lo ) method implemented in the wien2k package @xcite . local spin density approximation ( lsda ) @xcite and ceperley - alder parametrization @xcite were used . the atomic spheres radii were 2.41 atomic units ( a.u . ) and 2.17 a.u . for fe and te respectively , and the calculations were realized for 330 k - points in the ibz . the electronic structure of superconducting fe@xmath0te@xmath1s@xmath2 crystals ( figure [ bz](a ) ) was studied by means of arpes along the high symmetry directions @xmath3-m and @xmath3-x ( figure [ bz](b ) ) . radiation of linearly polarized photons with an energy of 40 ev was used . the spectra obtained along the @xmath3-m direction at 80 k ( figure [ bz ] ( c ) , ( d ) ) exhibit high intensity in the region of the @xmath3 point . for @xmath29-polarization a hole pocket is found , whereas for @xmath30-polarization the measurements reveal a hole like band and a feature with high intensity at @xmath4 . the nature of this high spectral intensity will be discussed further . photoelectron spectra obtained in the region of m with @xmath29-polarization reveal increased intensity near @xmath4 at the m point . the @xmath29-polarization is more favourable for the bands at m , similarly to the case of undoped fete @xcite . the spectra recorded with @xmath30-polarization do not reveal any bands in this region . near the x point no spectral intensity is found at low binding energy ( not shown ) . in particular , a replica of the band structure at @xmath3 is not found at x in contrast to the observations for undoped fe@xmath7te @xcite . this indicates that the sdw magnetic order is not seen in the fe@xmath0te@xmath1s@xmath2 system with arpes . ( color online ) ( a ) surface of fe@xmath0te@xmath1s@xmath2 single crystal exposed along ( 001 ) plane . ( b ) first brillouin zone for tetragonal fe@xmath7te@xmath8s@xmath31 with high symmetry points and directions . arpes intensity along the @xmath3-m ( z - a ) direction obtained at t=80k and photon energy h@xmath32=40 ev in ( c ) @xmath29-polarization and ( d ) @xmath30-polarization . ( e ) band structure of fe@xmath10te@xmath33s@xmath34 along high symmetry directions obtained by kkr - cpa calculations . ( f ) band structure of stoichiometric fete calculated with lapw+lo method . the distances between the high symmetry points are scaled to the real distances in k - space ( f ) or remain constant between the points ( e).,width=326 ] kkr - cpa calculations , which are destined for systems with disorder , were performed for fe@xmath10te@xmath33s@xmath34 ( figure [ bz](e ) ) . despite slightly higher s content than in the measured samples , the calculations should yield the overall effect of doping . the theoretically obtained spectra are broadened due to disorder , which should be reflected in the arpes data . lapw+lo calculations ( figure [ bz](f ) ) were realized for stoichiometric fete system as this approach can not deal with fractional atomic site occupancies . there is a qualitative agreement between the band structure obtained with these two methods ; in both cases three hole pockets are present at the @xmath3 point , two electron pockets are found at the m point , while there is no fs around the x point . the difference is observed at the m point , where the band seen below -0.6 ev for kkr - cpa is located below -1.2 ev for lapw+lo results , which is out of the scale for the figure [ bz](f ) . differences are also visible for the @xmath3-z direction . a dispersion along @xmath3-z is a matter of interest , as it may indicate whether the system is two - dimensional . in fact , weak dispersions or even lack of dispersion for certain bands are observed , what is seen in particular for the kkr - cpa approach . this means that this system may be considered as quasi two - dimensional to some extent . it is also noteworthy that the dispersions near @xmath3 obtained with kkr - cpa are characterized with lower slopes and higher band masses as compared to lapw+lo at low binding energies . the discrepancies between the obtained band structures may have arisen from different exchange - correlation potential and different modeling of atomic spheres in the approaches as well as due to the differences between the objects of the studies ; fe@xmath10te@xmath33s@xmath34 and fete . to obtain the agreement between the experiment and the theory the fermi energy for the calculated band structure needs to be shifted up by 0.11 ev and 0.10 ev for kkr - cpa and lapw+lo respectively . the band structure obtained from the calculations is generally consistent with the arpes results both along the @xmath3-m and @xmath3-x directions assuming that certain bands may be invisible in the experiment due to unfavourable matrix elements . out of the three hole pockets predicted by calculations at least two hole - like bands at @xmath3 are found in the experiment . theoretical results are also consistent with the spectra near m taken along the @xmath3-m direction ( figure [ bz](c ) ) , where a band moves towards @xmath4 when k approaches m , which is visible for @xmath29-polarization . the calculated electron pocket at m is not resolved in the experiment . theoretical dispersions along @xmath3-x confirm the absence of energy bands near @xmath4 at x. let us analyze the region of the @xmath3 point for fe@xmath0te@xmath1s@xmath2 , where the band structure appears to be different from that observed before for undoped non - superconducting fe@xmath7te @xcite . arpes studies performed at t=35 k include scans along m-@xmath3-m with @xmath30 and @xmath29 polarizations as well as along x-@xmath3-x with @xmath30 , @xmath29 , circular plus and circular minus polarizations ( figure [ gamma ] ( a)-(l ) ) . solid lines representing dispersions from kkr - cpa calculations ( figure [ bz](e ) ) are drawn on the experimental data in figure [ gamma ] ( a ) - ( f ) . they should be treated as guides to the eye as they are the results of fitting to the intensity map of kkr - cpa calculations . in order to trace the dispersions in the vicinity of @xmath4 the spectra were divided by the fermi - dirac distribution and are shown in figure [ gamma](m ) and ( n ) with binding energies determined from fitting energy distribution curves ( edcs ) or momentum distribution curves ( mdcs ) with the lorentzian function . the experimental and theoretical dispersions are compared in figure [ gamma ] ( o ) . a comparison of the band dispersions measured along the @xmath3-m ( figure [ gamma ] ( a),(b),(g),(h ) ) and @xmath3-x ( figure [ gamma ] ( c)-(f ) , ( i)-(l ) ) yields that they are quite similar at the @xmath3 point . for @xmath30-polarization a barely visible inner hole like band ( @xmath35 ) ( figure [ gamma ] ( m ) ) can be traced in both directions . the same polarization also yields a very flat quasiparticle band with strong intensity near the @xmath3 point ( @xmath36 ) . in fact , due to its high effective mass the dispersion was not measurable and the band exhibits practically constant binding energy determined to be 3 - 5 mev above the fermi level . the negligibility of the dispersion was confirmed by edcs shown in figure [ gamma ] ( p ) , which have approximately the same shape at @xmath3 and at @xmath370.05 @xmath28 . edcs from @xmath370.1 @xmath28 at the edges of @xmath36 seem to be more complex . their coherent part has the same binding energy but exhibits lower intensity . a contribution from another structure at higher binding energy is also observed . this structure may be evidence of a broadening of the quasiparticle band , incoherent spectral intensity or another hole band . integrating the edcs in the range @xmath37 0.1 @xmath28 over wave vector yields a peak with a width of 30 mev shown in figure [ gamma ] ( p ) . this narrow width , which is also characteristic of single edcs confirm the quasiparticle nature of this spectral intensity . raising the temperature to 70 k did not deliver any evidence of electron like dispersion ( not shown ) . on the other hand , the spectra obtained with @xmath29-polarization ( figure [ gamma](b ) , [ gamma](d ) , [ gamma](h ) , [ gamma](j ) ) show a dispersion ( @xmath38 ) , which looks like the outer hole pocket . ( color online ) energy bands in the @xmath3 point region for fe@xmath0te@xmath1s@xmath2 obtained by arpes along the m-@xmath3-m direction with ( a ) @xmath30 and ( b ) @xmath29 polarizations and along x-@xmath3-x , with ( c ) @xmath30 , ( d ) @xmath29 , ( e ) circular plus and ( f ) circular minus polarizations . the experimental dispersions are named as @xmath35 , @xmath36 and @xmath38 . theoretical dispersions obtained by kkr - cpa calculations ( solid lines ) are superimposed on the graphs . the spectra are shown as energy distribution curves ( edcs ) in ( g ) - ( l ) . the spectra from ( c ) and ( d ) divided by the fermi function are presented in ( m ) and ( n ) respectively . experimental band dispersions marked by black points result from energy or momentum distribution curve fitting . the extracted dispersions are compared to kkr - cpa ( for fe@xmath10te@xmath33s@xmath34 ) and lapw+lo ( for fete ) calculations ( o ) . panel ( p ) shows extracted edcs from ( c ) [ or ( i ) ] and the curve resulting from wave vector ( k ) integration of all edcs between @xmath39 and @xmath40 from ( c ) [ or ( i ) ] - red line ( dashed ) . all measurements were performed with incident photon energy of 40 ev at the temperature of 35 k.,width=595 ] it is rather clear that @xmath35 corresponds to the inner hole - like band in the calculations . however , the interpretation of @xmath36 and @xmath38 leaves certain ambiguity . the favoured scenario assumes that these features originate from the same band . this is supported by the circular polarization studies , which yield a continuous dispersion of @xmath36 and @xmath38 . moreover , such an interpretation is in agreement with the band structure calculations ( figure [ gamma](o ) ) as @xmath36 and @xmath38 match well the calculated middle hole band . however it has to be remarked that the experimental dispersion exhibits a more `` kink - like '' shape with mass renormalization near @xmath4 when compared to the theoretical one . it is noteworthy that this band changes its orbital character rather abruptly around the @xmath3 point , as @xmath36 and @xmath38 are sensitive to different polarizations in the experiment . one may still consider the other interpretation . the hypothesis that @xmath35 , @xmath36 and @xmath38 originate from three hole pockets , can also be compatible with our data . it may be supported by a possible similarity between s doped and se doped fe@xmath7te . the band structure at @xmath3 found in fete@xmath8se@xmath31 before @xcite consists of three hole like bands . in the case of fete@xmath41se@xmath42 @xcite one of the bands forms also a flat dispersion near @xmath4 with a narrow quasiparticle peak . an extension of this band is visible as a hole pocket . however , in our case , the hypothesis that @xmath36 originates from the third hole pocket , is not indicated directly by the data . importantly and independently of the interpretation @xmath36 remains flat and lies close to the chemical potential on a circle with a radius of approximately 0.15 @xmath28 . such a situation should result in a spike in the density of states close to @xmath4 called van hove singularity ( vhs ) . it is known as an important factor for induction or enhancement of superconductivity . it has been already suggested that vhs may play an important or even more universal role in the formation of superconductivity @xcite for a number of compounds . let us compare the spectra obtained for superconducting fe@xmath0te@xmath1s@xmath2 near the @xmath3 point with the literature results for undoped fe@xmath7te @xcite . the bands found with @xmath29-polarization by xia et al . @xcite are in relative agreement with our spectra . however , for @xmath30-polarization , the spectrum of fe@xmath7te consists of a hole pocket with no trace of the flat band at @xmath4 . on another hand the arpes studies of undoped fe@xmath43te @xcite and fe@xmath14te @xcite are characterized by broadened spectra with less clear band topography , which may be similar @xcite or rather different @xcite from fe@xmath0te@xmath1s@xmath2 results . it is known that bands in fe@xmath7te@xmath8se@xmath31 appear to be strongly renormalized @xcite when compared to ab - initio calculations . in the case of fe@xmath0te@xmath1s@xmath2 the inner hole pockets from kkr - cpa calculations fit the experimental spectra quite reasonably ( figure [ gamma](o ) ) and do not indicate strong mass renormalization . however , if the hypothesis of three hole pockets in the experiment was assumed , the agreement between the data and the calculations would be poorer . it is noteworthy that kkr - cpa calculations made for disordered fe@xmath10te@xmath33s@xmath34 and lapw+lo calculations performed for stoichiometric fete reveal different effective masses at @xmath4 ( figure [ gamma](o ) ) . this result shows that the estimation of band renormalization can be uncertain , as it depends on the used approach in band structure calculations . the kkr - cpa approach yields higher effective mass in the theoretical dispersions , what implies lower mass renormalization . the next important point is band dimensionality , which can be explored by a photon energy dependent study . therefore , the region of @xmath3 was investigated with energies between 22.5 ev and 50 ev ( figure [ photon ] ) . the outer part of the hole pocket ( @xmath38 ) can always be detected with @xmath29-polarization . the flat dispersion near @xmath3 ( @xmath36 ) can be seen for photon energies of 40 ev , 45 ev and 50 ev . on the other hand , its intensity is suppressed for 22.5 ev and 30 ev . there are two optional explanations for this fact : a dispersion along the wave vector component perpendicular to the surface ( @xmath44 ) or a photoionization cross section effect . to estimate the change of @xmath44 for the considered photon energy range one may use the free electron final state ( fefs ) model @xcite with a typical value of @xmath45=15 ev for the inner potential estimated in a case of iron pnictides @xcite . if the photon energy is increased from 22.5 ev to 50 ev the corresponding shift in @xmath44 would be 1.06 @xmath28 , which is approximately equal to the lattice constant in the reciprocal space c*=1.02 @xmath28 . an assumption of different @xmath45 values between 10 ev and 25 ev does not change the corresponding shift in @xmath44 considerably . therefore , if the fefs model is applicable , the spectra for 22.5 ev and 50 ev should refer to equivalent regions in the reciprocal lattice . in such a case different matrix elements could be the only explanation for the vanishing spectral intensity ( @xmath36 ) for lower photon energies . if the flat band is present for all @xmath44 values , it can be estimated that it covers about 3 @xmath6 of the brillouin zone volume . finally , eventual dispersion of @xmath38 as a function of @xmath44 was not found , so this band can be considered as two - dimensional . ( color online ) incident photon energy dependence of arpes spectra recorded for fe@xmath0te@xmath1s@xmath46 at t=35 k along x-@xmath3-x in the center of brillouin zone with the following photon energies h@xmath32 and polarizations : ( a ) 22.5 ev , @xmath30 , ( b ) 22.5 ev , @xmath29 , ( c ) 30 ev , @xmath30 , ( d ) 30 ev , @xmath29 , ( e ) 40 ev , @xmath30 , ( f ) 40 ev , @xmath29 , ( g ) 45 ev , @xmath30 , ( h ) 45 ev , @xmath29 , ( i ) 50 ev , @xmath30 , ( j ) 50 ev , @xmath29.,width=307 ] a photoelectron spectroscopy experiment realized in @xmath29 or @xmath30 geometry is able to determine the orbital wave function parity with respect to the mirror plane , which is defined by the positions of radiation source , sample and detector ( figure [ experiment ] ) @xcite . thus , possible orbital characters can be associated with the observed bands shown in figure [ gamma ] . in the first considered geometry the mirror plane is defined by the _ z _ axis perpendicular to the sample surface and the _ x _ axis corresponding to the @xmath3-m direction . the analyzer slit is oriented along this plane . the orientation of the fe - d orbitals dominating the vicinity of the fermi energy is similar to the case of the iron pnictides @xcite with the _ x _ and _ y _ axes pointing along corresponding @xmath3-m directions . @xmath30-polarized photons excite the states that are even with respect to the considered plane . consequently , the @xmath47 , @xmath48 and @xmath5 orbitals are allowed for the band @xmath35 along @xmath3-m ( figure [ gamma ] a , g ) . @xmath36 will be discussed separately as a special case related to the @xmath3 point , which was scanned four times with different geometries and polarizations . @xmath29-polarized radiation probes states with @xmath49 and @xmath50 orbital character , as they are odd with respect to the mirror plane ( figure [ experiment ] ) . hence , @xmath38 along @xmath3-m ( figure [ gamma ] b , h ) may be dominated by these orbital characters . a rotation of the sample such that the mirror plane is along the @xmath3-x direction changes the orbital parity related to the plane . along this direction the orbitals @xmath47 and @xmath49 equally contribute to bands as @xmath51 or @xmath52 . in this geometry measurements with @xmath30-polarization ( figure [ gamma ] c , i ) probing the bands with even symmetry indicate that @xmath35 can be dominated by @xmath51 , @xmath5 and @xmath50 . on the other hand the experiment with @xmath29-polarization ( figure [ gamma ] d , j ) reveals that @xmath38 should originate from @xmath48 and @xmath52 along @xmath3-x . schematic presentation of the arpes experiment with @xmath30-polarized photons ( electric field vector in the mirror plane ) and @xmath29-polarized photons ( electric field vector perpendicular to the mirror plane ) . for the sketched configuration @xmath29 polarized photons detect d@xmath53 and d@xmath54 orbitals whereas @xmath30-polarized radiation probes d@xmath55 , d@xmath56 and d@xmath57 orbitals.,width=403 ] finally , let us consider the @xmath36 spectrum . bands scanned along @xmath3-m with @xmath30-polarization ( figure [ gamma ] ( a , g ) ) can be composed of @xmath47 , @xmath48 and @xmath5 . however , the same @xmath3 point is also scanned along @xmath3-x with @xmath29-polarization ( figure [ gamma ] ( d , j ) ) . the later measurement yields no intensity at @xmath3 what indicates that @xmath48 and @xmath52 band characters are not present there . hence , only the @xmath5 remains as a dominant character for @xmath36 . similar reasoning for the @xmath3 point may be done using the spectra obtained with @xmath30-polarization along @xmath3-x ( figure [ gamma ] ( c , i ) ) permitting @xmath51 , @xmath5 and @xmath50 characters together with the other scan with @xmath29-polarization along @xmath3-m ( figure [ gamma ] ( b , h ) ) revealing the lack of intensity at @xmath3 . the last one indicates that @xmath49 and @xmath50 are not present at @xmath3 , what leads to the same conclusion that mainly @xmath5 character contributes to the @xmath36 spectrum . band structure of stoichiometric fete . contributions of ( a ) d@xmath56 , ( b ) d@xmath53 , ( c ) d@xmath55 and ( d ) d@xmath57/d@xmath54 orbital characters are represented by band widths ( fat bands).,width=585 ] the contribution of s- , p- and d- valence orbital characters was also estimated theoretically by means of lapw+lo method implemented in the wien2k package @xcite ( figure [ bands ] ) . the calculations were realized for stoichiometric fete . the results confirm that d - orbitals dominate the band structure in the vicinity of the fermi energy ( other orbital projections are not shown in figure [ bands ] ) . the hole bands @xmath35 and @xmath38 appearing around the @xmath3 point have their counterparts in the theoretical results . although it is not obvious to what extent the calculations for pure fete are reliable for fe@xmath0te@xmath1s@xmath2 , they can narrow down the list of possible band characters . the calculations yield that the @xmath35 band has mainly @xmath58 orbital character , while @xmath38 is dominated by @xmath58 and @xmath50 with some contribution of @xmath48 along @xmath3-x . this is in agreement with the experimental results obtained both along @xmath3-m and @xmath3-x directions . in contrast , the calculations for fete do not reveal the flat band at the fermi energy with dominant @xmath5 orbital character , which would correspond to @xmath38 . in this aspect they are not compatible with the experiment for fe@xmath0te@xmath1s@xmath2 . one may expect that s doping in fe@xmath7te@xmath8s@xmath31 system may have a particular effect on the @xmath5 orbital as it results in shrinking the @xmath59 lattice constant . the band structure of superconducting fe@xmath0te@xmath1s@xmath2 was studied along the @xmath3-x and @xmath3-m directions by arpes . an increased spectral intensity at @xmath4 is observed near the @xmath3 and m points . in particular , two hole bands ( @xmath35 and @xmath38 ) are found around @xmath3 with a high intensity quasiparticle peak ( @xmath36 ) located close to @xmath4 , with no evidence of dispersion . this latter feature has mainly @xmath5 orbital character and is interpreted as the maximum of the @xmath38 hole band or an evidence of another hole pocket . such a band structure yields a high density of states at the chemical potential , interpreted as a van hove singularity . measurements performed with variable photon energy show no dispersion of the @xmath38 hole band as a function of @xmath44 . hence , it is considered as two dimensional . the flat part of the band located at @xmath4 has a reduced intensity for the photon energies of 30 ev and 22.5 ev , which is attributed to a low photoionization cross - section . the band structure obtained from kkr - cpa calculations includes the broadening due to disorder and exhibits three hole pockets in @xmath3 and two electron pockets at m. further lapw - lo calculations performed for stoichiometric fete lead to a band topography , which is in reasonable agreement with the kkr - cpa results and the experiment for fe@xmath0te@xmath1s@xmath2 . the orbital characters calculated with the lapw - lo method agree with the experimental results for @xmath35 and @xmath38 dispersions but are inconsistent with the @xmath5 character observed for the flat @xmath36 spectrum . some authors ( h.s . , f.f . and f.r . ) acknowledge the support by the dfg through the for1162 . the study has been supported by polish national science centre grant 2011/01/b / st3/00425 . p.z . acknowledges use of the equipment at the umd nanoscale imaging spectroscopy and properties laboratory . the research leading to these results has received funding from the european community s seventh framework programme ( fp7/2007 - 2013 ) under grant agreement number 226716 99 hsu f c , luo j y , the k w , chen t k , huang t w , wu p m , lee y c , huang y l , chu y y , yan d c , and wu m k , 2008 _ proc . _ * 105 * , 14262 . fang m h , pham h m , qian b , liu t j , vehstedt e k , liu y , spinu l , and mao z q , 2008 _ phys . b _ * 78 * , 224503 . mizuguchi y , tomioka f , tsuda s , yamaguchi t , and takano y , 2009 _ appl . b _ * 94 * , 012503 . mizuguchi y and takano y , 2010 _ j. phys . japan _ * 79 * , 102001 . guo j g , jin s f , wang g , wang s c , zhu k x , zhou t t , he m , and chen x l , 2010 _ phys . b _ * 82 * , 180520 . miao h , richard p , tanaka y , nakayama k , qian t , umezawa k , sato t , xu y - m , shi y b , xu n , wang x - p , zhang p , yang h - b , xu z - j , wen j s , gu g - d , dai x , hu j - p , takahashi t , and ding h , 2012 _ phys . rev . b _ * 85 * , 094506 . mou d x , liu s y , jia x w , he j f , peng y y , zhao l , yu l , liu g d , he s l , dong x l , zhang j , wang h d , dong c h , fang m h , wang x y , peng q j , wang z m , zhang s j , yang f , xu z y , chen c t , zhou x j , 2011 _ phys . * 106 * , 107001 . zhao l , mou d , liu s , jia x , he j , peng y , yu l , liu x , liu g , he s , dong x , zhang j , he j b , wang d m , chen g f , guo j g , chen x l , wang x , peng q , wang z , zhang s , yang f , xu z , chen c , zhou x j , 2011 _ phys . rev . b _ * 83 * , 140508 . blaha p , schwarz k , madsen g , kvasnicka d and luitz j , ( 2001 ) wien2k , an augmented plane wave + local orbitals program for calculating crystal properties ( karlheinz schwarz , tech . wien , austria ) . thirupathaiah s , dejong s , ovsyannikov r , drr h a , varykhalov a , follath r , huang y , huisman r , golden m s , zhang yu - zhong , jeschke h o , valenti r , erb a , gloskovskii a , and fink j , 2010 _ phys . b _ * 81 * , 104512 .

the electronic structure of superconducting fe@xmath0te@xmath1s@xmath2 has been studied by angle resolved photoemission spectroscopy ( arpes ) . experimental band topography is compared to the calculations using the methods of korringa - kohn - rostoker ( kkr ) with coherent potential approximation ( cpa ) and linearized augmented plane wave with local orbitals ( lapw+lo ) . the region of the @xmath3 point exhibits two hole pockets and a quasiparticle peak close to the chemical potential ( @xmath4 ) with undetectable dispersion . this flat band with mainly @xmath5 orbital character is formed most likely by the top of the outer hole pocket or is an evidence of the third hole band . it may cover up to 3 @xmath6 of the brillouin zone volume and should give rise to a van hove singularity . studies performed for various photon energies indicate that at least one of the hole pockets has a two - dimensional character . the apparently nondispersing peak at @xmath4 is clearly visible for 40 ev and higher photon energies , due to an effect of photoionisation cross section rather than band dimensionality . orbital characters calculated by lapw+lo for stoichiometric fete do not reveal the flat @xmath5 band but are in agreement with the experiment for the other dispersions around @xmath3 in fe@xmath0te@xmath1s@xmath2 .

rs oph is one of the well - observed recurrent novae and is suggested to be a progenitor of type ia supernova . it has undergone its sixth recorded outburst on 2006 february 12 @xcite and many observational results are reported ( see other papers in this proceedings ) . it s @xmath6 magnitude light curve has been obtained throughout the outburst @xcite [ the numerical table is provided in @xcite in this volume ] , which shows a mid - plateau phase that lasts 45 - 75 days from the optical peak followed by a quick decrease ( figure 1 ) . rs oph has also been observed with x - ray satellites . we analyzed the _ xrt observations available in the heas - arc database and extracted the count rate in the energy band of 0.3 - 0.55 kev binned in 2000 s. ( see * ? ? ? * for more details ) . the supersoft x - ray ( ssx ) light curve is plotted in figure 1 ( see also table 1 ) . the light curve rises at about 30 days after the optical peak and shows a long plateau phase that lasts as long as about 50 days corresponding to a long mid - plateau phase of optical light curve . during the nova outburst hydrogen - rich envelope around the white dwarf ( wd ) expands to a giant size and strong wind mass - loss occurs . in such stages dynamical calculation codes often encounter numerical difficulties , so we can not calculate light curves . for example , one must take off the outermost lagrange mesh points , which prevents us accurately determining the wind mass - loss rate and calculating the resultant evolution speed of novae . we have calculated light curve models based on the optically thick wind theory @xcite , which is a quasi - evolution euler code in which the wind mass - loss rate is accurately obtained as an eigenvalue of a boundary value problem . the photospheric temperature and luminosity are also accurately calculated . therefore , up to now , the optically thick wind is only the method that can follow the theoretical light curves of novae . to explain the ssx phase and the optical light curves of rs oph , we have included effects of heat exchange between the hydrogen - rich envelope and a helium layer underneath . hydrogen burning produces hot helium ash which accumulates underneath the burning zone because convection may descend quickly after the optical peak . this helium layer grows in mass with time and behave as a heat reserver . in the later phase of the outburst heat flows upward from the hot helium layer , which keeps hydrogen - rich envelope hot enough to emit ssx in a long time . after calculating many models for two parameters , i.e. , the wd mass and the hydrogen content of the envelope , we obtain a best fit model as shown in fig 1a ( see * ? ? ? * for details ) . this model reproduces the optical light curve and explains reasonably well the x - ray count rate . figure 1b and table 2 show that the wind mass - loss stops when the ssx count rate increases . the total luminosity @xmath7 is almost constant until day 80 when nuclear burning extinguishes and helium ash layer becomes too cool to provide heat any more . in this way the duration of the ssx phase can be explained only if we assume hot helium ash underneath the hydrogen layer . this helium layer accumulates on the wd although some part of the hydrogen - rich matter is blown off by the wind . therefore , we conclude that the wd mass is growing though the 2006 outburst . we summarize our results as follows ; \2 . the accreted matter during 21 years before the outburst is estimated from the envelope mass at the optical peak to be @xmath8 , @xmath9 of which is ejected by the wind and the rest @xmath10 accumulates on the wd . therefore the net growth rate of the wd is @xmath11yr@xmath2 . the durations of the mid - plateau phase of optical and the peak plateau phase of ssx suggest the presence of a helium layer which accumulates on the wd . therefore , the wd mass of rs oph is now growing . time & count & time & count & time & count & time & count & time & count & time & count + & rate & & rate & & rate & & rate & & rate & & rate + [ day ] & [ s@xmath2 ] & [ day ] & [ s@xmath2 ] & [ day ] & [ s@xmath2]&[day ] & [ s@xmath2]&[day ] & [ s@xmath2 ] & [ day ] & [ s@xmath2 ] + 3.53 & -0.20 & 36.74 & 1.64 & 40.52 & 1.56 & 47.09 & 2.07 & 52.67 & 2.10 & 64.59 & 2.00 + 11.33 & -0.69 & 36.77 & 1.59 & 40.54 & 1.47 & 47.16 & 2.10 & 52.74 & 2.08 & 66.12 & 2.00 + 11.40 & -0.95 & 36.79 & 1.54 & 40.59 & 1.52 & 47.37 & 2.12 & 52.81 & 2.06 & 66.19 & 2.00 + 11.47 & -0.78 & 36.81 & 1.47 & 40.61 & 1.53 & 47.42 & 2.08 & 52.86 & 2.09 & 66.26 & 1.98 + 13.94 & -0.93 & 36.84 & 1.50 & 40.68 & 1.72 & 47.44 & 2.14 & 52.93 & 1.99 & 67.12 & 1.99 + 15.96 & -0.86 & 36.86 & 1.49 & 41.07 & 1.12 & 47.49 & 2.10 & 52.97 & 2.09 & 67.14 & 1.99 + 18.53 & -1.02 & 36.91 & 1.43 & 41.12 & 1.62 & 47.51 & 2.12 & 52.99 & 2.11 & 67.18 & 1.99 + 18.57 & -0.80 & 36.93 & 1.46 & 41.14 & 1.70 & 47.83 & 2.10 & 53.04 & 2.11 & 67.21 & 1.95 + 26.33 & -0.51 & 36.98 & 1.22 & 41.19 & 1.74 & 47.90 & 2.12 & 53.06 & 2.12 & 67.25 & 1.99 + 26.35 & -0.51 & 37.00 & 1.20 & 41.21 & 1.74 & 48.02 & 2.09 & 53.18 & 2.10 & 67.28 & 1.94 + 29.34 & 1.07 & 37.05 & 1.11 & 41.26 & 2.00 & 48.04 & 2.09 & 53.20 & 2.11 & 67.32 & 1.97 + 29.36 & 1.06 & 37.12 & 1.37 & 41.28 & 2.02 & 48.09 & 2.09 & 53.25 & 2.09 & 67.35 & 1.86 + 30.22 & 0.48 & 37.14 & 1.39 & 41.33 & 2.04 & 48.11 & 2.12 & 53.27 & 2.09 & 67.51 & 2.02 + 30.24 & 0.59 & 37.18 & 1.39 & 41.35 & 2.04 & 48.16 & 2.08 & 53.32 & 2.12 & 67.53 & 2.02 + 32.35 & 1.21 & 37.21 & 1.41 & 41.40 & 2.04 & 48.18 & 2.13 & 53.34 & 2.14 & 68.67 & 1.99 + 32.37 & 1.19 & 37.25 & 1.58 & 41.42 & 2.07 & 48.23 & 2.14 & 53.39 & 1.79 & 68.74 & 1.96 + 33.16 & 1.78 & 37.32 & 1.52 & 41.47 & 1.97 & 48.30 & 2.12 & 53.41 & 1.80 & 69.66 & 1.96 + 33.25 & 1.86 & 37.39 & 1.64 & 42.46 & 2.09 & 48.37 & 2.11 & 53.46 & 2.10 & 69.68 & 1.96 + 33.30 & 1.95 & 37.42 & 1.56 & 42.53 & 2.03 & 48.43 & 2.10 & 53.48 & 2.10 & 69.73 & 1.97 + 33.32 & 1.88 & 37.46 & 1.81 & 43.13 & 1.90 & 48.50 & 2.07 & 53.53 & 2.09 & 69.75 & 1.95 + 33.37 & 1.88 & 37.49 & 1.79 & 43.20 & 1.87 & 48.57 & 2.13 & 53.55 & 2.06 & 70.59 & 1.95 + 33.39 & 1.94 & 37.53 & 1.81 & 43.27 & 1.98 & 48.90 & 2.10 & 53.60 & 2.04 & 70.61 & 1.97 + 33.43 & 1.94 & 37.60 & 1.84 & 43.34 & 1.97 & 49.04 & 2.10 & 53.62 & 1.97 & 70.66 & 1.96 + 33.46 & 1.94 & 37.67 & 1.92 & 43.41 & 1.92 & 49.11 & 2.09 & 53.67 & 2.05 & 70.68 & 1.96 + 33.50 & 1.85 & 37.74 & 1.85 & 43.48 & 1.78 & 49.18 & 2.09 & 53.69 & 1.99 & 72.35 & 1.90 + 33.53 & 1.88 & 37.79 & 1.88 & 43.55 & 1.94 & 49.22 & 2.08 & 53.74 & 1.88 & 72.37 & 1.87 + 33.57 & 1.92 & 37.81 & 1.50 & 43.94 & 1.69 & 49.24 & 2.10 & 53.78 & 2.03 & 72.42 & 1.89 + 33.60 & 1.86 & 38.13 & 2.10 & 44.01 & 1.91 & 49.29 & 2.05 & 53.80 & 2.01 & 72.49 & 1.88 + 33.64 & 1.18 & 38.32 & 2.13 & 44.08 & 1.91 & 49.31 & 2.07 & 53.85 & 1.79 & 73.30 & 1.91 + 33.67 & 0.96 & 38.34 & 2.11 & 44.15 & 1.81 & 49.36 & 2.07 & 53.87 & 2.05 & 73.34 & 1.89 + 33.71 & 0.37 & 38.39 & 2.14 & 44.20 & 1.67 & 49.38 & 2.08 & 53.92 & 2.09 & 73.37 & 1.87 + 33.74 & 0.34 & 38.41 & 2.11 & 44.22 & 1.76 & 49.43 & 2.06 & 53.94 & 2.03 & 73.41 & 1.89 + 33.78 & 0.27 & 38.46 & 2.10 & 44.27 & 1.91 & 49.45 & 2.04 & 53.99 & 2.10 & 73.43 & 1.86 + 33.90 & 1.34 & 38.48 & 2.07 & 44.29 & 1.90 & 49.50 & 2.06 & 54.01 & 2.11 & 74.22 & 1.82 + 33.97 & 1.66 & 38.53 & 1.97 & 44.34 & 1.43 & 49.52 & 2.06 & 54.06 & 2.08 & 74.29 & 1.79 + 34.04 & 1.94 & 38.60 & 1.92 & 44.36 & 1.73 & 49.82 & 2.10 & 54.13 & 1.91 & 74.36 & 1.80 + 34.11 & 1.98 & 38.67 & 2.02 & 44.41 & 1.93 & 49.85 & 2.12 & 54.18 & 2.08 & 74.43 & 1.87 + 34.18 & 1.99 & 38.74 & 2.05 & 44.43 & 1.97 & 50.03 & 2.12 & 54.20 & 2.12 & 74.48 & 1.87 + 34.20 & 2.01 & 38.78 & 2.02 & 44.48 & 1.95 & 50.05 & 2.13 & 54.24 & 1.78 & 74.50 & 1.88 + 34.24 & 1.68 & 38.85 & 1.54 & 44.50 & 1.97 & 50.10 & 2.09 & 54.27 & 1.78 & 75.17 & 1.87 + 34.27 & 1.56 & 38.92 & 0.84 & 44.55 & 1.99 & 50.17 & 2.08 & 54.31 & 2.09 & 75.22 & 1.86 + 34.31 & 1.28 & 38.99 & 0.92 & 45.15 & 1.92 & 50.24 & 2.07 & 54.34 & 2.10 & 75.24 & 1.85 + 34.34 & 1.45 & 39.06 & 1.33 & 45.17 & 1.90 & 50.31 & 2.04 & 54.41 & 2.10 & 77.23 & 1.81 + 34.36 & 1.57 & 39.13 & 1.70 & 45.22 & 2.09 & 50.38 & 2.09 & 54.45 & 2.11 & 77.25 & 1.79 + 34.38 & 1.69 & 39.34 & 1.55 & 45.29 & 2.11 & 50.52 & 2.16 & 54.48 & 2.12 & 77.30 & 1.81 + 34.43 & 1.90 & 39.41 & 1.01 & 45.36 & 1.98 & 51.51 & 2.12 & 57.60 & 2.03 & 77.37 & 1.81 + 34.45 & 1.95 & 39.48 & 0.88 & 45.43 & 2.12 & 51.54 & 2.11 & 57.62 & 2.05 & 79.15 & 1.74 + 34.50 & 2.07 & 39.52 & 1.19 & 45.47 & 1.90 & 51.84 & 2.14 & 58.62 & 2.03 & 79.18 & 1.74 + 34.52 & 2.04 & 39.55 & 1.20 & 45.49 & 1.85 & 51.91 & 2.11 & 58.64 & 1.99 & 80.52 & 1.67 + 35.05 & 1.86 & 39.59 & 1.24 & 45.56 & 2.14 & 51.93 & 2.12 & 59.15 & 2.11 & 80.56 & 1.68 + 36.26 & 1.51 & 39.62 & 1.27 & 45.82 & 2.09 & 51.98 & 2.12 & 59.87 & 2.07 & 80.59 & 1.68 + 36.30 & 1.51 & 39.66 & 1.16 & 45.89 & 2.00 & 52.05 & 2.11 & 59.89 & 2.08 & 80.63 & 1.67 + 36.33 & 1.52 & 39.68 & 1.08 & 46.03 & 2.09 & 52.12 & 2.12 & 61.63 & 2.06 & 80.66 & 1.68 + 36.37 & 1.54 & 39.75 & 1.41 & 46.07 & 2.02 & 52.18 & 2.10 & 61.70 & 2.05 & 85.15 & 1.36 + 36.40 & 1.53 & 39.82 & 1.33 & 46.10 & 2.02 & 52.25 & 2.12 & 62.16 & 2.01 & 85.19 & 1.35 + 36.44 & 1.59 & 39.85 & 1.58 & 46.14 & 2.09 & 52.30 & 2.09 & 62.18 & 2.00 & 85.22 & 1.35 + 36.47 & 1.62 & 39.87 & 1.61 & 46.17 & 2.09 & 52.32 & 2.12 & 62.23 & 2.05 & 87.14 & 1.17 + 36.51 & 1.50 & 39.99 & 1.56 & 46.21 & 2.08 & 52.37 & 2.12 & 62.25 & 2.04 & 87.16 & 1.18 + 36.54 & 1.43 & 40.05 & 1.66 & 46.24 & 2.08 & 52.39 & 2.14 & 62.58 & 2.08 & 87.21 & 1.17 + 36.58 & 1.38 & 40.12 & 1.55 & 46.28 & 2.08 & 52.44 & 2.12 & 62.65 & 2.07 & 87.28 & 1.17 + 36.61 & 1.20 & 40.19 & 1.44 & 46.30 & 2.09 & 52.46 & 2.13 & 63.18 & 2.02 & 87.35 & 1.15 + 36.65 & 1.47 & 40.26 & 1.11 & 46.35 & 1.97 & 52.53 & 2.08 & 63.25 & 2.04 & 91.03 & 0.70 + 36.68 & 1.41 & 40.33 & 0.87 & 46.37 & 1.97 & 52.58 & 2.10 & 64.18 & 2.04 & 91.10 & 0.74 + 36.70 & 1.70 & 40.40 & 1.01 & 46.42 & 2.09 & 52.60 & 2.09 & 64.24 & 1.83 & 93.57 & 0.43 + 36.72 & 1.65 & 40.47 & 1.53 & 46.49 & 2.06 & 52.65 & 2.10 & 64.52 & 1.98 & 93.90 & 0.36 + lllllll time & @xmath14&@xmath15 & @xmath16 ( mass loss rate)&@xmath17 & @xmath18 & @xmath19 + [ day ] & [ k]&[cm ] & [ @xmath20yr@xmath2 ] & [ cm s@xmath2 ] & [ erg s@xmath2 ] & [ cm s@xmath21 + 9.8 & 4.86 & 11.04 & -4.754 & 7.990 & 38.38 & 4.181 + 22.1 & 5.09 & 10.60 & -5.287 & 8.087 & 38.40 & 5.060 + 27.3 & 5.16 & 10.46 & -5.507 & 8.096 & 38.41 & 5.342 + 32.6 & 5.29 & 10.20 & -5.778 & 7.971 & 38.42 & 5.846 + 34.8 & 5.37 & 10.05 & -5.952 & 7.815 & 38.43 & 6.151 + 38.2 & 5.50 & 9.79 & -6.490 & 7.405 & 38.43 & 6.679 + 38.9 & 5.55 & 9.69 & -6.820 & 7.154 & 38.44 & 6.873 + 39.1 & 5.56 & 9.67 & -6.921 & 7.067 & 38.44 & 6.918 + 39.5 & 5.63 & 9.53 & 0.000 & 0.000 & 38.44 & 7.191 + 40.5 & 5.82 & 9.15 & 0.000 & 0.000 & 38.44 & 7.948 + 44.9 & 5.95 & 8.89 & 0.000 & 0.000 & 38.44 & 8.471 + 50.7 & 6.03 & 8.73 & 0.000 & 0.000 & 38.43 & 8.796 + 60.6 & 6.10 & 8.58 & 0.000 & 0.000 & 38.41 & 9.093 + 71.7 & 6.13 & 8.51 & 0.000 & 0.000 & 38.39 & 9.239 + 78.1 & 6.14 & 8.44 & 0.000 & 0.000 & 38.30 & 9.368 + 80.8 & 6.13 & 8.43 & 0.000 & 0.000 & 38.24 & 9.399 + 85.0 & 6.12 & 8.41 & 0.000 & 0.000 & 38.16 & 9.425 + 89.5 & 6.09 & 8.40 & 0.000 & 0.000 & 38.02 & 9.445 + 94.4 & 6.04 & 8.40 & 0.000 & 0.000 & 37.82 & 9.460 + 97.5 & 6.01 & 8.39 & 0.000 & 0.000 & 37.70 & 9.466 + 100.3 & 5.99 & 8.39 & 0.000 & 0.000 & 37.58 & 9.469 + 116.4 & 5.85 & 8.39 & 0.000 & 0.000 & 37.03 & 9.480 +

the recurrent nova rs ophiuchi , one of the candidates for type ia supernova progenitors , underwent the sixth recorded outburst in february 2006 . we report a complete light curve of supersoft x - ray that is obtained for the first time . a numerical table of x - ray data is provided . the supersoft x - ray flux emerges about 30 days after the optical peak and continues until about 85 days when the optical flux shows the final decline . such a long duration of supersoft x - ray phase can be naturally understood by our model in which a significant amount of helium layer piles up beneath the hydrogen burning zone during the outburst , suggesting that the white dwarf mass is effectively growing up . we have estimated the white dwarf mass in rs oph to be @xmath0 and its growth rate to be about @xmath1 yr@xmath2 in average . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ mariko kato,@xmath3 izumi hachisu,@xmath4 gerardo juan manuel luna@xmath5 + + + _ @xmath4univ . of tokyo , tokyo 153 - 8902 , japan _ + _ @xmath5university of sapaulo , 05508 - 900 sao paulo , brazil _ + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

arnold s cat map is a linear area - preserving map @xmath0 on a torus in phase space formed by identifying the boundaries of the interval @xmath1 $ ] in both the coordinate @xmath2 and the momentum @xmath3 directions @xcite . ( because of this the area of the torus is characterized by planck s constant which takes on the values @xmath4 , where @xmath5 is the number of sites in both the coordinate and the momentum directions in the phase space . ) from time step @xmath6 to @xmath7 it is given by @xmath8 where @xmath9 guarantees area preservation . the degree of chaos depends on the choice of @xmath0 . the eigenvalues of @xmath0 are either both real or both imaginary . in the latter case , @xmath0 is elliptic , the motion becomes periodic and no sensitive dependence on the initial condition is observed . when @xmath0 is hyperbolic , the motion is chaotic . quantized cat map is studied in detail by hannay and berry @xcite . the matrix has to assume a special form in order to yield nontrivial values of the progagator for the map . we choose @xmath10 for the elliptic and the hyperbolic cases respectively . for the special choice of the matrix elements @xmath11 made above the propagator takes on the simple forms , @xmath12 , ~~ u_2(j+1 , j ) = \sqrt{\frac{i}{\cal n}}\exp [ \frac{i}{\hbar } ( q_{j}^{2}- q_{j}q_{j+1}+q_{j+1}^{2})].\ ] ] since each iteration describes a permutation among sites , each site belongs to a periodic orbit . thus the quantum dynamics follows the classical way , resulting in the recurrence of the wave function ( or equivalently , the wigner function @xcite ) . we now couple the system linearly to a bath of @xmath13 harmonic oscillators with coordinates @xmath14 and momentum @xmath15 ( @xmath16 ) described by the hamiltonian @xmath17 and the interaction hamiltonian @xmath18 @xmath19 where @xmath2 is the coordinate of the system and @xmath20 is the coupling constant of @xmath2 to the @xmath21th oscillator in the bath . by integrating out the bath variables , we get the reduced density matrix , @xmath22.\ ] ] where @xmath23 is the classical action of the system which appears as the exponent of the propagator in ( 1.3 ) . @xmath24 , and @xmath25 are the actions for bath and interaction , respectively . the propagator @xmath26 for the reduced density matrix from time steps @xmath6 to @xmath7 is @xmath27,\ ] ] in a path - integral representation @xcite , where @xmath28\ ] ] is the influence action . here @xmath29 , and @xmath30 are the dissipation and noise kernels respectively @xcite . if we consider the simplest case of an ohmic bath at high temperature @xmath31 @xcite , and consider times shorter than the relaxation time , then we obtain a gaussian form for the influence functional , with @xmath32 where the noise kernel becomes local @xmath33 and @xmath34 is the damping coefficient . the unit - time propagator becomes @xmath35 \rangle.\ ] ] here @xmath36 is a gaussian white noise given by @xmath37\ ] ] where @xmath38 denotes statistical average over noise realization @xmath36 . for the elliptic map , we get @xmath39.\ ] ] and for the hyperbolic map , @xmath40\ ] ] the wigner function is defined as @xmath41 where @xmath42 is the momentum conjugate to @xmath43 . the propagator @xmath44 for the wigner function is @xmath45 this is reduced to the form of the classical cat map . for the elliptic case , @xmath46 for the hyperbolic case , @xmath47 without noise , quantum evolution follows classical permutation @xcite the phase space is divided by a finite number of different periodic orbits and the period is known to increase roughly proportional to @xmath5 with some irregular oscillation . when coupled to a bath , the cat map is exposed to a gaussian noise in each time step . the discretized noise induces transitions between different periodic orbits in an irregular way . interaction with an environment blurs the recurrence of physical quantities in the quantum map . fig.1 shows @xmath48 , the linearized entropy ( with the reversed sign ) for various cases . if there is no interaction with the environment , the entropy is constant for both regular and chaotic cases . quantum recurrence is evident even when the system is chaotic . when interaction sets in , @xmath48 decays exponentially , showing that the system rapidly decoheres . the rate of decoherence is much faster in chaotic systems than in regular systems @xcite . it suggests that recurrence would be less evident in a decohering chaotic system . in fig . 2 , we show the mean displacement of points in the phase space as a function of time steps . this is defined by @xmath49 , where @xmath50 and @xmath51 are the displacements from the initial phase space points , and @xmath52 denotes averaging over noise distributions . in the chaotic case , we see that recurrence disappears with just a small amount of noise ( fig . 2a ) whereas in the regular case , the same amount of noise does not alter the qualitative picture of recurrence ( fig . 2b ) . in both cases , the decohered quantum system behaves close to the classical picture in which the regular and chaotic dynamics are clearly distinguished . in spite of the discreteness of the points on the torus , the system behaves effectively classically due to the influence of the environment . the kicked rotor is one of the most intensively studied models from both the quantum and classical point of view @xcite . the hamiltonian of the kicked rotor is given by @xmath53 which describes a one - dimensional rotor subjected to a delta - functional periodic kick at @xmath54 . here @xmath55 is the angle of the rotor with period @xmath56 , @xmath57 is the moment of inertia , @xmath42 is the angular momentum , and @xmath44 is the strength of the kick which measures the nonlinearity . when @xmath58 , the system becomes chaotic over the entire phase space . the quantum dynamics of the kicked rotor is given by the corresponding schrdinger equation @xmath59 where @xmath60 is the wave function of the rotor . denoting @xmath61 as the wave function @xmath62 at each discrete time @xmath63 , and integrating ( 2.2 ) from @xmath6 to @xmath7 , we obtain @xmath64 \exp [ -i \frac{k\cos x}{\hbar } ] \psi_{j}(x ) \ ] ] the quantum kicked rotor ( qkr ) is known to exhibit dynamical localization . after some relaxation time scale , the wave function becomes exponentially localized in the momentum space @xcite . this may be interpreted as a particle moving in a lattice with a quasi - random potential . this heuristic picture seems to justify the analogy between the quantum kicked rotor to the tight binding model with an exponentially decaying hopping parameter which is known to show anderson localization @xcite . dynamical localization in this context arises from the suppression of classical diffusive behavior by the quantum dynamics . however , as shown by ott , et.al . @xcite , a small external noise can break the localization . sufficient amount of noise would induce the quantum system to exhibit classical diffusive behavior . dittrich and graham studied this problem @xcite by coupling the system to a zero temperature harmonic oscillator bath and analysed solutions to the master equation . cohen and fishman presented the most detailed study of this problem for an ohmic bath @xcite . here we want to approach these issues from an environment - induced decoherence point of view @xcite . we begin by calculating the density matrix for the kicked rotor coupled to an environment . we introduce a linear coupling of the system momentum @xmath42 with each oscillator coordinate @xmath65 in the bath in the form @xmath66 ( here @xmath67 without the subscript @xmath21 denote the system coordinate and momentum variables ) . as before , we assume an ohmic bath and examine the time period where dissipation is small . under these assumptions , the unit time propagator for the wave function @xmath68 is given by @xmath69 \exp [ -\frac{i}{\hbar } k\cos x ] \exp [ -\frac{i}{\hbar } \xi p ] \ ] ] where , as before , the noise term @xmath70 arises from using a gaussian identity in the integral transform of the term involving the noise kernel in the influence functional . summing over all noise realizations @xmath71 gives the desired reduced density matrix , @xmath72 where @xmath73 loss of quantum coherence is measured by the density matrix becoming approximately diagonal . @xmath48 can be expressed as @xmath74 where @xmath75 denotes the statistical average of all possible noise histories of two independent noises @xmath76 defined at each time interval from @xmath6 to @xmath7 . at high temperatures @xmath77 are reduced to two time - uncorrelated independent gaussian white noises defined at each time step . we see that there is a close relation between the breaking of dynamical localization and quantum decoherence . in fig.3 we plot the linearized entropy @xmath48 versus the energy @xmath78 . this shows that delocalization occurs as quantum coherence breaks down , suggesting that delocalization and decoherence occurs by the same mechanism . as the nonlinearity parameter @xmath44 increases , the system decoheres more rapidly . at the same time , the amount of delocalization measured by the diffusion constant increases . this may be explained in the following way : because the coupling is through the momentum , the noise term does not involve any nonlinearity . the time scale for the system to lose coherence is given by @xmath79 , where @xmath80 is the thermal de broglie wavelength , and @xmath81 is the relevant momentum scale . after this time , noise will destroy the quantum coherence between such momentum separations . in the kicked rotor case , localization will occur due to the coherence around @xmath82 , where @xmath83 is the localization length . since @xmath84 , this gives @xmath85 . this shows that nonlinearity increases the rate of decoherence . the relation between the diffusion constant @xmath86 and the noise strength is given in @xcite . for our case , @xmath87 and for weak noises , we can consider the particle as undergoing a random walk with hopping parameter @xmath88 . then @xmath89 . the wigner function is often used to examine the quantum to classical transition . the wigner function at time @xmath54 is defined as @xmath90 where @xmath91 . from ( 2.3 ) , the unit - time propagator for the wigner function of the qkr is found to be @xmath92 where @xmath93 measures the effect of the kick . we can see the effects of quantum corrections is seen more clearly if we expand @xmath94 in orders of @xmath95 : @xmath96 the first exponential contains the classical propagator and the second contains quantum corrections of even orders of @xmath95 . thus we get @xmath97 where the wigner function with the new arguments depicts classical evolution . this map alone is the source of streching and folding of volume in phase space which signify classical chaos . if the initial system wavefunction is described by a gaussian wave packet with width @xmath98 , we would expect to see a classical - like evolution of the packet at short times . when the width of the contracting wave packet gets so small as comparable to @xmath95 , the effect of quantum corrections from higher @xmath95 order terms in ( 3.4 ) set in . by comparing the classical and quantum terms , we see that quantum corrections will become important when @xmath99 . here @xmath100 , where the lyapunov exponent @xmath101 . thus we can deduce the ehrenfest time when the wigner function or the expectation value of any observable follow classical trajectories . ] for qkr to be @xmath102 note that in the continuum case , this definition gives us a different time scale for each term in the expansion @xcite . the major effect of the bath ( at times short compared with the relaxation time ) is the appearance of a diffusion term in ( 3.4 ) , @xmath103 competition amongst the three terms with different physical origins is apparent : the first term in ( 3.5 ) is the quantum diffusion term , the second is the quantum correction term , and the third is purely classical evolution . as discussed by zurek and paz @xcite , if d is sufficiently large , the effect of quantum corrections becomes inconspicuous . in this case , the diffusion term traces out a small scale oscillating behavior before quantum corrections have a chance to change classical evolution . then one may expect the time evolution of the wigner function to be like that of classical evolution with noise . the role of quantum diffusion is to add some gaussian averaging so that the contracting direction in phase space will be suppressed while it does not affect the stretching direction . as long as the width of the wave packet is large such that the first term is negligible , the evolution should be liouvillian ( time reversible if we assume infinite measurement precision ) . furthermore , we expect that after the width of the packet along the contracting direction becomes comparable to the diffusion generated width ( in the gaussian wave packet ) , the dynamics will start showing irreversible behavior arising from coarse graining ( as distict from irrreversibility from instability ) . consequently , entropy should increase in this regime . in fig . 4a , we plot the von neumann entropy for the dynamics of ( 3.5 ) . we can see three qualitatively different regimes : i. the liouville regime : the entropy is constant and the dynamics is time reversible . the decohering regime : the entropy keeps increasing due to coarse graining . the finite size regime : due to the bounded nature of the phase space , the entropy shows saturation . our result from quantitative analysis seems to confirm the qualitative description of zurek and paz @xcite who used the inverted harmonic oscillator potential as a generic source of instability . since the phase space in their model is not bounded they do not see regime iii . similar features appear in the quantum cap map ( fig . 4b ) in this case , the full quantum dynamics can be calculated in a simple way . resemblance with the result of a classical rotor with noise is obvious . however , in this case , the stable entropy is smaller than the maximum value which may be explained as a finite ( phase space ) size effect . + * acknowledgement * we thank drs . shmul fishman and juan pablo paz for explaining their work and drs . ed ott and richard prange for general discussions . research is supported in part by the national science foundation under grant phy91 - 19726 . blh gratefully acknowledges support from the general research board of the graduate school of the university of maryland and the dyson visiting professor fund at the institute for advanced study , princeton . + * figure captions * + * figure 1 * the linearized entropy ( with reversed sign ) @xmath104 is plotted here as a function of time . if there is no environment , the entropy is constant for both hyperbolic and elliptic cases , indicating the purity of the state . for the hyperbolic map , even though classically this system is strongly chaotic , the corresponding quantum system does not show chaotic behavior . this situation changes drastically when the system interacts with a thermal bath : entropy keeps increasing due to coarse graining . note that in the hyperbolic case ( solid line ) the rate of entropy increase is greater than in the elliptic case ( dotted line ) . n = 50 is used here ( also in fig.2 ) . + * figure 2 * the mean phase space point displacement is shown . when there is no environment ( dotted line ) , the system shows recurrence in both hyperbolic ( a ) and elliptic ( b ) cases . in the presence of an environment , the hyperbolic map loses the recurrence behavior ( solid line ) under a gaussian noise with @xmath105 and maintains a near - constant value , indicating the ergodicity of the classical map . on the other hand , the ellptic map still shows recurrence with the same amount of noise , suggesting classical periodicity . + * figure 3 * @xmath104 ( solid line , left scale ) and @xmath106 ( dashed line , right scale ) are plotted against time for @xmath107 and @xmath108 . the upper solid line and the lower dashed line correspond to the case when there is noise , with @xmath109 . as the noise strength increases to @xmath110 , the decoherence time shortens , and @xmath104 decays rapidly ( the lower solid line ) . this accompanies the increase of diffusive behavior in @xmath106 ( upper dashed line ) . + * figure 4 * the von neumann entropy is plotted versus time for ( a ) the quantum kicked rotor with an environment . here , @xmath108 , @xmath105 and @xmath111 . entropy stays at zero ( reversible dynamics ) until a transition regime , after which the dynamics becomes irreversible . ( b ) the quantum cat map , with the same parameters and the same amount of noise . we see the same qualitative feature as in the qkr case . b. l. hu , j. p. paz and y. zhang , phys . d45 , 2843 ( 1992 ) ; b. l. hu , j. p. paz and y. zhang , phys . rev . d47 , 1576 ( 1993 ) ; b. l. hu and yuhong zhang , in _ quantum dynamics of chaotic systems _ eds . j. m. yuan , d. h. feng and g. m. zaslavsky ( gordon and breach science publishers , langhorne , 1993 ) . a. tameshitit and j. e. sipe , in _ quantum dynamics of chaotic systems _ eds j. m. yuan , d. h. feng , g. m. zaslavsky ( gordon and breach science publishers , langhorne , 1993 ) . a45 , 8280 ( 1992 ) ; a47 , 1697 ( 1993 ) f. g. casati , b. v. chirikov , f. m. izrailev , and j. ford , in _ stochastic behavior in classical and quantum hamiltonian systems _ , lecture notes in physics vol.93 , eds f. g. casati , j. ford ( springer - verlag , berlin , 1979 ) . s. fishman , d. r. grempel , and r. e. prange , phys . * 49 * , 509 ( 1982 ) ; phys . rev . * a36 * , 289 ( 1987 ) ; d. r. grempel , r. e. prange , and s. fishman , phys . lett . * 49 * , 833 ( 1982 ) ; phys . rev . * a29 * , 1639 ( 1984 ) . w. h. zurek , phys . d24 , 1516 ( 1981 ) ; d26 , 1862 ( 1982 ) ; e. joos and h. d. zeh , z. phys . b59 , 223 ( 1985 ) ; a. o. caldeira and a. j. leggett , phys . a31 , 1059 ( 1985 ) ; w. g. unruh and w. h. zurek , phys . d40 , 1071 ( 1989 ) ; w. h. zurek , prog . 89 , 281 ( 1993 ) ; w. h. zurek and j. p. paz , phys . lett . * 72 * , 2508 ( 1994 ) .

decoherence in quantum systems which are classically chaotic is studied . it is well - known that a classically chaotic system when quantized loses many prominent chaotic traits . we show that interaction of the quantum system with an environment can under general circumstances quickly diminish quantum coherence and reenact some characteristic classical chaotic behavior . we use the feynman - vernon influence functional formalism to study the effect of an ohmic environment at high temperature on two classically - chaotic systems : the linear arnold cat map ( qcm ) and the nonlinear quantum kicked rotor ( qkr ) . features of quantum chaos such as recurrence in qcm and diffusion suppression leading to localization in qkr are destroyed in a short time due to environment - induced decoherence . decoherence also undermines localization and induces an apparent transition from reversible to irreversible dynamics in quantum chaotic systems . body of paper c i u plus 1000pt minus 1000pt # 1 # 1= to # 1= to = 8.5 in = 6.5 in = 0.in = 0.in = 0.in addtoresetequationsection

the observation and theoretical modeling of weak polarization signatures in spectral lines are opening a new window on the investigation of the weak magnetism of the solar atmosphere ( see , e.g. , the recent reviews by @xcite ) . to this aim , it is important to investigate carefully within the framework of the quantum theory of polarization ( e.g. , landi deglinnocenti 1983 ) the observable effects of the atomic polarization of the energy levels involved in the line transitions of interest , including their subtle modification by the presence of magnetic fields . in this respect , in a recent letter ( @xcite , hereafter paper i ) , we reported on an interesting property of the polarizability of the levels of the d@xmath1 line of : in spite of the fact that those levels can both be aligned , atomic _ alignment _ is a condition of population imbalances between the zeeman substates of a level , such that the total populations of substates with different values of @xmath6 are different . one speaks instead of atomic _ orientation _ when , for a given value of @xmath6 , the substates labeled by @xmath7 and @xmath8 have different populations . see , e.g. , @xcite , or the recent review by @xcite . ] when proper account is taken of the additional quantum numbers introduced by the hyperfine structure ( hfs ) of , the alignment is drastically reduced for fields larger than @xmath2 , and practically vanishes for @xmath9 , _ irrespective _ of the relative directions of the magnetic field and of the incident radiation . accordingly , any contribution to the linear polarization in the core of d@xmath1 that arises from atomic alignment is suppressed for magnetic fields sensibly larger than @xmath2 , so the only expected linear - polarization signal for such field strengths must be due to the transverse zeeman effect ( see fig . 2 of paper i ; the reader should note how the stokes-@xmath10 signature of single - scattering events taking place in the presence of a vertical magnetic field changes from antisymmetric for @xmath11 to symmetric for @xmath12 ) . in paper i , we were concerned mainly with a detailed calculation of the polarizability of the levels , and with the consequences it bears for our understanding of the magnetic - field distribution and topology in the solar atmosphere . in the present work , we focus instead on the investigation of the atomic physics that is behind the polarization properties of those lines . to this end , we follow our approach of paper i , and apply the quantum theory of line formation in the limit of complete frequency redistribution ( crd ) and in the collisionless regime , as developed by @xcite , to investigate the statistical equilibrium ( se ) of an ensemble of atoms illuminated by anisotropic radiation ( see also @xcite ) . the hypothesis of crd corresponds to the requirement that the incident radiation field coming from the underlying photosphere , and illuminating the scattering atoms , be spectrally flat over an interval much larger than the energy separation between atomic levels whose wavefunctions sensibly overlap ( leading to the phenomenon of quantum interferences ) . in the case of the d@xmath1 and d@xmath13 lines forming in the solar atmosphere , this is a good assumption only if we neglect the quantum interferences between the upper levels of d@xmath1 and d@xmath13 . more specifically , these are interferences between the levels p@xmath0 and p@xmath5 pertaining to the same atomic term . whereas the role of these so - called _ super - interferences _ is important for a correct interpretation of line polarization in the wings of d@xmath1 and d@xmath13 , the line - core polarization of those lines , which was the subject of the investigation of paper i , is expected to be largely unaffected by them . in 2 , we summarize our qualitative description of the polarization properties of the levels of ( see paper i ) , and introduce some useful new concepts and terminology . in [ sec : polar.quant ] , we put those concepts on a more quantitative basis , and provide an algebraic proof that the alignment of the levels of d@xmath1 is suppressed when a magnetic regime of complete decoupling of the angular momenta @xmath3 and @xmath4 is reached in the excited state p@xmath5 . finally , in the conclusive section , we provide further arguments to illuminate this interesting phenomenon . the stable isotope of sodium has a nuclear spin @xmath14 , therefore we must take into account the role of hfs in the solution of the se problem of . hfs was already indicated by @xcite as the only possible mechanism allowing for the existence of atomic alignment in the levels of the d@xmath1 line . in fact , levels with total angular momentum @xmath15 can not be aligned , whereas both hyperfine levels @xmath16 and @xmath17 , into which a level @xmath15 splits in the coupling process with a nuclear spin @xmath14 , can be aligned . for this reason , it is convenient to introduce the concept of _ intrinsic polarizability _ ( ip ) , for those levels whose values of @xmath18 allow the presence of atomic alignment , and of _ extrinsic polarizability _ ( ep ) , for those levels that can carry atomic alignment only through the `` internal '' @xmath19 quantum numbers , because of the presence of hfs . ( what distinguishes the roles of @xmath18 and @xmath19 as quantum numbers , in this context , is the assumption we made at the beginning , that quantum interferences can exist only between different @xmath19 levels , but not between different @xmath18 levels . ) in this sense , we can speak of ep only in the cases of @xmath20 and @xmath15 . therefore , both levels of the d@xmath1 line of are ep , whereas the upper level of d@xmath13 is ip , because @xmath21 . this nomenclature has a direct link with the physics of the interaction processes of the atom with the incident radiation field . we speak of ip of an atomic level when this level has the possibility of absorbing the multipole order @xmath22 of the polarization tensor of the incident radiation field ( @xcite ; see also @xcite ) , expressed in the irreducible spherical tensor representation , @xmath23 ( @xmath24 ) . in particular , if we assume that the incident radiation field is unpolarized , and has cylindrical symmetry around the local solar vertical through the scattering centre , only the multipole orders @xmath25 ( intensity ) and @xmath22 ( anisotropy ) are present in the radiation - field tensor . in this case , it is found that an ep level can only absorb the multipole order @xmath25 , so there is no atomic polarization directly induced by the incident radiation field . any atomic alignment ( @xmath22 , in the irreducible spherical representation of the density matrix ) that such level can show when proper account is taken of its sub - structure associated with hfs can only come from the transfer of atomic alignment from other atomic levels that are instead ip . in the case of , for instance , if the two levels of d@xmath1 were isolated ( i.e. , not radiatively connected with other levels in the atom ) , no atomic alignment could be created , even accounting for the presence of hfs . because of the presence of the upper level p@xmath5 of d@xmath13 in the se problem of , instead , transfer of atomic polarization from such ip level to the lower level of d@xmath1 can occur , via the radiative de - excitation associated with the formation of the d@xmath13 line . once ep has been created in the level s@xmath0 , this can be transferred via absorption processes to the upper level of d@xmath1 as well . in our case , the two levels of d@xmath1 manifest their ep because of the alignment induced onto the corresponding hfs levels , with @xmath26 ( see fig . 1 of paper i ; also fig . [ fig : alignment ] introduced below ) . on the other hand , the transfer of atomic alignment from an ip level to an ep level can be inhibited under particular conditions . for the three - level model of the atom considered here , and for the prescribed radiation field , we determined that the atomic polarization in the two ep levels vanishes when the ip level p@xmath5 reaches the regime of the complete paschen - back effect , in which the zeeman splittings of the @xmath19 levels due to the local magnetic field become much larger than the hfs separations between those levels . in this regime , the hfs coupling of the electronic and nuclear angular momenta , @xmath3 and @xmath4 , of the atoms in the excited state p@xmath5 , is `` relaxed '' by the presence of the strong magnetic field , through the electronic zeeman effect .. ] [ to understand the meaning of such decoupling process , we must observe that , in the regime of complete paschen - back effect , and assuming the direction of @xmath27 as the quantization axis , @xmath28 becomes a conserved quantity ( rigorously , an element of the complete set of commuting observables of the atomic system ) , along with @xmath29 . because @xmath30 must be conserved as well , both @xmath31 and @xmath32 become good quantum numbers , so the eigenvectors of the atomic system take the form @xmath33 . ] the inhibition of the transfer of atomic alignment from an ip level to an ep level for increasing magnetic strengths is clearly illustrated by the results presented in paper i. in figure [ fig : alignment ] , we reproduce similar results . we calculated the atomic alignment of the levels of d@xmath1 and d@xmath13 for magnetic strengths between @xmath34 and @xmath35 . a vertical field ( i.e. , aligned along the symmetry axis of the radiation cone from the photosphere illuminating the scattering atom ) was chosen , in order to clarify that the obtained trend of the alignment against the magnetic field strength is _ not _ due to hanle - effect depolarization . as we see , atomic alignment in the levels of d@xmath1 is drastically reduced for fields larger than @xmath2 , and practically vanishes for fields of the order of @xmath36 or larger . in figure [ fig : orientation ] , we show analogous results for the atomic orientation ( @xmath37 , in the irreducible spherical representation of the density matrix ; see note [ note : atompol ] for a description of atomic orientation ) , for magnetic strengths between @xmath38 and @xmath39 . also in this case , the orientation of the levels of the d@xmath1 line practically vanishes for @xmath9 . ( we note that , for @xmath40 , the level p@xmath5 approaches the regime of complete paschen - back effect . in fact , for @xmath41 , the typical zeeman splitting is already 10 times larger than the typical hfs separation for that level . ) on the other hand , as suggested by the work of @xcite concerning optical - pumping processes in cadmium , a sufficient condition for the vanishing of atomic alignment in the ep level is that the hfs frequency separation of the ip level be negligible with respect to the einstein @xmath42-coefficient of the atomic transition . this condition is very general , as it holds regardless of the magnetic - field strength ( in particular , it is valid also for zero magnetic fields ) . in the case of , the hfs frequency separation of the ip level p@xmath5 is comparable with the einstein @xmath42-coefficient of the d@xmath13 line . for this reason , transfer of atomic alignment from the ip upper level to the ep lower level is possible when @xmath43 , since @xmath3 and @xmath4 are still significantly coupled in the ip level p@xmath5 ( cf . also paper i , end of 3 ) . these results suggest that the inhibition of the transfer of atomic alignment from an ip level to an ep level should be regarded as an aspect of the so - called _ principle of spectroscopic stability _ applied to the ip level : whenever the hyperfine structure of the ip level becomes negligible , whether because a magnetic field is present which is strong enough to reach the complete paschen - back regime for that level , or because the hfs separation of that level is much smaller than its radiative width , the transfer of alignment from the ip level to the ep level is inhibited , so the ep level behaves as if the atomic hfs were not present at all . the reason for this is hidden in the complexity of the se problem , and it is addressed in the following section . we consider an ip level with total electronic angular momentum @xmath18 . we assume that this level can only interact with ep levels in the atom . beyond this restriction , the atomic system can be arbitrary , so the following formalism applies also for atoms other than . if a nuclear spin is present , of angular momentum @xmath44 , the density matrix for the ip level in the irreducible spherical tensor representation is ( cf . @xcite ) @xmath45 we write explicitly @xmath46 where @xmath47 in the previous equation , @xmath48 are clebsh - gordan coefficients , which can be expressed in terms of @xmath49 symbols as @xmath50 substitution of eq . ( [ eq : ket ] ) into eq . ( [ eq : rhoff ] ) , using eq . ( [ eq : cg ] ) , gives @xmath51 we now make the assumption that the electronic spin and the nuclear spin are decoupled ( or very weakly coupled ) when the atom is in the ip level . as anticipated in the previous section , this can be the case if the hfs separation is much smaller than the natural width of that level , or , in the presence of a magnetic field , if the level is in the regime of complete paschen - back effect . in either case , the atomic density matrix for the ip level can be factorized as @xmath52 we introduce at this point the formalism of the irreducible spherical tensors for both @xmath53 and @xmath54 , @xmath55 substitution of eq . ( [ eq : rho.fact ] ) into eq . ( [ eq : rhoff.1 ] ) , using eqs . ( [ eq : rhoj ] ) and ( [ eq : rhoi ] ) , gives @xmath56 finally , this equation must be substituted into eq . ( [ eq : rhokq ] ) . we then obtain an expression which involves the contraction over all magnetic quantum numbers of a product of five @xmath49 symbols . this contraction can be evaluated using , e.g. , eq . ( 14 ) , p. 456 , of @xcite , yielding the expression @xmath57 as a particular case , if nuclear polarization is absent ( @xmath58 ) , eq . ( [ eq : rhokq.ji ] ) reduces to @xmath59 in this case , the ( electronic ) atomic polarization of the @xmath18 level translates _ directly _ ( i.e. , with the same @xmath60 and @xmath10 ) into the atomic polarization of the @xmath61 pair . as an application of the former development , we consider a two - level atom @xmath62 endowed with hfs . neglecting stimulated emission for simplicity , the se equations for the two levels read ( @xcite ) @xmath63 and @xmath64 to understand how atomic polarization is created in an ep level , assuming that the other level is ip , we must consider explicitly the expressions of the transfer rates for absorption and spontaneous emission processes , respectively , @xmath65 and @xmath66 the relaxation rate due to spontaneous emission , @xmath67 , is completely diagonal , so it can only relate each of the elements @xmath68 to itself . the relaxation rate due to absorption , @xmath69 , is a necessary ingredient of this demonstration . however , the only fact we will rely upon is the presence in that rate of the @xmath70 symbol @xmath71 the rate @xmath72 , in both eqs . ( [ eq : seu ] ) and ( [ eq : sel ] ) , describes magnetic and hfs depolarization . the importance of this rate is that it accounts for the conversion mechanism of atomic alignment ( @xmath22 ) into atomic orientation ( @xmath37 ) discussed by @xcite . this is related to the fact that , in the algebraic expression of the rate ( not given here ) , @xmath73 and @xmath74 ( cf . [ [ eq : seu ] ] ) or @xmath75 and @xmath76 ( cf . [ [ eq : sel ] ] ) can have different parity . if the radiation illuminating the atom is not circularly polarized ( which is the case of the present discussion ) , this conversion mechanism is the only process capable of creating orientation in the atomic system ( see , e.g. , @xcite ) . on the other hand , this mechanism is only effective when quantum interferences between different @xmath19 levels are important , which corresponds to a regime of magnetic fields such that level crossing between @xmath19 levels can occur . therefore , for magnetic fields such that the upper level approaches the regime of complete paschen - back effect ( @xmath40 ) , the conversion of atomic alignment into atomic orientation is drastically reduced ( see fig . [ fig : orientation ] ) . for this reason , the role of the rate @xmath72 is not of immediate concern for the following arguments . we first consider the case in which @xmath77 is the ip level . when this level is in a regime of very weak coupling between @xmath3 and @xmath4 ( whether because the hfs separation is much smaller than @xmath78 , or because a magnetic field is present that is strong enough to establish a regime of complete paschen - back effect in that level ) , the irreducible components of the density matrix for that level , @xmath79 , can be written according to eq . ( [ eq : rhokq.ji ] ) . it is then found that the double summation over @xmath80 and @xmath81 in eq . ( [ eq : sel ] ) can be performed algebraically . this corresponds to a contracted product of a @xmath82 symbol with three @xmath70 symbols , which is evaluated using , e.g. , eq . ( 36 ) , p. 471 , of @xcite . the result is that the overall contribution of the transfer rate @xmath83 to eq . ( [ eq : sel ] ) is proportional to the product ( notice that @xmath84 ) @xmath85 since @xmath86 for the ep level , the former product vanishes unless @xmath87 . in particular , to create alignment in the ep lower level ( @xmath88 ) , either both electronic and nuclear orientations ( @xmath89 ) or only nuclear alignment ( @xmath90 ) must be present when the atom is in the excited state @xmath77 . to convince ourselves that these conditions can not be met , let us assume that initially ( i.e. , before irradiation ) atomic polarization is completely absent , in particular @xmath91 . since the level @xmath92 is ep , it is only sensitive ( through the relaxation rate @xmath69 ; cf . the @xmath70 symbol [ [ eq:6j.ra ] ] ) to the intensity of the incident radiation field , so lower - level polarization ( @xmath93 ) can not be created directly by irradiation . therefore , when irradiation begins , from eqs . ( [ eq : seu ] ) and ( [ eq : ta ] ) we see that the prescribed radiation field ( @xmath94 ) can only induce atomic alignment in the upper level ( besides populating it ) , because of the selection rule introduced by the @xmath49 symbol in eq . ( [ eq : ta ] ) . since the atom was initially unpolarized , and by assumption the electronic and nuclear systems are decoupled in the excited state , @xmath77 , the atomic alignment of the upper level can only be electronic . in fact , electric - dipole transitions can not affect the nuclear system , so the nuclear zeeman sublevels remain naturally populated in all cases of interest , even if strong @xmath18-@xmath44 coupling is present in the ep level . from this argument , we conclude that @xmath95 , and @xmath96 , as a result of the excitation process . as anticipated above , we can dismiss the alignment - to - orientation conversion mechanism as a possible source of upper - level orientation ( @xmath97 ) , because of the assumed regime of weak @xmath18-@xmath44 coupling . also , upper - level alignment ( @xmath98 ) can not be transferred in the de - excitation process , because the product ( [ eq : contr1 ] ) vanishes . therefore , nuclear polarization can never be created in this regime , and eq . ( [ eq : rhokq.ji.0 ] ) applies to the upper level . under these conditions , the product ( [ eq : contr1 ] ) vanishes identically for @xmath93 , so lower - level polarization can not be created . this is in agreement with the results of paper i , and of figures 1 and 2 in this paper . in summary , when the ip upper level is in a regime of very weak @xmath18-@xmath44 coupling , the creation of atomic alignment in the ep lower level ( @xmath88 ) by transfer of atomic alignment from the ip upper level ( @xmath98 ) is inhibited . in the case of , this implies that the ground level @xmath99 can not be aligned , and consequently also the upper level p@xmath0 of d@xmath1 must have zero alignment , as illustrated in paper i and by figure [ fig : alignment ] in this paper . lower - level orientation ( @xmath100 ) can in principle be created directly by irradiation , if @xmath101 , although it requires that the incident radiation be circularly polarized ( @xmath102 ; cf . the @xmath70 symbol [ [ eq:6j.ra ] ] ) . in our case , because of the prescribed radiation field , lower - level orientation can only be created by the transfer of atomic orientation from the upper level ( @xmath97 ) , which is not inhibited in principle . on the other hand , the alignment - to - orientation conversion mechanism in the upper level becomes very inefficient for very weak @xmath18-@xmath44 coupling ( see fig . [ fig : orientation ] ) , so also upper - level orientation can only be created if the incident radiation field is circularly polarized . we checked our conclusion that eq . ( [ eq : rhokq.ji.0 ] ) must apply to the ip upper level , in the regime of very weak @xmath18-@xmath44 coupling , against the numerical results of paper i ( cf . also fig . [ fig : alignment ] in this paper ) . in particular , we verified that the ratio of the quantities @xmath103 and @xmath104 for the upper level @xmath105 of ( @xmath14 ) in the strong - field limit ( @xmath106 ; see rightmost panels of fig . 1 in paper i ; also fig . [ fig : alignment ] in this paper ) is correctly reproduced by eq . this equation also accounts for the curious vanishing of the quantity @xmath107 in the same limit , which is due to the fact that @xmath108 vanishes identically because of the ( non - trivial ) nullity of the @xmath70 symbol in eq . ( [ eq : rhokq.ji.0 ] ) . within the same approximation of the two - level atom @xmath62 , we now assume that the decoupling of @xmath3 and @xmath4 is reached first in the lower level , while strong coupling is still present in the upper level . this time we assume that @xmath92 is the ip level , whereas @xmath77 is the ep level . since we assumed that the lower level is in a regime of very weak coupling between @xmath3 and @xmath4 , the irreducible components of the density matrix for that level , @xmath109 , can be written according to eq . ( [ eq : rhokq.ji ] ) . it is then found that the double summation over @xmath110 and @xmath111 in eq . ( [ eq : seu ] ) can be performed algebraically . this corresponds to a contracted product of two @xmath82 symbols with two @xmath70 symbols that is evaluated using , e.g. , eq . ( 37 ) , p. 471 , of @xcite . the result is that the overall contribution of the transfer rate @xmath112 to eq . ( [ eq : seu ] ) is proportional to the sum @xmath113 since @xmath114 for the ep level , this sum is limited to @xmath115 . again , we assume that the atomic polarization is absent before irradiation . because the lower level is ip , lower - level alignment can be created when irradiation begins . however , since @xmath3 and @xmath4 are decoupled in the lower level , nuclear polarization remains zero ( @xmath96 ) , so all the alignment of the lower level must be electronic ( @xmath116 ) . under these conditions , the sum ( [ eq : contr2 ] ) is restricted to @xmath117 only , because the first @xmath82 symbol in the sum ( [ eq : contr2 ] ) vanishes for @xmath118 unless @xmath119 is an odd integer . therefore atomic polarization in the upper level ( @xmath120 ) can never be created , because of nullity of the second @xmath82 symbol in the sum ( [ eq : contr2 ] ) . this shows , in particular , that the concept of ep is also valid for an upper level . in this case , the ep upper level is sensitive to the anisotropy of radiation ( @xmath121 ) through the transfer rate @xmath112 , but nonetheless creation of alignment in the upper level through the absorption of that anisotropy is not possible when the ip lower level is in a regime of very weak @xmath18-@xmath44 coupling , because of the selection rules implied by the sum ( [ eq : contr2 ] ) . upper - level orientation ( @xmath122 ) is not excluded in principle , if @xmath123 , although it can only be created by transfer of atomic orientation from the lower level ( @xmath124 ; see eq . [ [ eq : contr2 ] ] ) . however , when the lower level is in the regime of weak @xmath18-@xmath44 coupling , its orientation can only be due to the presence of circular polarization in the incident radiation field . in this paper we demonstrated analytically that the presence of @xmath18-@xmath44 coupling in the ip level p@xmath5 of is a necessary condition for the transfer of atomic alignment from that level to the ep ground level s@xmath0 . we based our demonstration on the quantum theory of line formation , as developed by @xcite , and assuming unpolarized incident radiation without spectral structure over the frequency intervals encompassing the hfs components of the atomic transitions of interest . under these conditions , we relied on the argument that nuclear polarization can not be created in an atom having only one ip level , if @xmath3 and @xmath4 are completely decoupled in that level , because the assumed incident radiation can not induce directly nuclear transitions in the atom . it follows , from the results of 3 , that atomic polarization can not be created in the ep levels , when @xmath3 and @xmath4 are completely decoupled in the ip level . we can further strengthen this argument by showing that the possibility of nuclear polarization actually resides in the presence of @xmath18-@xmath44 coupling in the ip level , whereas the presence of @xmath18-@xmath44 coupling in the ep level is not relevant . to this purpose , we repeated the calculation of figure [ fig : alignment ] after artificially zeroing the hfs separation in the level s@xmath0 of . the results of this calculation are shown in figure [ fig : small.hfs ] . since @xmath3 and @xmath4 are completely decoupled in the `` modified '' level s@xmath0 , the factorization ( [ eq : rho.fact ] ) always applies to this level . on the other hand , any atomic alignment in this modified ep level requires the presence of nuclear polarization ( cf . [ [ eq : rhokq.ji ] ] ) , since the electronic angular momentum of the level is @xmath15 . such nuclear polarization in the ep level s@xmath0 can only come from the atomic polarization of the ip level p@xmath5 ( which is transferred to the ep level via radiative de - excitation ) , since it is not possible for the prescribed radiation field to directly create atomic polarization in the ep level . from the results of figure [ fig : small.hfs ] , it is evident that the nuclear polarization in the modified level s@xmath0 vanishes when the regime of complete paschen - back effect is reached in the level p@xmath5 , and eq . ( [ eq : rho.fact ] ) also applies to that level . comparing the results of figures [ fig : alignment ] and [ fig : small.hfs ] , we see that the suppression of @xmath18-@xmath44 coupling in the level s@xmath0 does not alter substantially the se of the model atom . on the basis of these arguments , it seems safe to conclude that , even in the real case , nuclear polarization can not be created in the atom , when the regime of complete paschen - back effect is reached in the ( only ) ip level . finally , we must emphasize that the presence of atomic alignment in the upper level of the d@xmath1 line induces a characteristic _ antisymmetric _ signature in the core of the stokes-@xmath10 profile resulting from the scattering of the anisotropic radiation illuminating the atom ( see fig . 2 of paper i ) . this applies particularly to the optically thin `` prominence case '' considered in paper i , where the scattering polarization is solely due to the emission events following atomic excitation by the anisotropic radiation . currently we are investigating to what extent such antisymmetric signature can be modified through dichroism and radiative transfer effects , because of the presence of atomic alignment in the ground level of ( see , e.g. , @xcite ; for the observable effects of dichroism and ground - level polarization on the 10830 multiplet , see @xcite ) . in this respect , it is interesting to note that spectropolarimetric observations of the d - lines obtained with thmis in quiet regions close to the solar limb show an antisymmetric signature in the fractional linear polarization @xmath125 of the d@xmath1 line ( see fig . 1 of @xcite , which was adapted from @xcite ; see also @xcite ) . there seems to be an indication of a similar antisymmetric signature in the @xmath125 atlas of @xcite , which was obtained with the polarimeter zimpol - ii attached to the gregory coud telescope ( gct ) of irsol at locarno ( italy ) . on the contrary , analogous observations that @xcite had obtained previously with the polarimeter zimpol - i attached to the mcmath - pierce facility of the national solar observatory show almost symmetric profiles with a central positive peak ( see their fig . 3 ) . observations of quiet solar regions were obtained in march 1998 ( i.e. , two years earlier than the above mentioned thmis observations ) , when the sun had not yet reached the maximum of its magnetic activity cycle . ] as shown in paper i , for single - scattering events , one should expect a _ shape of the stokes-@xmath10 signature in the core of the d@xmath1 line for magnetic fields @xmath12 ( see fig . 2 of paper i ; note that such symmetric signature would change its sign if we considered , say , a horizontal canopy - like field instead of the vertical field assumed for the calculation of that figure ) . nevertheless , we think that the above mentioned linear - polarization observations of the d@xmath1 line in very quiet regions of the solar disk , with thmis and zimpol , both have the same physical origin , i.e. , atomic alignment in the levels of the d@xmath1 line . now that we understand how the ground level of becomes polarized , and how its polarization is modified by the presence of weak magnetic fields , it will be worthwhile to investigate the sodium polarization problem by means of full radiative transfer simulations , taking also into account the quantum interferences among the two upper levels of the `` enigmatic '' d - lines . the authors are grateful to philip judge and arturo lpez ariste ( both of hao ) , and to rafael manso sainz ( universit di firenze , italy ) , for reading the manuscript , and for helpful comments and suggestions . they also thank maurizio landi deglinnocenti ( italian council for national research ) for helpful discussions about the principle of spectroscopic stability during the early stages of this work . thanks are also due to jan stenflo and co - workers for some useful discussions and clarifications concerning their spectropolarimetric observations . one of the authors ( j.t.b . ) acknowledges the support of the spanish ministerio de ciencia y tecnologa through project aya2001 - 1649 . gandorfer , j. 2000 , _ the second solar spectrum : a high spectral resolution polarimetric survey of scattering polarization at the solar limb in graphical representation _ , vdf hochschulverlag ag an der eth zrich

in a recent letter ( @xcite ) , we showed the remarkable result that the atomic alignment of the levels p@xmath0 and s@xmath0 of the d@xmath1 line of is practically destroyed in the presence of magnetic fields sensibly larger than @xmath2 , irrespectively of the field direction . in this paper , we demonstrate analytically that this property is a consequence of the decoupling of the electronic and nuclear angular momenta , @xmath3 and @xmath4 , in the excited state p@xmath5 , which is achieved when the zeeman splitting from the local magnetic field becomes much larger than the typical hyperfine separation for that level .

the question of the origin of @xmath0 violation remains one of the outstanding puzzles in particle physics . although @xmath0 violation has now been seen in a number of processes in the kaon and @xmath6-meson systems @xcite , it is still far from clear whether its explanation lies exclusively within the picture provided by the standard model @xcite . to pin down the sources of @xmath0 violation , it is essential to observe it in many other processes . hyperon nonleptonic decays provide an environment where it is possible to make additional observations of @xmath0 violation @xcite . currently , there are @xmath0-violation searches in such processes being conducted by the hypercp ( e871 ) collaboration at fermilab . its main reactions of interest are the decay chain @xmath7 @xmath8 and its antiparticle counterpart @xcite . a different , but related , system also being studied by hypercp involves the spin-@xmath9 hyperon @xmath10 , namely the sequence @xmath11 @xmath8 and its antiparticle process @xcite . for each of these decays , the decay distribution in the rest frame of the parent hyperon with known polarization @xmath12 has the form @xmath13 where @xmath14 is the final - state solid angle , @xmath15 is the unit vector of the daughter - baryon momentum , and @xmath16 is the parameter relevant to the @xmath0 violation of interest . in the case of @xmath17 the hypercp experiment is sensitive to the _ sum _ of @xmath0 violation in the @xmath18 decay and @xmath0 violation in the @xmath19 decay , measuring @xcite @xmath20 where @xmath21 are the @xmath0-violating asymmetries in @xmath22 and @xmath23 respectively . similarly , the observable it measures in @xmath24 is @xmath25 @xcite . on the theoretical side , @xmath0 violation in @xmath26 and @xmath27 has been extensively studied @xcite . in contrast , the literature on @xmath0 violation in @xmath18 decays is minimal , perhaps the only study being ref . @xcite which deals with the partial - rate asymmetry in @xmath28 there is presently no data available or experiment being done on this rate asymmetry . in view of the upcoming measurement of @xmath29 by hypercp , it is important to have theoretical expectations of this observable . clearly , the information to be gained from @xmath29 will complement that from @xmath30 . since the estimates of @xmath2 and @xmath31 within and beyond the standard model ( sm ) have been updated very recently in refs . @xcite , in this paper we focus on @xmath1 . we begin in sec . [ observables ] by relating the observables of interest in @xmath22 to the strong and @xmath0-violating weak phases in the decay amplitudes . we discuss the role played by final - state interactions in this decay , which not only affect @xmath1 , but also cause its partial - rate asymmetry to be nonvanishing , thereby providing another @xmath0-violating observable . in sec . [ strong_phases ] , we employ heavy - baryon chiral perturbation theory ( @xmath32pt ) to calculate @xmath33- and @xmath34-wave amplitudes for baryon - meson scattering in channels with isospin @xmath35 and strangeness @xmath36 we use the derived amplitudes in a coupled - channel @xmath37-matrix formalism to determine the strong parameters needed in evaluating the @xmath0-violating asymmetries . in sec . [ a_sm ] , we estimate the asymmetries within the standard model . working in the framework of @xmath32pt , we calculate the weak phases by considering factorizable and nonfactorizable contributions to the matrix elements of the leading penguin operator . subsequently , we compare the resulting @xmath1 with @xmath2 , which was previously evaluated , as both asymmetries appear in @xmath29 . in sec . [ a_np ] , we address contributions to the @xmath0-violating asymmetries from possible new physics , taking into account constraints from @xmath0 violation in the kaon system . specifically , we consider contributions induced by chromomagnetic - penguin operators , which in certain models can be enhanced compared to the sm effects . sec . [ conclusion ] contains our conclusions . the amplitudes for @xmath38 and @xmath39 each contain parity - conserving @xmath33-wave and parity - violating @xmath34-wave components , with the former being empirically known to be dominant @xcite . they are related to the parameters @xmath40 and @xmath41 by @xmath42 where @xmath43 and @xmath44 @xmath45 and @xmath46 are the @xmath33- and @xmath34-wave components , respectively , for the @xmath10 @xmath47 decay . since both @xmath18 and @xmath19 have @xmath48 each of these decays is an exclusively @xmath49 transition . before writing down the amplitudes in terms their phases , we note that the strong phases in @xmath50 are not generated by the strong rescattering of @xmath51 alone . watson s theorem for elastic unitarity @xcite does not apply here , though it does in the cases of @xmath26 and @xmath52 final - state interactions also allow @xmath53 to contribute , yielding additional strong phases as well as weak ones , because the channel @xmath54 is open at the scattering energy @xmath55 since the @xmath56 rates overwhelmingly dominate the @xmath18 width @xcite , we expect other contributions via final - state rescattering to be negligible . the requirements of @xmath57 invariance and unitarity provide us with a relationship between the amplitudes for @xmath58 and its antiparticle counterpart . thus , with @xmath59 denoting the amplitude corresponding to @xmath60 being in a state with orbital angular momentum @xmath61 , we have @xmath62 where @xmath63 is the element of the strong @xmath64-matrix associated with the @xmath61 partial - wave of @xmath65 and only the @xmath35 component of the @xmath66 state is involved in the second term . assuming that the @xmath66 and @xmath51 channels are the only ones open , we can express the @xmath64-matrix as @xcite @xmath67 where @xmath68 is the inelasticity factor and @xmath69 denotes the phase shift in @xmath70 clearly @xmath71 is unitary , and each partial - wave has its own @xmath71 . now , since @xmath68 is expected to be close to and smaller than 1 , it is convenient to introduce a parameter @xmath72 defined by @xmath73 and so @xmath72 is positive and small . consequently , for @xmath74 and @xmath75 , to first order in @xmath76 we have @xcite @xmath77 where @xmath78 and @xmath79 are real , associated with @xmath80 and @xmath81 denote the corresponding weak phases in the @xmath49 amplitudes . putting together the results above , and keeping only the terms at lowest order in small quantities , we obtain @xmath82 where we have made use of the expectation that @xmath83 , @xmath84 , and @xmath85 are also small . unlike the strong phases in @xmath19 and @xmath86 decays , there are no data currently available for @xmath87 , and so we will calculate them here . to estimate the weak phases @xmath88 , we will consider contributions coming from the sm as well as from possible new physics . as for @xmath78 and @xmath79 , we will extract their approximate values from data shortly , under the assumption of no final - state interactions and no @xmath0 violation . now , the presence of the @xmath76 terms with additional weak and strong phases in the decay amplitudes in eq . ( [ pd ] ) implies that the rate of @xmath89 @xmath90 evaluated in the rest frame of @xmath18 , is no longer identical to that of @xmath91 hence these decays yield another @xmath0-violating observable , namely the partial - rate asymmetry @xmath92 it follows that to leading order @xmath93 we will also estimate this asymmetry below . was evaluated under the assumption that @xmath94 since @xmath95 results from the interference of @xmath33-wave amplitudes , a future measurement of it will probe @xmath0 violation in the underlying parity - conserving interactions . we note that the strong parameters entering eq . ( [ deltao ] ) , and the second and third terms in eq . ( [ ao ] ) , are not the strong phases , but @xmath96 . before ending this section , we determine the values of @xmath78 and @xmath79 which are needed in eqs . ( [ ao ] ) and ( [ deltao ] ) , and also in evaluating the weak phases . to do so , we apply the measured values of @xmath16 and @xmath97 , as well as of the masses involved , in the corresponding formulas , as those in eqs . ( [ alpha ] ) and ( [ width ] ) , assuming that the strong and weak phases are zero . the experimental values of @xmath97 for @xmath98 are well determined , but those of @xmath16 are not @xcite . hypercp is currently also measuring @xmath40 , in @xmath89 with much better precision , and has reported @xcite preliminary results of @xmath99 and @xmath100 applying the pdg averaging procedure @xcite to all the experimental results , including the preliminary ones from hypercp , yields the average @xmath101 which we adopt in the following . in the case of @xmath102 we use the data given by the pdg @xcite , and also @xmath103 to project out the @xmath49 amplitudes . thus we extract @xmath104 all in units of @xmath105 , with @xmath106 being the fermi coupling constant . to calculate the strong parameters needed in eq . ( [ ao ] ) , we take a @xmath37-matrix approach @xcite . furthermore , we include the contributions of other @xmath60 states with @xmath35 and @xmath107 namely @xmath108 and @xmath109 , which are coupled to @xmath51 and @xmath66 through unitarity constraints . although at @xmath110 the @xmath108 and @xmath109 channels are below their thresholds , it is important to incorporate their contributions to the open ones . such kinematically closed channels have been shown to have sizable influence on the open ones in some other cases @xcite . the @xmath37 matrix for the four coupled channels can be written as @xmath111 where the subscripts `` o '' and `` c '' refer to open and closed channels , respectively , at @xmath112 thus @xmath113 are all 2@xmath1142 matrices in this case and @xmath115 now , it is convenient to introduce the matrix @xmath116 where @xmath117 is the 2@xmath1142 unit matrix and @xmath118 with @xmath119 being the magnitude of the cm three - momentum in @xmath60 scattering , implying that @xmath120 and @xmath121 are purely imaginary at @xmath112 the elements of @xmath71 in eq . ( [ s ] ) can then be evaluated using @xcite @xmath122 where @xmath123 for the @xmath37-matrix elements , we make the simplest approximation by adopting the partial - wave amplitudes @xmath124 at leading order in chiral perturbation theory , namely @xmath125 before deriving them , we remark that time - reversal invariance of the strong interaction implies @xmath126 the chiral lagrangian that describes the interactions of the lowest - lying mesons and baryons is written down in terms of the lightest meson - octet , baryon - octet , and baryon - decuplet fields @xcite . the meson and baryon octets are collected into @xmath127 matrices @xmath128 and @xmath6 , respectively , and the decuplet fields are represented by the rarita - schwinger tensor @xmath129 , which is completely symmetric in its su(3 ) indices ( @xmath130 ) . the octet mesons enter through the exponential @xmath131 where @xmath132 is the pion - decay constant . in the heavy - baryon formalism @xcite , the baryons in the chiral lagrangian are described by velocity - dependent fields , @xmath133 and @xmath134 . for the strong interactions , the lagrangian at lowest order in the derivative and @xmath135 expansions is given by @xcite @xmath136 \right\rangle \nonumber \\ & & \!\ ! -\,\ , \bar{t}_v^\mu\ , { \rm i}v\cdot{\cal d } t_{v\mu}^ { } + \delta m\ , \bar{t}_v^\mu t_{v\mu}^ { } + { \cal c } \left ( \bar{t}_v^\mu { \cal a}_\mu^ { } b_v^ { } + \bar{b}_v^ { } { \cal a}_\mu^ { } t_v^\mu \right ) \nonumber \\ & & \!\ ! + \,\ , \frac{b_d^{}}{2 b_0^ { } } \left\langle \bar b_v^ { } \left\ { \chi_+^ { } , b_v^ { } \right\ } \right\rangle + \frac{b_f^{}}{2 b_0^ { } } \left\langle \bar b_v^ { } \left [ \chi_+^ { } , b_v^ { } \right ] \right\rangle + \frac{b_0^{}}{2 b_0^ { } } \left\langle \chi_+^ { } \right\rangle \left\langle \bar b_v^ { } b_v^ { } \right\rangle \nonumber \\ & & \!\ ! + \,\ , \frac{c}{2 b_0^{}}\ , \bar t_v^\mu \chi_+^ { } t_{v\mu}^ { } - \frac{c_0^{}}{2 b_0^ { } } \left\langle \chi_+^ { } \right\rangle \bar t_v^\mu t_{v\mu}^ { } \,\,+\,\ , \mbox{$\frac{1}{4}$ } f^2 \left\langle \chi_+^ { } \right\rangle \,\,+\,\ , \cdots \,\,,\end{aligned}\ ] ] where @xmath137 denotes @xmath138 in flavor - su(3 ) space , and we have shown only the relevant terms . in the first two lines , @xmath139 is the spin operator and @xmath140 with further details given in ref . the last two lines of @xmath141 contain @xmath142 with @xmath143 which explicitly breaks chiral symmetry . we will take the isospin limit @xmath144 and consequently @xmath145 the constants @xmath34 , @xmath146 , @xmath147 , @xmath148 , @xmath149 , @xmath150 , @xmath151 are free parameters which can be fixed from data . in the center - of - mass ( cm ) frame , the @xmath33-wave amplitude for @xmath152 with total angular - momentum @xmath153 has the form @xmath154 \hat{k}{}'\cdot\hat{k } + \left [ f_{b\phi\to b'\phi'}^{(p , j=\frac{1}{2 } ) } - f_{b\phi\to b'\phi'}^{(p , j=\frac{3}{2 } ) } \right ] { \rm i}\bm{\sigma}\cdot\hat{k}{}'\times\hat{k } \right\ } \chi_{b}^ { } \,\ , , \nonumber \\\end{aligned}\ ] ] where @xmath155 is the cm energy , @xmath156 and @xmath157 are the pauli spinors of the baryons , @xmath158 and @xmath159 denote the unit vectors of the momenta of @xmath6 and @xmath160 , respectively , and @xmath161 are the partial - wave amplitudes . at lowest order in @xmath32pt , the @xmath162 amplitude arises from the lagrangian in eq . ( [ ls ] ) , and the pertinent diagrams are displayed in fig . [ pwave ] . the amplitudes in the @xmath35 channels are then extracted using the @xmath35 states in eq . ( [ |xp > ] ) and @xmath163 which follow a phase convention consistent with the structure of the @xmath128 and @xmath133 matrices . we write the results as @xmath164 where the expressions for @xmath165 corresponding to the four channels have been collected in appendix [ pd ] . diagrams contributing to the @xmath33-wave @xmath162 amplitude for @xmath152 at leading order in @xmath32pt . in all figures , a dashed line denotes a meson field , a single ( double ) solid - line denotes an octet - baryon ( decuplet - baryon ) field , and each solid vertex is generated by @xmath141 in eq . ( [ ls ] ) . ] since a @xmath34-wave amplitude has to be at least of second order in momentum , @xmath166 , it can not arise from the lagrangian in eq . ( [ ls ] ) alone . also required is the lagrangian involving baryons at second order in the derivative expansion , namely @xmath167 b_v^ { } \,+\ , \frac{1}{2 m_0^{}}\ , \bar{t}{}_v^\mu\ , \bigl[{\cal d}^2-(v\cdot{\cal d})^2 \bigr ] t_{v\mu}^ { } \,\,+\,\ , \cdots \,\,,\end{aligned}\ ] ] where @xmath168 is the octet - baryon mass in the chiral limit , and we have shown only the relevant terms . these are two of the relativistic - correction terms in the @xmath166 lagrangian , and so their coefficients are fixed . in the cm frame , the @xmath34-wave amplitude for @xmath152 has the form @xmath169 \bigl [ \mbox{$\frac{3}{2}$ } \bigl ( \hat{k}{}'\cdot \hat{k}\bigr)^2 - \mbox{$\frac{1}{2}$ } \bigr ] \right . \nonumber\\ & & \hspace*{6em } + \left . \left [ f_{b\phi\to b'\phi'}^{(d , j=\frac{3}{2 } ) } - f_{b\phi\to b'\phi'}^{(d , j=\frac{5}{2 } ) } \right ] \bigl ( 3\hat{k}{}'\cdot\hat{k } \bigr)\ , { \rm i}\bm{\sigma}\cdot\hat{k}{}'\times\hat{k } \right\ } \chi_b^ { } \,\,.\end{aligned}\ ] ] the leading nonzero contribution to this amplitude for @xmath162 comes from diagrams shown in fig . [ dwave ] . the resulting @xmath35 partial - wave amplitudes are given by @xmath170 where the expressions for @xmath171 corresponding to the four channels have also been collected in appendix [ pd ] . diagrams for the leading nonzero contribution to the @xmath34-wave @xmath162 amplitude for @xmath172 each hollow vertex is generated by @xmath173 in eq . ( [ ls ] ) . ] numerically , we adopt the tree - level values @xmath174 and @xmath175 extracted from hyperon semileptonic decays @xcite , as well as @xmath176 from the strong decays @xmath177 after nonrelativistic quark models @xcite , which predict @xmath178 and @xmath179 both well satisfied by the adopted @xmath34 , @xmath146 , and @xmath147 values . ] we also employ @xmath180 @xmath181 value comes from simultaneously fitting the tree - level formulas for the octet - baryon masses and the sigma term , @xmath182 all derived from eq . ( [ ls ] ) , to the measured masses and the empirical value @xcite @xmath183 and the isospin - averaged masses @xmath184 all in units of mev . thus , putting together all the results above and setting @xmath185 from the @xmath33- and @xmath34-wave @xmath71-matrices we obtain @xmath186 which are pertinent to eqs . ( [ ao ] ) and ( [ deltao ] ) . the effects of the closed channels turn out to be significant on @xmath187 and @xmath188 . excluding the @xmath108 and @xmath109 channels would lead to @xmath189 and @xmath190 the closed channels have minor effects on the @xmath34-wave parameters . since the numbers in eq . ( [ deltapd ] ) proceed from the leading nonzero amplitudes in @xmath32pt , part of the uncertainties in these predictions comes from our lack of knowledge about the higher - order contributions , which are presently incalculable . to get an idea of how they might affect our results , we redo the calculation using the one - loop values @xmath191 @xmath192 and @xmath193 @xcite , finding @xmath194 @xmath195 @xmath196 and @xmath197 the differences between the two sets of results then provide an indication of the size of this part of the uncertainties . another part is due to our lack of knowledge about the reliability of our @xmath37-matrix approximation . a comparison of @xmath37-matrix results in @xmath198 scattering with experiment suggests that this approach gives results with the correct order - of - magnitude and sign @xcite . for these reasons , we may conclude that @xmath199 we will employ these numbers in evaluating the asymmetries . to calculate the @xmath0-violating phases , we will work in the framework of heavy - baryon @xmath32pt . the amplitude for the weak decay @xmath58 in the heavy - baryon approach has the general form @xmath201 where @xmath202 is the four - momentum of @xmath128 , and the superscripts refer to the @xmath33- and @xmath34-wave components of the amplitude . in the rest frame of @xmath18 , these components are related to the @xmath43 and @xmath44 amplitudes by @xmath203 we will follow the usual prescription for estimating a weak phase @xcite , namely , first calculating the imaginary part of the amplitude and then dividing it by the real part of the amplitude extracted from experiment under the assumption of no strong phases and no @xmath0 violation . within the sm , the weak interactions responsible for hyperon nonleptonic decays are described by the short - distance effective @xmath204 hamiltonian @xcite @xmath205 where @xmath206 are the elements of the cabibbo - kobayashi - maskawa ( ckm ) matrix @xcite , @xmath207 are the wilson coefficients , and @xmath208 are four - quark operators whose expressions can be found in ref . @xcite . in this case , the weak phases @xmath209 of eq . ( [ ao ] ) proceed from the @xmath0-violating phase residing in the ckm matrix , and its elements appearing in @xmath210 above can be expressed in the wolfenstein parametrization @xcite as @xmath211 at lowest order in @xmath212 . as is well known , @xmath213 transforms mainly as @xmath214 under su(3@xmath215@xmath114su(3@xmath216 rotations . it is also known from experiment that the octet term dominates the 27-plet term @xcite . we , therefore , assume in what follows that within the sm the decays of interest are completely characterized by the @xmath217 , @xmath49 interactions . the leading - order chiral lagrangian for such interactions is @xcite @xmath218 \right\rangle + h_c^{}\ , \bar t_v^\mu\ , \xi^\dagger h \xi\ , t_{v\mu}^ { } \,\,+\,\ , { \rm h.c . } \,\,,\end{aligned}\ ] ] where the 3@xmath1143-matrix @xmath219 selects out @xmath220 transitions , having elements @xmath221 and the parameters @xmath222 contain the weak phases of interest . these phases are induced primarily by the imaginary part of @xmath223 associated with the penguin operator @xmath224 , and this is due to its chiral structure and the relative size of @xmath225 . in order to relate the imaginary part of @xmath222 to @xmath225 , we use the results of ref . @xcite , obtained from factorizable and nonfactorizable contributions . accordingly , we have @xmath226 all in units of @xmath227 from @xmath228 together with @xmath141 , we can derive the diagrams displayed in fig . [ pwave_sm ] , which represent the leading - order contributions to the @xmath33-wave transitions in @xmath229 and yield the amplitudes @xmath230 diagrams representing standard - model contributions to the leading - order @xmath33-wave amplitude for @xmath231 each square represents a weak vertex generated by @xmath228 in eq . ( [ lw_sm ] ) . ] applying eq . ( [ imh ] ) in @xmath232 then leads to @xmath233 where @xmath234 are the central values of @xmath235 in eq . ( [ px , dx ] ) . the uncertainties in these predictions are due to our neglect of higher - order terms that are presently incalculable and to our lack of knowledge on the reliability of the matrix - element calculation . therefore , we assign an error of 100@xmath236 to these ratios , as was similarly done in ref . @xcite for the weak phases in @xmath26 and @xmath52 thus , using @xmath237 and @xmath238 as in ref . @xcite , we obtain @xmath239 the @xmath240 result is comparable in size to that estimated in ref . @xcite using the vacuum - saturation method . value in ref . @xcite . ] turning now to the @xmath34-wave phases , we note that the expression for the @xmath241 term in eq . ( [ i m ] ) implies that @xmath228 , in conjunction with @xmath141 and @xmath173 , can not solely give rise to diagrams for the @xmath34-wave components . rather , the weak lagrangian that can generate the leading nonzero contributions to this term must have the dirac structure @xmath242 which is of @xmath166 . the @xmath34-wave amplitude at @xmath166 can also receive contributions from so - called tadpole diagrams , each being a combination of a strong @xmath243 vertex , generated by a lagrangian having the structure @xmath244 and a @xmath245-vacuum vertex coming from a weak lagrangian of @xmath246 . unfortunately , at present the parameters of these strong and weak lagrangians of @xmath166 are incalculable . the best that we can do is to make a crude estimate based on naive dimensional analysis @xcite . thus , since the lowest - order chiral lagrangian yielding @xmath247 is of @xmath248 , whereas that yielding @xmath249 is of @xmath166 , and since @xmath250 in hyperon nonleptonic decays , we expect that @xmath251 where @xmath252 is the chiral - symmetry breaking scale . it is worth remarking here that for @xmath253 @xcite this naive expectation is compatible with the value of @xmath254 from eq . ( [ px , dx ] ) , in which the @xmath255 number is determined largely by the preliminary data from hypercp @xcite . for these reasons , we make the approximation @xmath256 for the magnitude of the phase , where @xmath257 comes from eq . ( [ phip_sm ] ) . since @xmath258 as quoted in eq . ( [ px , dx ] ) is poorly determined , we take the further approximation @xmath259 for its magnitude in order to estimate @xmath260 . all this leads to @xmath261 the errors that we quote in @xmath81 are obviously not gaussian and simply indicate the ranges resulting from our calculation . putting together the numbers from eqs . ( [ px , dx ] ) , ( [ deltapd ] ) , ( [ phip_sm ] ) , and ( [ phid_sm ] ) in eq . ( [ ao ] ) yields @xmath262 we note that the second term on the right - hand side of eq . ( [ ao ] ) , which would vanish if the @xmath54 rescattering were ignored , has turned out to be the largest one . this is due to @xmath240 and @xmath188 being much larger than @xmath263 and @xmath264 , respectively , as well as to @xmath83 being small . for the partial - rate asymmetry in eq . ( [ deltao ] ) , we find @xmath265 this is comparable to the corresponding asymmetry in @xmath266 @xcite , but larger than those in octet - hyperon decays @xcite . since the asymmetry measured by hypercp is the sum @xmath267 it is important to know how @xmath1 compares with @xmath2 . the sm contribution to @xmath2 has been evaluated most recently to be @xmath268 @xcite . thus within the standard model @xmath1 is smaller than @xmath2 , but not negligibly so , and the resulting @xmath29 has a value within the range @xmath269 for this observable , hypercp expects to have a statistical precision of @xmath270 @xcite , and so its measurement will unlikely be sensitive to the sm effects . here we evaluate @xmath1 and @xmath95 arising from possible physics beyond the standard model . in particular , we consider contributions generated by the chromomagnetic - penguin operators ( cmo ) , which in some new - physics models could be significantly larger that their sm counterparts @xcite . the relevant effective hamiltonian can be written as @xcite @xmath271 where @xmath272 and @xmath273 are the wilson coefficients , and @xmath274 are the cmo , with @xmath275 being the gluon field - strength tensor , @xmath276 the gluon coupling constant , and @xmath277 since various new - physics scenarios may contribute differently to the coefficients of the operators , we will not focus on specific models , but will instead adopt a model - independent approach , only assuming that the contributions are potentially sizable , in order to estimate bounds on the resulting asymmetries as allowed by constraints from kaon measurements . the chiral lagrangian proceeding from the cmo has to respect their symmetry properties . under @xmath278@xmath114@xmath279 rotations @xmath280 and @xmath281 transform as @xmath282 and @xmath283 respectively . moreover , under a @xmath284 transformation ( a @xmath0 operation followed by interchanging the @xmath285 and @xmath44 quarks ) @xmath280 and @xmath281 change into each other . these symmetry properties are also those of the quark densities @xmath286 of which the lowest - order chiral realization has been derived in ref . @xcite . from this realization , we can infer the leading - order chiral lagrangian induced by the cmo , namely @xmath287 \right\rangle + \beta_0^ { } \left\langle h\sigma^\dagger \right\rangle \left\langle \bar{b}{}_v^ { } b_v^ { } \right\rangle \nonumber \\ & & \!\ ! + \,\ , \tilde\beta_d^ { } \left\langle \bar{b}{}_v^ { } \left\ { \xi h\xi , b_v^ { } \right\ } \right\rangle + \tilde\beta_f^ { } \left\langle \bar{b}{}_v^ { } \left [ \xi h\xi , b_v^ { } \right ] \right\rangle + \tilde\beta_0^ { } \left\langle h\sigma \right\rangle \left\langle \bar{b}{}_v^ { } b_v^ { } \right\rangle \nonumber \\ & & \!\ ! + \,\ , \beta_c^{}\ , \bar{t}{}_v^\alpha\ , \xi^\dagger h\xi^\dagger\ , t_{v\alpha}^ { } - \beta_0 ' \left\langle h\sigma^\dagger \right\rangle \bar{t}{}_v^\alpha t_{v\alpha}^ { } + \tilde\beta_c^{}\ , \bar{t}{}_v^\alpha\ , \xi h\xi\ , t_{v\alpha}^ { } - \tilde\beta_0 ' \left\langle h\sigma \right\rangle \bar{t}{}_v^\alpha t_{v\alpha}^ { } \nonumber \\ & & \!\ ! + \,\ , \beta_\varphi^{}\ , f^2 b_0^ { } \left\langle h\sigma^\dagger \right\rangle \,\,+\,\ , \tilde\beta_\varphi^{}\ , f^2 b_0^ { } \left\langle h\sigma \right\rangle \,\,+\,\ , { \rm h.c . } \,\,,\end{aligned}\ ] ] where @xmath288 @xmath289 are parameters containing the coefficient @xmath290 @xmath291 . the part of this lagrangian without the decuplet - baryon fields was first written down in ref . @xcite . from @xmath292 along with @xmath293 , we derive the diagrams shown in fig . [ pwave_np ] , which represent the lowest - order contributions induced by the cmo to the @xmath33-wave transitions in @xmath294 we remark that each of the three diagrams in the figure is of @xmath248 in the @xmath135 expansion , and that fig . [ pwave_sm ] does not include the meson - pole diagram because within the sm it contributes only at next - to - leading order . the amplitudes following from fig . [ pwave_np ] are @xmath295 where @xmath296 and we have used @xmath297 derived from eq . ( [ ls ] ) . and @xmath27 cases @xcite , each of the two amplitudes in eq . ( [ ap_np ] ) vanishes if we set @xmath298 @xmath299 and @xmath300 with @xmath301 being a constant , take the limit @xmath302 and use the relations @xmath303 and @xmath304 both derived from eq . ( [ ls ] ) . this satisfies the requirement implied by the feinberg - kabir - weinberg theorem @xcite that the operator @xmath305 can not contribute to physical decay amplitudes @xcite , and thus serves as a check for the formulas in eq . ( [ ap_np ] ) . ] diagrams representing chromomagnetic - penguin contributions to the leading - order @xmath33-wave amplitude for @xmath231 each square represents a weak vertex generated by @xmath306 in eq . ( [ lw_np ] ) . ] in order to estimate the weak phases in @xmath1 , we need to determine the parameters @xmath307 in terms of the underlying coefficient @xmath308 which is the combination corresponding to parity - conserving transitions . from the effective hamiltonian in eq . ( [ hw_np ] ) and the chiral lagrangian in eq . ( [ lw_np ] ) , we can derive the one - particle matrix elements @xmath309 since there is presently no reliable way to determine these matrix elements from first principles , we employ the mit bag model to estimate them . the results for @xmath310 have already been derived in ref . @xcite using the bag - model calculations of ref . @xcite and are given by @xmath311 where @xmath312 , @xmath313 , and @xmath314 are bag parameters . for @xmath315 , extending the work of ref . @xcite we find @xmath316 numerically , we take @xmath317 for the octet baryons , @xmath318 for the decuplet baryons , and @xmath319 for the mesons , after refs . in addition , as in ref . @xcite , we have @xmath320 and @xmath321 for both the baryons and mesons . it follows that @xmath322 we note that @xmath323 here is the wilson coefficient at the low scale @xmath324 and hence already contains the qcd running from the new - physics scales . we also note that the bag - model numbers in eq . ( [ beta_i ] ) are comparable in magnitude to the natural values of the parameters as obtained from naive dimensional analysis @xcite , @xmath325 where we have chosen @xmath326 the differences between the two sets of numbers provide an indication of the level of uncertainty in estimating the matrix elements .. this will be taken into account in our results below . applying eq . ( [ beta_i ] ) in @xmath232 then leads to the cmo contributions @xmath327 where , as in the @xmath26 and @xmath27 cases @xcite , we have assigned an error of 200@xmath236 to each of these numbers to reflect the uncertainty due to our neglect of higher - order terms that are presently incalculable and the uncertainty in estimating the matrix elements above . for the @xmath34-wave phases , we have here the same problem in estimating them as in the standard - model case , and so we have to resort again to dimensional arguments . thus , since the @xmath34-wave amplitude is parity violating , we have @xmath328 where @xmath329 is the combination corresponding to parity - violating transitions . putting together the numbers from eqs . ( [ px , dx ] ) , ( [ deltapd ] ) , ( [ phip_np ] ) , and ( [ phid_np ] ) in eq . ( [ ao ] ) , we find @xmath330 as in the sm result , the second term in @xmath1 dominates these numbers . for the partial - rate asymmetry , we obtain @xmath331 we can now write down the contribution of the cmo to the sum of asymmetries @xmath332 being measured by hypercp . the most recent evaluation of their contribution to @xmath2 has been done in ref . @xcite , the result being @xmath333 evidently , @xmath334 is much smaller than , though still not negligible compared to , @xmath335 . summing the two asymmetries yields @xmath336 since the cmo also contribute to the @xmath0-violating parameters @xmath337 in kaon mixing and @xmath338 in kaon decay , which are now well measured , it is possible to obtain bounds on @xmath339 and @xmath340 using the @xmath337 and @xmath338 data . as discussed in ref . @xcite , the experimental values @xmath341 and @xmath342 @xcite imply that @xmath343 then , from eqs . ( [ delta_o^np ] ) and ( [ a_ol^np ] ) , it follows that @xmath344 the upper limits of these ranges well exceed those within the sm in eqs . ( [ delta_o^sm ] ) and ( [ a_ol^sm ] ) , but the largest size of @xmath339 is still an order of magnitude below the expected sensitivity of hypercp @xcite . this , nevertheless , implies that a nonzero measurement by hypercp would be an unmistakable signal of new physics . we have evaluated the sum of the @xmath0-violating asymmetries @xmath1 and @xmath2 occurring in the decay chain @xmath17 which is currently being studied by the hypercp experiment . the dominant contribution to @xmath1 has turned out to be due to final - state interactions via @xmath345 we have found that both within and beyond the standard model @xmath1 is smaller than @xmath2 , but not negligibly so . taking a model - independent approach , we have also found that contributions to @xmath346 from possible new - physics through the chromomagnetic - penguin operators are allowed by constraints from kaon data to exceed the sm effects by up to two orders of magnitude . in summary , @xmath347 since the sm contribution is well beyond the expected reach of hypercp , a finding of nonzero asymmetry would definitely indicate the presence of new physics . in any case , the upcoming data on @xmath29 will yield information which complements that to be gained from the measurement of @xmath30 in @xmath348 finally , we have shown that the contribution of @xmath349 also causes the partial - rate asymmetry @xmath95 in @xmath22 to be nonvanishing , thereby providing another means to observe @xmath0 violation in this decay . this asymmetry and that in @xmath266 tend to be larger than the corresponding asymmetries in octet - hyperon decays and hence are potentially useful probes of @xmath0 violation in future experiments . since @xmath95 results from the interference of @xmath33-wave amplitudes , a measurement of it will probe the underlying parity - conserving interactions . numerically , we have found @xmath350 where the bound on the contribution of the cmo arises from the constraint imposed by @xmath337 data . i would like to thank g. valencia for helpful discussions and comments . i am also grateful to e.c . dukes and l .- c . lu for experimental information . this work was supported in part by the lightner - sams foundation . for the four coupled channels , the @xmath359 factors are @xmath360 @xmath361 @xmath362 @xmath363 and the @xmath364 factors @xmath365 @xmath366 @xmath367 @xmath368 where @xmath369 is the energy of @xmath128 in the final state . we note that contributions to the propagators from the @xmath370 and quark - mass terms in eq . ( [ ls ] ) have been implicitly included in these results . christenson _ et al . _ , phys . * 13 * , 138 ( 1964 ) ; a. alavi - harati _ et al . _ [ ktev collaboration ] , _ ibid . _ * 83 * , 22 ( 1999 ) ; v. fanti _ et al . _ [ na48 collaboration ] , phys . b * 465 * , 335 ( 1999 ) ; b. aubert _ et al . _ [ babar collaboration ] , phys . lett . * 87 * , 091801 ( 2001 ) ; k. abe _ et al . _ [ belle collaboration ] , _ ibid . _ * 87 * , 091802 ( 2001 ) . see , e.g. , a. j. buras , arxiv : hep - ph/0402191 and references therein . s. okubo , phys . rev . * 109 * , 984 ( 1958 ) ; a. pais , phys . lett . * 3 * , 242 ( 1959 ) ; t.brown , s.f . tuan , and s. pakvasa , _ ibid . _ * 51 * , 1823 ( 1983 ) ; l.l . chau and h.y . cheng , phys . b * 131 * , 202 ( 1983 ) ; j.f . donoghue and s. pakvasa , phys . lett . * 55 * , 162 ( 1985 ) . luk , arxiv : hep - ex/9803002 . luk _ et al . _ [ e756 and hypercp collaborations ] , arxiv : hep - ex/0005004 . lu [ hypercp collaboration ] , aip conf . * 675 * ( 2003 ) 251 ; talk given at the meeting of the division of particles and fields of the american physical society , philadelphia , pennsylvania , 5 - 8 april 2003 . donoghue , x .- he , and s. pakvasa , phys . d * 34 * , 833 ( 1986 ) . iqbal and g.a . miller , phys . d * 41 * , 2817 ( 1990 ) ; x .- g . he , h. steger , and g. valencia , phys . b * 272 * , 411 ( 1991 ) ; n.g . deshpande , x .- he , and s. pakvasa , _ ibid . _ * 326 * , 307 ( 1994 ) . d. chang , x .- he , and s. pakvasa , phys . lett . * 74 * , 3927 ( 1995 ) ; x .- g . he and g. valencia , phys . d * 52 * , 5257 ( 1995 ) ; x .- g . he , h. murayama , s. pakvasa , and g. valencia , _ ibid . _ * 61 * , 071701 ( 2000 ) ; c .- h . chen , phys . b * 521 * , 315 ( 2001 ) ; j .- h . jiang and m .- l . yan , j. phys . g * 30 * , b1 ( 2004 ) . j. tandean and g. valencia , phys . d * 67 * , 056001 ( 2003 ) . j. tandean , phys . d * 69 * , 076008 ( 2004 ) . j. tandean and g. valencia , phys . b * 451 * , 382 ( 1999 ) . k. hagiwara _ et al . _ [ particle data group collaboration ] , phys . d * 66 * , 010001 ( 2002 ) . watson , phys . * 95 * , 228 ( 1954 ) . see , e.g. , h. pilkuhn , _ the interactions of hadrons _ ( wiley , new york , 1967 ) . l. wolfenstein , phys . d * 43 * , 151 ( 1991 ) . oller , e. oset , and a. ramos , prog . phys . * 45 * , 157 ( 2000 ) . j. tandean , a.w . thomas , and g. valencia , phys . d * 64 * , 014005 ( 2001 ) ; references therein . j. bijnens , h. sonoda , and m.b . wise , nucl . * b261 * , 185 ( 1985 ) . e. jenkins and a.v . manohar , phys . b * 255 * , 558 ( 1991 ) ; _ ibid . _ * 259 * , 353 ( 1991 ) . in _ effective field theories of the standard model _ , edited by u .- meissner ( world scientific , singapore , 1992 ) . e. jenkins , nucl . * b368 * , 190 ( 1992 ) ; n. kaiser , p.b . siegel , and w. weise , _ ibid . * a594 * , 325 ( 1995 ) ; n. kaiser , t. waas , and w. weise , _ ibid . _ * a612 * , 297 ( 1997 ) ; j. caro ramon , n. kaiser , s. wetzel , and w. weise , _ ibid . _ * a672 * , 249 ( 2000 ) ; c.h . lee , g.e . brown , d .- min , and m. rho , _ ibid . _ * a585 * , 401 ( 1995 ) ; j.w . bos _ et al . d * 51 * , 6308 ( 1995 ) ; _ ibid . _ * 57 * , 4101 ( 1998 ) ; g. mller and u .- g . meissner , nucl . phys . * b492 * , 379 ( 1997 ) . a. abd el - hady , j. tandean , and g. valencia , nucl . phys . a * 651 * , 71 ( 1999 ) . j. gasser , h. leutwyler , and m.e . sainio , phys . b * 253 * , 252 ( 1991 ) ; * 253 * , 260 ( 1991 ) . m.n . butler , m.j . savage , and r.p . springer , nucl . b * 399 * , 69 ( 1993 ) . c. dukes [ hypercp collaboration ] , talk given at da@xmath371ne 2004 : physics at meson factories , laboratori nazionali di frascati , italy , 7 - 11 june 2004 . n. cabibbo , phys . * 10 * , 531 ( 1963 ) ; m. kobayashi and t. maskawa , prog . phys . * 49 * , 652 ( 1973 ) . l. wolfenstein , phys . lett . * 51 * , 1945 ( 1983 ) . see , e.g. , j.f . donoghue , e. golowich , and b.r . holstein , _ dynamics of the standard model _ ( cambridge university press , cambridge , 1992 ) . e. jenkins , nucl . * b375 * , 561 ( 1992 ) . a. manohar and h. georgi , nucl . phys . * b234 * 189 ( 1984 ) ; h. georgi and l. randall , _ ibid . _ * b276 * 241 ( 1986 ) ; s. weinberg , phys . lett . * 63 * , 2333 ( 1989 ) . buras , arxiv : hep - ph/0307203 . f. gabbiani , e. gabrielli , a. masiero , and l. silvestrini , nucl . b * 477 * , 321 ( 1996 ) ; a. masiero and h. murayama , phys . lett . * 83 * , 907 ( 1999 ) ; x .- g . he and g. valencia , phys . d * 61 * , 075003 ( 2000 ) ; g. colangelo , g. isidori , and j. portoles , phys . b * 470 * , 134 ( 1999 ) ; j. tandean , phys . d * 61 * , 114022 ( 2000 ) ; j. tandean and g. valencia , _ ibid . _ * 62 * , 116007 ( 2000 ) . buras _ et al . _ , nucl . b * 566 * , 3 ( 2000 ) . g. feinberg , p. kabir , and s. weinberg , phys . * 3 * , 527 ( 1959 ) . donoghue and b.r . holstein , phys . d * 33 * , 2717 ( 1986 ) ; j.f . donoghue , e. golowich , and b.r . holstein , phys . * 131 * , 319 ( 1986 ) . donoghue , e. golowich , and b.r . holstein , phys . d * 15 * , 1341 ( 1977 ) ; j.f . donoghue , e. golowich , b.r . holstein , and w.a . ponce , _ ibid . _ * 23 * , 1213 ( 1981 ) . t. degrand , r.l . jaffe , k. johnson , and j.e . kiskis , phys . d * 12 * , 2060 ( 1975 ) .

the sum of the @xmath0-violating asymmetries @xmath1 and @xmath2 in the decay sequence @xmath3 @xmath4 is presently being measured by the e871 experiment . we evaluate contributions to @xmath1 from the standard model and from possible new physics , and find them to be smaller than the corresponding contributions to @xmath2 , although not negligibly so . we also show that the partial - rate asymmetry in @xmath5 is nonvanishing due to final - state interactions . taking into account constraints from kaon data , we discuss how the upcoming result of e871 and future measurements may probe the various contributions to the observables . smu - hep-04 - 06

the concept of elementary excitations and the diagrammatic perturbation - theoretic methods borrowed from quantum field theory have given us , over the past decades , many powerful insights into the behavior of materials . in a number of cases , however , these concepts and methods do nt seem to work . in previous papers@xcite , we presented results on a nonperturbative extension of the magnetic interaction model , which had until then been extensively used in the context of diagrammatic approaches . these latter applications were successful in many respects : in the eliashberg approximation , the magnetic interaction model correctly anticipated the pairing symmetry of the cooper state in the copper oxide superconductors@xcite and is consistent with spin - triplet p - wave pairing in superfluid @xmath0 [ for a recent review see , e.g. , ref . one also gets the correct order of magnitude of the superconducting and superfluid transition temperature @xmath1 when the model parameters are inferred from experiments in the normal state of the above systems . however , in ref.@xcite it was found that when the model was treated nonperturbatively and one approached the border of magnetic long - range order , the quasiparticle spectrum showed qualitative changes not captured by the eliashberg approximation . in ref.@xcite , we raised the possibility that these qualitative changes , namely the opening of a pseudogap in the quasiparticle spectrum , were intrinsically nonperturbative in nature . in this paper , we examine this possibility by comparing the nonperturbative results to various kinds of perturbation - theoretic approximations . the paper is organized as follows . in the next section we describe the model as well as the various perturbation - theoretic approximations to be compared to the monte carlo calculations . section iii contains the results of the nonperturbative and diagrammatic calculations . section iv contains a discussion of the results and finally we give a summary and outlook the model and its motivation have been extensively discussed in ref.@xcite . here we only give the definitions relevant to the present discussion . we consider particles on a two - dimensional square lattice whose hamiltonian in the absence of interactions is @xmath2 where @xmath3 is the tight - binding hopping matrix , @xmath4 the chemical potential and @xmath5 , @xmath6 respectively create and annihilate a fermion of spin orientation @xmath7 at site @xmath8 . we take @xmath9 if sites @xmath8 and @xmath10 are nearest neighbors and @xmath11 if sites @xmath8 and @xmath10 are next - nearest neighbors . to introduce interactions between the particles , we couple them to a dynamical molecular ( or hubbard - stratonovich ) field . it is instructive to consider two different types of molecular fields . in the first instance , we consider a vector hubbard - stratonovich field that couples locally to the fermion spin density . we also consider the case of a scalar field that couples locally to the fermion number density . this case corresponds to a coupling to charge - fluctuations or , within the approximation we are using here , `` ising''-like magnetic fluctuations where only longitudinal modes are present . the hamiltonians at imaginary time @xmath12 for particles coupled to the fluctuating exchange or scalar dynamical field are then @xmath13 where @xmath14 and @xmath15 are the real vector exchange and scalar hubbard - stratonovich fields respectively , and @xmath16 the coupling constant . the reason for the choice of an extra factor @xmath17 in eq . ( [ ham1 ] ) becomes clear later . since we ignore the self - interactions of the molecular fields , their distribution is gaussian and given by@xcite @xmath18 = { 1\over z } \exp\bigg(-\sum_{{\bf q},\nu_n } { { \bf m}({\bf q},i\nu_n)\cdot { \bf m}(-{\bf q},-i\nu_n ) \over 2\alpha({\bf q},i\nu_n)}\bigg ) \label{probm } \\ z = \int d{\bf m}\exp\bigg(-\sum_{{\bf q},\nu_n } { { \bf m}({\bf q},i\nu_n)\cdot { \bf m}(-{\bf q},-i\nu_n ) \over 2\alpha({\bf q},i\nu_n)}\bigg ) \label{normm}\end{aligned}\ ] ] in the case of a vector exchange molecular field and @xmath19 = { 1\over z}\exp\bigg(-\sum_{{\bf q},\nu_n } { \phi({\bf q},i\nu_n)\phi(-{\bf q},-i\nu_n ) \over 2\alpha({\bf q},i\nu_n)}\bigg ) \label{probphi } \\ z = \int d\phi\exp\bigg(-\sum_{{\bf q},\nu_n } { \phi({\bf q},i\nu_n)\phi(-{\bf q},-i\nu_n ) \over 2\alpha({\bf q},i\nu_n)}\bigg ) \label{normphi}\end{aligned}\ ] ] in the case of a scalar hubbard - stratonovich field . in both cases @xmath20 since the dynamical molecular fields are periodic functions in the interval @xmath21 $ ] . the fourier transforms of the molecular fields are defined as @xmath22\big ) \label{fourierm } \\ \phi_{\bf r}(\tau ) & = & \sum_{{\bf q},\nu_n } \phi({\bf q},i\nu_n ) \exp\big(-i[{\bf q}\cdot{\bf r}-\nu_n\tau]\big ) \label{fourierp}\end{aligned}\ ] ] we consider the case where there is no long - range magnetic or charge order . the average of the dynamical molecular fields must then vanish and their gaussian distributions eqs . ( [ probm],[probphi ] ) are completely determined by their variance @xmath23 , which we take to be @xmath24 where @xmath25 is the number of allowed wavevectors in the brillouin zone . then @xmath26 where @xmath27 denotes an average over the probability distributions eq . ( [ probm ] ) and eq . ( [ probphi ] ) for the vector and scalar cases respectively . in order to compare the scalar and vector molecular fields , we take the same form for their correlation function @xmath28 and parametrize it as in refs.@xcite . in what follows , we set the lattice spacing @xmath29 to unity . for real frequencies , we have @xmath30 where @xmath31 and @xmath32 are the correlation wavevectors or inverse correlation lengths in units of the lattice spacing , with and without strong correlations , respectively . let @xmath33 we consider commensurate charge fluctuations and antiferromagnetic spin fluctuations , in which case the parameters @xmath34 and @xmath35 in eq . ( [ chiml ] ) are defined as @xmath36 where @xmath37 is a characteristic temperature . we also consider the case of ferromagnetic spin - fluctuations , where the parameters @xmath34 and @xmath35 in eq . ( [ chiml ] ) are given by @xmath38 @xmath28 is related to the imaginary part of the response function @xmath39 , eq . ( [ chiml ] ) , via the spectral representation @xmath40 to get @xmath28 to decay as @xmath41 as @xmath42 , as it should , we introduce a cutoff @xmath43 and take @xmath44 for @xmath45 . a natural choice for the cutoff is @xmath46 . in our model , the single particle green s function is the average over the probability distributions @xmath47 $ ] ( eq . ( [ probm ] ) ) or @xmath48 $ ] ( eq . ( [ probphi ] ) ) of the fermion green s function in a dynamical vector or scalar field . @xmath49\ ; g(i\sigma\tau ; j\sigma'\tau'|[{\bf m } ] ) \label{gm } \\ { \cal g}(i\sigma\tau ; j\sigma'\tau ' ) & = & \int d\phi\ ; { \cal p}[\phi]\ ; g(i\sigma\tau ; j\sigma'\tau'|[\phi ] ) \label{gphi}\end{aligned}\ ] ] where @xmath50\;or \;[\phi ] ) = -\big < t_\tau\{\psi_{i\sigma}(\tau ) \psi^\dagger_{j\sigma'}(\tau')\}\big > \label{gfield}\ ] ] is the single particle green s function in a dynamical molecular field and is discussed at length in ref.@xcite . in evaluating expressions eqs . ( [ gm],[gphi ] ) one is summing over all feynman diagrams corresponding to spin or charge - fluctuation exchanges@xcite . the diagrammatic expansion of the green s function , eq . ( [ gm ] ) is shown pictorially in fig . 1 . since in our model no virtual fermion loops are present , there is no fermion sign problem@xcite . in this paper we compare the results of the monte carlo simulations to various diagrammatic approximations for the same model . we denote by @xmath51 and @xmath52 the bare and dressed quasiparticle propagators respectively . they are given by @xmath53 where @xmath54 is the quasiparticle self - energy and @xmath55 the tight - binding dispersion relation obtained from fourier transforming the hopping matrix @xmath3 in eq . ( [ ham0 ] ) and @xmath4 the chemical potential . we consider four approximations to the quasiparticle self - energy @xmath54 whose diagrammatic representations are shown in fig . 2 . in fig . 2a , the self - energy is approximated by first order perturbation theory in the exchange of magnetic or charge fluctuations and denoted @xmath56 . 2b shows the eliashberg approximation in which the self - energy denoted @xmath57 is given by the first order self - consistent ( or brillouin - wigner ) perturbation theory . the expressions for @xmath56 or @xmath57 in the case of quasiparticles coupled to magnetic or charge fluctuations are identical ( this is the reason for our choice of the factor @xmath17 in eq . ( [ ham1 ] ) ) and given by @xmath58 where @xmath59 and @xmath60 are the bare and dressed quasiparticle green s functions defined in eq . ( [ gbare ] ) and eq . ( [ gdressed ] ) respectively . 2c shows the diagrammatic expansion corresponding to second order perturbation theory and we denote the self - energy corresponding to that approximation @xmath61 . the second order self - consistent approximation to the quasiparticle self - energy , denoted @xmath62 , is shown diagrammatically in fig . 2d . the expressions for @xmath61 and @xmath62 now depend on whether the quasiparticles are coupled to the vector hubbard - stratonovich field ( magnetic fluctuations ) or scalar hubbard - stratonovich field ( charge fluctuations ) , because vertex corrections in the two cases do not have the same coefficient or even the same sign . the expressions for @xmath61 and @xmath62 for quasiparticles coupled to magnetic fluctuations are given by @xmath63 in the case of the scalar hubbard - stratonovich field , or coupling to charge fluctuations , the corresponding expressions are @xmath64 in eqs . ( [ 2ptm],[2scm],[2ptc],[2scc ] ) , @xmath59 and @xmath60 are the bare and dressed quasiparticle green s functions defined in eq . ( [ gbare ] ) and eq . ( [ gdressed ] ) respectively . the strength of the coupling to the magnetic or charge flucutations can be parametrized by a dimensionless mass renormalization parameter @xmath65 , which is defined as @xmath66 the fermi surface averages are given by @xmath67 in practice , we compute the fermi surface average with a discrete set of momenta and we replace the delta function by a finite temperature expression @xmath68 where @xmath69 is the fermi function . note that @xmath70 as @xmath71 . we have used @xmath72 and @xmath73 in all of our calculations . the finite temperature effectively means that van hove singularities will be smeared out . note that the fermi surface average that appears in @xmath65 , eq . ( [ lambda1 ] ) plays a role similar to that of @xmath74 in the case of phonon mediated superconductivity . one therefore expects @xmath75 to indicate the crossover between weak and strong coupling . the quasiparticle dispersion relation for the two - dimensional square lattice is obtained from eq . ( [ ham0 ] ) . we measure all energies and temperatures in units of the nearest - neighbor hopping parameter @xmath76 . we set the next - nearest - neighbor hopping parameter @xmath77 . the chemical potential is adjusted so that the electronic band filling is @xmath78 . the dimensionless parameters describing the molecular field correlations are @xmath79 , @xmath80 , @xmath32 and @xmath31 . we chose a representative value for @xmath81 , and set @xmath82 as in the earlier work@xcite . for an electronic bandwidth of @xmath83 , @xmath84k . we only consider one value of the coupling constant @xmath85 . in the random phase approximation , the magnetic instability would be obtained for a value of @xmath79 of the order of 10 . we consider what happens to the quasiparticle spectrum at a fixed temperature @xmath86 as the inverse correlation length @xmath31 changes , as in ref.@xcite . all the calculations were done on a 8 by 8 spatial lattice . in the monte carlo calculations we used 41 imaginary time slices , or equivalently 41 matsubara frequencies for the molecular fields , @xmath87 and @xmath88 ( @xmath20 , with @xmath89 ) . in the diagrammatic calculations , we used between 40 to 60 fermion matsubara frequencies . by analytic continuation of the single particle green s function @xmath90 one can obtain the quasiparticle spectral function @xmath91 and the tunneling density of states @xmath92 , where @xmath93 is the retarded single particle green s function . the imaginary time monte carlo data is analytically continued with the maximum entropy method@xcite , using the same methodology as in the earlier work@xcite . we used 10000 monte carlo samples grouped into 100 bins of 100 samples each . we always use a flat default model in the maximum entropy calculations . to provide a fair comparison between diagrammatic and nonperturbative calculations , one should use the same analytic continuation method ( with the same parameters ) in all cases . therefore , we generated 100 noisy measurements by adding gaussian random noise to the results of the diagrammatic calculations and analytically continued @xmath94 using the maximum entropy method as well , with the same default model as in the corresponding analytic continuation of the monte carlo data . the scheme is not perfect , however . while the variance of the gaussian noise added to the diagrammatic green s functions was chosen such that the statistical uncertainty of the average over the 100 noisy samples was identical to that in the corresponding monte carlo green s function , the correlations in the errors for different values of @xmath12 present in the monte carlo results can not be easily modeled . the gaussian random numbers added to the diagrammatic green s function were therefore taken to be independent of each other , and thus the noise in the diagrammatic and monte carlo green s functions did not have identical statistical properties . in spite of this , the present scheme is almost certainly better than the alternatives . 3,4 and 5 show the comparison , for different values of @xmath95 , between the nonperturbative calculations of the quasiparticle green s function @xmath90 , spectral function @xmath96 and tunneling density of states @xmath97 and those obtained from the approximations @xmath56 , @xmath61 , @xmath57 , and @xmath62 to the quasiparticle self - energy . 3 shows our results for @xmath98 . for this value of the inverse correlation length squared , the mass renormalization parameter @xmath99 . the coupling to the antiferromagnetic spin - fluctuations is therefore weak . not surprisingly , the quasiparticle green s function , spectral function and tunneling density of states obtained from the various diagrammatic approximation agree well with the monte carlo results . at @xmath100 , the difference between the nonperturbative green s function @xmath101 and its diagrammatic approximations is of the order of 0.001 t for all values of @xmath12 . there is virtually no difference between the straightforward perturbation - theoretic calculations of the spectral function and their self - consistent counterparts , in first and second order , which is expected for weak coupling . thus the small difference in the spectral functions @xmath96 at @xmath100 , seen in fig 3b , to the extent that they are not an artifact of the analytic continuation , must come from the vertex corrections . since the first order spectral functions are slightly sharper than the second order ones , the first order vertex corrections result in an increased spin - fluctuation interaction , as pointed out in refs.@xcite . the monte carlo spectral function is also somewhat broader than the diagrammatic calculations , and provided again that it is not an artifact of the analytic continuation , this suggests that the higher order diagrams lead to a further increase of the spin - fluctuation interaction . the results for @xmath102 are shown in fig . 4 . this value of @xmath95 gives a mass renormalization parameter @xmath103 . one is now in the intermediate coupling regime . the quasiparticle green s function @xmath90 and spectral function @xmath96 at @xmath100 as well as the tunneling density of states obtained from the various diagrammatic approximations agree qualitatively with the monte carlo results . there are , however , noticeable quantitative differences , not surprisingly much more so than for @xmath98 . the largest difference between the green s functions obtained from the diagrammatic approximations and the nonperturbative calculations is now bigger than the width of the lines and is roughly an order of magnitude ( 0.01 t ) larger than for @xmath98 , which is not unexpected since the mass renormalization parameter is also about an order of magnitude greater for @xmath102 than for @xmath98 . with the above caveat regarding the analytic continuation , one can make a few additional remarks . first of all , there is now a difference between the straightforward perturbation - theoretic results and the self - consistent calculations of the spectral function @xmath96 at @xmath100 , both at first and second order . in particular , the second order self - consistent spectral function is slightly broader than the first order self - consistent one , an indication that the first order vertex correction leads to an enhancement of the effective spin - fluctuation interaction , in agreement with refs.@xcite . the nonperturbative @xmath104 is broader than the second order self - consistent result , which would imply the higher order vertex corrections are further enhancing the magnetic interaction . note , that the second order perturbation - theoretic @xmath104 is slightly broader than its self - consistent counterpart ( the dressing of green s functions tends to reduce the effect of interactions ) and agrees very well with the monte carlo result . this may be due to a cancellation of errors ( or the analytic continuation procedure ) since the agreement between the nonperturbative tunneling density of states @xmath97 and the second order perturbation - theoretic @xmath97 is not as good . as @xmath105 , the quasiparticle mean free path becomes of the order of the magnetic correlation length for some wavevectors near the fermi surface , the quasiparticles then ca nt tell there is no long - range order , and this marks the onset of pseudogap behavior@xcite . for @xmath106 , the mass renormalization parameter is @xmath107 . one is therefore in the strong coupling regime . the results of our calculations for @xmath106 are shown in fig . 5 . the developing pseudogap in the spectral function @xmath96 at @xmath100 ( fig . 5b ) and in the tunneling density of states @xmath97 ( fig . 5e ) found in the nonperturbative monte carlo calculations is not seen in any of the diagrammatic approximations considered here , which therefore fail qualitatively . given that one is in the strong coupling regime @xmath108 , the breakdown of perturbation theory should not come as a surprise . the maximum difference in the quasiparticle green s function @xmath90 between the nonperturbative and diagrammatic calculations is now of the order 0.1 t , and hence an order of magnitude larger than for @xmath102 and a couple of orders of magnitude larger than in the weak coupling regime with @xmath98 . it is therefore not suprprising that the quasiparticle spectra that give rise to these rather different imaginary time green s functions turn out to show qualitative differences . note that for @xmath106 , there is nearly as much difference between the perturbation - theoretic and self - consistent approximations of the same order as there are between calculations of the same type at first and second order . figs . 6 - 9 show our results for the quasiparticle green s function @xmath90 , spectral function @xmath96 and tunneling density of states @xmath97 for several values of @xmath95 . we start with @xmath98 , for which the mass renormalization parameter @xmath99 for coupling to ferromagnetic spin - flucutations . in this weak coupling regime , fig . 6 shows that the results of the various diagrammatic calculations are in good agreement with the monte carlo results . as in the corresponding antiferromagnetic case , at @xmath100 , the difference between the nonperturbative green s function @xmath101 and its diagrammatic approximations is of the order of 0.001 t for all values of @xmath12 . moreover , there is virtually no difference between the straightforward perturbation - theoretic calculations of the spectral function and their self - consistent counterparts , in first and second order . thus the small difference in the spectral functions @xmath96 at @xmath100 , seen in fig . 6b must come from the vertex corrections . since the first order spectral functions are slightly sharper than the second order ones , the first order vertex corrections result in an increased spin - fluctuation interaction . given the smallness of the difference between the first and second order results and the ill - posed nature of the analytic continuation problem , one should take the above remark with some degree of caution . in ref.@xcite , however , it was shown that the increase in the effective interaction induced by the first order vertex correction is due to the spin dependence of the interaction , and thus should occur for quasiparticles coupled to either antiferromagnetic or ferromagnetic fluctuations . our analytically continued results are at least consistent with this . from fig . 6b , one also sees that the monte carlo spectral function is slightly broader than the first or second order results , as in the corresponding antiferromagnetic case . with the above caveat on the nature of the analytic continuation problem , this would suggest the higher order diagrams not included in our perturbation - theoretic approximations lead to a further enhancement of the magnetic interaction , as in the corresponding antiferromagnetic case . for @xmath102 , the mass renormalization parameter @xmath109 and one is therefore in an intermediate coupling regime . fig . 7 shows that for this value of @xmath95 , the diagrammatic approximations all qualitatively agree with the monte carlo results . the quantitative agreement is , not surprisingly , not as good as in the weak coupling limit with @xmath98 . one notes a number of similarities between the results of fig . 6 and the corresponding antiferromagnetic case , shown in fig . 4 : ( i ) the second order perturbation theory results for @xmath96 give the best agreement with the nonperturbative calculation , ( ii ) since the spectral function in either second order calculation , which include vertex corrections , is slightly broader in @xmath110 than the corresponding first order result , we conclude that first order vertex corrections lead to an enhancement of the effective quasiparticle interaction , which is what is expected on the basis of the arguments made in refs.@xcite ( iii ) the spectral function obtained by monte carlo sampling of the gaussian dynamical molecular fields is slightly broader than the second order results , which to the extent this is not an artifact of the maximum entropy analytic continuation is an indication that higher order spin - fluctuation exchanges not included in the diagrammatic approximations considered lead to a further enhancement of the effective quasiparticle interaction . the dynamical exponent @xmath111 is larger for ferromagnetic than antiferromagnetic spin fluctuations . hence the effective dimension @xmath112 in the ferromagnetic versus @xmath113 in the antiferromagnetic case and the standard theory of quantum critical phenomena@xcite leads one to expect weaker corrections for higher effective dimensions . the perturbative calculations qualitatively fail at @xmath105 in the antiferromagnetic case and on the basis of the above arguments one would expect that the breakdown of perturbation theory in the case of ferromagnetic fluctuations , if it happens , would occur for a smaller value of @xmath95 or larger values of the mass renormalization parameter @xmath65 . indeed , at @xmath106 , @xmath114 and therefore one is in the strong coupling regime . 8 shows that while for this value of @xmath95 the diagrammatic calculations still agree qualitatively with the monte carlo results , unsurprisingly there are larger quantitative differences than in the case @xmath102 shown in fig . 7 . our results for @xmath115 , for which the mass renormalization parameter @xmath116 are shown in fig . 9 . the spectral function @xmath96 obtained from the nonperturbative monte carlo calculations shows a double peak structure . this has been interpreted in ref.@xcite as an effective spin - splitting of the quasiparticle spectrum induced by the local ferromagnetic order . in looking at the evolution of the spectral function @xmath96 as @xmath95 is decreased , one first sees a broadening of @xmath96 and then , the broad quasiparticle peak splits into two . the monte carlo calculations show very little suppression of the quasiparticle spectral weight or density of states between the two split peaks . a look at figs . 7c , 8c , and 9e reveals that for @xmath117 , @xmath118 and depends very little on @xmath95 . this is is sharp contrast to the case of antiferromagnetic fluctuations discussed in the previous section . this difference is to be expected of course , since the antiferromagnetic state is gapped while the ferromagnetic state is not . it is clear that none of the diagrammatic approximations considered here reproduce this spin - splitting of the broad quasiparticle peak in @xmath96 and tunneling density of states @xmath97 well . in fact , the first order perturbation theoretic result shows a strong suppression of the tunelling density of states , which clearly does nt describe the precursor to the ferromagnetic state well , and therefore can be considered to fail qualitatively . we observe that the first order perturbation theoretic calculation failed to show a suppression of the tunneling density of states in the antiferromagnetic case where it is obtained in the nonperturbative calculations as expected ( see previous subsection ) but does show such a pseudogap in the ferromagnetic case where it is nt expected and does nt appear in the nonperturbative calculations . it therefore qualitatively fails in both cases . another clear sign that not all is well with the perturbation expansion is the large quantitative differences between the one - loop and two - loop results in fig.9 , something that could be expected at @xmath116 . in the view of the differences between the imaginary time green s function @xmath90 obtained from the monte carlo simulations and those of the various perturbation - theoretic approximations shown in figs . 9a and 9c which are of the order of 0.1 t , one would expect the spectral functions that produce these rather different imaginary time green s functions to be rather different themselves . the results of our calculations of the quasiparticle green s function @xmath90 , spectral function @xmath96 and tunneling density of states @xmath97 for several values of @xmath95 are shown in figs . 10 , 11 and 12 . for the model studied here , the mass renormalization parameter @xmath65 is the same for charge and antiferromagnetic fluctuations . therefore the results of the calculations for @xmath98 shown in fig . 10 correspond to @xmath119 , namely the coupling to the charge fluctuations is weak . the agreement between the monte carlo results and those of the various diagrammatic approximations is good . as seen in fig . 10a , the difference between the nonperturbative imaginary time green s function and its perturbative approximations at @xmath100 is less than the width of the line and of the order of 0.001 t for all imaginary times @xmath12 . if one compares the results of the perturbation - theoretic calculations at first and second order , one sees from fig . 10b that there is virtually no difference between the spectral functions @xmath96 obtained by straightforward perturbation theory or the self - consistent calculation at either first or second order . hence the slight difference bewteen the first and second order calculations , to the extent they are nt an artifact of the analytic continuation , must come from vertex corrections . in contrast to the case of coupling to antiferromagnetic fluctuations , the spectral functions at second order are slightly narrower than their first order counterpart . this suggests the first order vertex correction acts to reduce the effective charge fluctuation interaction , in agreement with the arguments presented in ref.@xcite . moreover , the nonperturbative @xmath96 at @xmath100 is slightly broader than the second order results , which would indicate that the higher order diagrams lead to an enhancement of the effective charge fluctuation interaction , as in the case of a coupling to antiferromagnetic fluctuations . while this observation is made on the basis of analytically continued results , it is consistent with the results for other values of @xmath95 presented below , where the enhancement of the effective charge fluctuation mediated interaction by higher than second order diagrams can be shown to occur on general grounds . 11 shows the the quasiparticle green s function @xmath90 , spectral function @xmath96 at @xmath100 and tunneling density of states @xmath97 for @xmath102 , for which @xmath103 . for this value of @xmath95 corresponding to an intermediate coupling regime , the reader will notice that the results of the second order perturbation theory ( self - consistent or not ) are not displayed in the figures . the reason is that both second order approximations , @xmath61 and @xmath62 , for the model parameters considered here , effectively violate causality requirements in that the eliashberg renormalization factor @xmath120 becomes less than one . in terms of the quasiparticle self - energy @xmath54 , @xmath121 where @xmath122 is the imaginary part of the retarded self - energy and we have made use of the spectral representation for the self - energy @xmath123 . causality demands that the retarded green s function be analytic in the upper - half complex frequency plane and therefore that the imaginary part of the retarded self - energy be always less than or equal to zero ( @xmath124 ) for all values of @xmath125 . this in turn means that @xmath126 for all values of @xmath125 . one can write the second order eliashberg renormalization factor @xmath127 , where @xmath128 is the change in @xmath120 coming from the @xmath129 order diagrams . @xmath130 is always greater than zero and therefore poses no problem as far as the condition @xmath126 is concerned . in the charge - fluctuation case , as was explained in ref.@xcite , the first order vertex correction has the opposite sign compared to the spin - fluctuation case , and leads to a suppression of the effective quasiparticle interaction . the enhancement of the quasiparticle spin - fluctuation vertex comes from the transverse magnetic fluctuations that manage to overcome the reduction of the effective coupling due to the longitudinal fluctuations . because of this cancellation effect , not only is the sign of the first order vertex correction different in the magnetic case , it is also smaller in magnitude than in the charge - fluctuation case , under otherwise similar conditions , as can be seen from the factor 1/3 in eqs . ( [ 2ptm],[2scm ] ) not present in the corresponding charge - fluctuation case in eqs . ( [ 2ptc],[2scc ] ) . the different sign of the vertex corrections in the charge and magnetic cases means that while in the magnetic case @xmath131 and at second order @xmath132 is always @xmath133 , in the charge fluctuation case @xmath134 . for @xmath117 , we find that the second order contribution to the eliashberg renormalization factor is greater in magnitude than the first order contribution , @xmath135 . note that the nonperturbative calculations always satisfy @xmath136 , and the problem only arises in the perturbative approximation and is a sign that , for @xmath117 , the perturbation expansion for the charge - fluctuation case is quite badly behaved , possibly even more so than for magnetic fluctuations . aso , the fact that the nonperturbative calculations always satisfy @xmath136 is a proof that the higher than second order diagrams contribute to an enhancement of the charge - fluctuation interaction for these values of @xmath95 . fig . 11b shows that the spectral function @xmath96 obtained from the nonperturbative calculations is noticeably sharper than those produced by the first order self - consistent calculations . this means that for @xmath85 and @xmath82 the first and higher order vertex corrections suppress the effective quasiparticle interaction . it is therefore not surprising that there are no qualitative differences between the nonperturbative and diagrammatic calculations . there are quantitative differences , however , and these are more pronounced than in the case of a coupling of quasiparticles to antiferromagnetic spin - fluctuations for the same value of @xmath65 shown in fig . 4 . finally , 12 shows our results for @xmath115 , for which @xmath137 , hence in the strong coupling regime . the difference between the nonperturbative imaginary time green s function and its first order diagrammatic approximations seen in figs . 12b and 12d is a clear indication of the breakdown of perturbation theory . but even in this strong coupling regime , a cdw - precursor pseudogap in the spectral function @xmath96 , which can be expected to occur on general grounds@xcite is not seen . the pseudogap effects in the charge fluctuation case thus require a stronger coupling still ( larger coupling constant @xmath138 or smaller value of @xmath95 ) . in ref.@xcite , we showed that the magnetic pseudogap induced by a coupling to antiferromagnetic spin - fluctuations and the spin - splitting of the quasiparticle peak induced by a coupling to ferromagnetic spin fluctuations were not captured by the first order self - consistent , or eliashberg , approximation . the main result of this paper , is that these phenomena also lie beyond the two magnetic - fluctuation exchange theories ( self - consistent or not ) , which contain first order vertex corrections . while this does obviously not constitute a proof , these results are consistent with the conjecture expressed in ref.@xcite that the pseudogap effects found in the monte carlo calculations are intrinsically nonperturbative in nature . since the calculations reported here show that the first order vertex corrections alone do not produce a magnetic pseudogap , the physics of that state must then mainly come from the higher order spin - fluctuation exchange processes . the results presented here and in ref.@xcite also indicate that a cdw pseudogap induced by coupling to the scalar dynamical molecular field ( eq . ( [ ham2 ] ) ) must also originate from high order charge - fluctuation exchange processes . close enough to a second order cdw transition , the diverging cdw correlation length is bound to exceed the characteristic length scale for quasiparticles and the calculations of ref.@xcite showed that when this happens a pseudogap opens in the quasiparticle spectrum . the first order vertex correction ca nt produce the pseudogap state , since as we have seen , in the case of charge fluctuations it leads to a suppression of the interaction . in fact we even found that for the range of model parameters considered here , the second order diagrams more than cancel the contribution from the first order terms leading to a second order eliashberg renormalization parameter @xmath139 , which is inconsistent with causality requirements . moreover , we expect this `` over - cancellation '' effect to get worse as @xmath95 gets smaller than the lowest value considered here , @xmath115 . since @xmath120 must be @xmath133 when all the diagrams are summed up , as in the monte carlo simulations , one can conclude that the higher than second order terms must give a contribution @xmath140 to @xmath141 which is positive . therefore , higher order charge - fluctuation exchange processes produce an enhancement of the effective quasiparticle interaction , as in the magnetic case , and it must be through this enhancement of the effective interaction that a pseudogap can appear in the quasiparticle spectrum on the border of long - range cdw order . these observations lead one to a unified picture of the pseudogap state found in our model of quasiparticles coupled to spin or charge fluctuations . when the dynamical molecular field correlation length exceeds the characteristic length scale for quasiparticles , either the thermal de broglie wavelength@xcite or mean free path@xcite , the quasiparticles effectively see long - range order and this marks the onset of the pseudogap state . this state must be produced by high order spin or charge - fluctuation exchanges which contain subtle quantum mechanical coherence effects . in the magnetic fluctuation case , the first order vertex correction favors the pseudogap state , while in the charge fluctuation case it suppresses it . this implies one has to be closer to the boundary of long - range charge order to observe a pseudogap than one has to be to the boundary of magnetic long - range order , under otherwise similar conditions . as the dynamical molecular field correlation length increases , the mass renormalization parameter @xmath65 gets larger , and therefore the many - body effects become stronger . our results show that the agreement between the results of the monte carlo simulations and the perturbation - theoretic results gets worse as @xmath65 increases , and that not surprisingly , the perturbation - theoretic calculations break down when one enters the strong coupling regime @xmath108 , where the pseudogap is found . a more rigourous analysis of the relevance of the effective quasiparticle interactions as @xmath95 increases or as the energy scales are decreased would require a renormalization group ( rg ) treatment@xcite . recent rg calculations@xcite on the border of the ferromagnetic state indicate that the quasiparticle interactions are indeed relevant in @xmath142 , and the rg flows to strong coupling as the energy cutoff is decreased . one would like to understand what property of the full vertex function @xmath143 is responsible for the appearance of the pseudogap and seems to be missing in the first order approximation to @xmath144 . our physical picture of the pseudogap state emerging from quantum mechanical coherence effects contained in high order feynman diagrams is to be contrasted with the results of refs.@xcite where a suppression of the quasiparticle tunneling density of states at the fermi level is obtained in the single spin or charge - fluctuation exchange approximation . this effect is typically obtained with relatively large magnetic or charge correlation lengths . in our model , the calculations reported here and in ref.@xcite show that as one approaches the border of magnetic long - range order , @xmath145 , the multiple spin - fluctuation exchange processes become important long before a suppression of the quasiparticle tunneling density of states at the fermi level is seen in the first order perturbation - theoretic and self - consistent calculations . indeed , pseudogap effects are only obtined in our calculations when the dimensionless mass renormalization parameter @xmath108 , i.e in the strong coupling regime where one does nt expect diagrammatic perturbation theory to give reliable approximations . the above finding is likely to be valid more generally , since the intuitive arguments for the physical origin of the pseudogap@xcite lead one to expect the breakdown of migdal s theorem to be a generic occurence near a spin or charge instability . there is also an important difference between a vertex correction induced pseudogap and a single - fluctuation exchange pseudogap . in the latter case , there is no essential distinction bewteen spin and charge fluctuations , in that at the single - fluctuation exchange level , for a given fluctuation spectrum the spin and charge - fluctuation theories of the quasiparticle spectral function can be made identical by an appropriate scaling of the coupling constant to the molecular field . this is no longer the case when vertex corrections are included , since these actually depend on the nature of the hubbard - stratonovich field , in our case vector versus scalar . the distinction could turn out to be essential , since we find , for a range of model parameters , that a pseudogap is observed for quasiparticles coupled to spin fluctuations but not in the corresponding charge - fluctuation case . moukouri et al.@xcite have developed a many - body theory of the precursor pseudogap to the mott transition in the half - filled hubbard model . their theory is inspired by the fluctuation exchange approximation ( flex)@xcite in which bare spin and charge susceptibilities are used to build up the effective quasiparticle interaction , corresponding to @xmath146 in our model . the key respect in which the theory of moukouri et al.@xcite differs from flex is that the coupling to spin and charge fluctuations are not given by the bare on - site coulomb repulsion , but by renormalized parameters determined self - consistently in such a way that an exact relationship between the single and two - particle green s functions is satisfied . this last step goes beyond perturbation theory and it is therefore plausible that the precursor pseudogap to the mott transition seen in the monte carlo simulations of the half - filled hubbard model@xcite is also nonperturbative in origin . the analog of their scheme for the present model would be the use of the first order perturbation theory approximation for the quasiparticle self - energy @xmath147 , eq . ( [ 1pt ] ) and a simultaneous renormalization of the coupling constant @xmath16 and the correlation wavevector @xmath95 . a renormalization of the coupling constant @xmath16 could account for all vertex corrections provided they are local in space and time . one can indeed get a pseudogap in the tunneling density of states with the first order perturbation theory approximation to @xmath148 ( eq . ( [ 1pt ] ) ) , as in refs.@xcite , provided @xmath95 is renormalized to lower values and @xmath16 renormalized to higher ones . one would thus have to renormalize the model to stronger coupling , roughly to values of @xmath149 . it should be clear that in this regime , first order perturbation theory is not controlled . one would also naively expect a sensible renormalization scheme that goes beyond the one - loop level to lead to renormalized values of @xmath95 larger than the bare value . the renormalized theory should be further away from the magnetic instability than the one - loop approximation rather than closer to it , since ideally one would like the improved theory to satisfy the mermin - wagner theorem in two dimensions . if @xmath95 were to be increased by the renormalization scheme , in order to obtain a pseudogap in @xmath97 one would likely need a large renormalization of the coupling @xmath16 and such a scheme for the present model does not look promising to the author . however , it is important to note that the model studied here , although similar in some respects , is actually different than the one considered in ref.@xcite . the renormalization scheme proposed by moukouri et al . which works well for the hubbard model need not necessarily apply to other theories . in ref.@xcite , we pointed out that in the case of quasiparticles coupled to ferromagnetic spin fluctuations , our results are at variance with expectations based on the standard theory of quantum critical phenomena@xcite . since the dynamical exponent @xmath150 , in @xmath151 spatial dimensions , the effective dimension is @xmath112 and is greater than the upper critical dimension @xmath152 above which one would expect the first order theory to be at least qualitatively correct . but our nonperturbative results show that at least for small enough @xmath95 , the first order theory qualitatively breaks down . al@xcite have shown that the quasiparticle interactions are indeed relevant in the rg sense for ferromagnetic fluctuations in @xmath142 , a result consistent with our findings . for antiferromagnetic and charge fluctuations , @xmath113 , the marginal case , and hence the qualitative breakdown of the first order approximation is nt necessarily inconsistent with the standard theory . however , the scaling relations derived in ref.@xcite rely on the applicability of perturbation theory . if the pseudogap effects are indeed intrinsically nonperturbative in nature , a conjecture that is consistent with the present work , it opens the possibility that the physics in the proximity of a quantum critical point is dominated by nonperturbative quantum mechanical effects and therefore even richer than anticipated in the earlier work@xcite . a number of new ideas in this field have recently been proposed@xcite and a discussion of some fundamental problems associated with quantum critical points can be found in ref.@xcite . we studied a nonperturbative formulation of the magnetic interaction model , in which quasiparticles are coupled to a gaussian distributed dynamical molecular exchange field . far from the magnetic boundary , the multiple magnetic fluctuation exchange processes do not bring about qualitative changes to the quasiparticle spectrum . but as one gets closer to the border of long - range magnetic order , we find , for a range of model parameters , that migdal s theorem does nt apply and the quasiparticle spectrum is qualitatively different from its eliashberg approximation . moreover , we find that going one step beyond the single spin - fluctuation exchange approximation and including first order vertex corrections , self - consistently or not , does nt help to reproduce the qualitative changes seen in the nonperturbative calculations . near the magnetic boundary , the simple perturbation expansion shows signs it is not well behaved , since the second order results differ greatly from their first order counterparts . the self - consistent , or renormalized perturbation expansion , which effectively consists in a reordering of the diagrammatic perturbation theory , is better behaved in that the differences between first and second order are much less pronounced . however , even if the renormalized perturbation expansion converges , our results show that it is quite likely to converge to the wrong answer , which could be explained if the original perturbation expansion is divergent . the intuitive argument for the onset of pseudogap behavior@xcite , namely that if the distance quasiparticles can travel during their lifetime becomes shorter than the molecular field correlation length , these quasiparticles effectively see long - range order , does not explain the failure of the single spin - fluctuation exchange approximation . as we pointed out in ref.@xcite , one can get in the regime where the mean - free path gets shorter than @xmath153 in the eliashberg approximation , but fail to observe a pseudogap in this regime . since we have not been able to produce a good fit to the monte carlo simulations by including either first order or vertex corrections that are local in space and time , i.e by a renormalization of the coupling constant @xmath16 to the molecular field , we conjecture that the physical origin of the pseudogap state found in the present calculations lies in non - local vertex corrections produced by high order spin - fluctuation exchanges . these vertex corrections effectively induce a quasiparticle coupling to the dynamical molecular field that is non - local in both space and time . the above conjecture raise the question of what essential property of the vertex function is not captured by its first order approximation . a study of the vertex function along the same lines as the work reported here for the single particle green s function should provide further insights into this problem . i would like to thank p. coleman , j.r . cooper , p.b . littlewood , g.g . lonzarich , j. loram , and d. pines for discussions on this and related topics . we acknowledge the support of the epsrc , the newton trust and the royal society . bickers , d.j . scalapino , and s.r . white , phys . rev . lett . * 62 * , 961 ( 1989 ) ; t. moriya , y. takahashi and k. ueda , j. phys . . jpn . * 52 * , 2905 ( 1990 ) ; p. monthoux , a.v . balatsky and d. pines , phys . * 67 * , 3448 ( 1991 ) .

we study a model of quasiparticles on a two - dimensional square lattice coupled to gaussian distributed dynamical molecular fields . we consider two types of such fields , a vector molecular field that couples to the quasiparticle spin - density and a scalar field coupled to the quasiparticle number density . the model describes quasiparticles coupled to spin or charge fluctuations , and is solved by a monte carlo sampling of the molecular field distributions . the nonperturbative solution is compared to various approximations based on diagrammatic perturbation theory . when the molecular field correlations are sufficiently weak , the diagrammatic calculations capture the qualitative aspects of the quasiparticle spectrum . for a range of model parameters near the magnetic boundary , we find that the quasiparticle spectrum is qualitatively different from that of a fermi liquid , in that it shows a double peak structure , and that the diagrammatic approximations we consider fail to reproduce , even qualitatively , the nonperturbative results of the monte carlo calculations . this suggests that the magnetic pseudogap induced by a coupling to antiferromagnetic spin - fluctuations and the spin - splitting of the quasiparticle peak induced by a coupling to ferromagnetic spin - fluctuations lie beyond diagrammatic perturbation theory . while a pseudogap opens when quasiparticles are coupled to antiferromagnetic fluctuations , such a pseudogap is not observed in the corresponding charge - fluctuation case for the range of parameters studied , where vertex corrections are found to effectively reduce the strength of the interaction . this suggests that one has to be closer to the border of long - range order to observe pseudogap effects in the charge - fluctuation case than for a spin - fluctuation induced interaction under otherwise similar conditions . the diagrammatic approximations that contain first order vertex corrections show the enhancement of the spin - fluctuation induced interaction and the suppression of the effective interaction in the charge - fluctuation case . however , for the range of model parameters considered here , the multiple spin or charge - fluctuation exchange processes not included in the diagrammatic approximations considered are found to be important , especially for quasiparticles coupled to charge fluctuations .

as widely discussed in many papers , the spectrum of the cosmic microwave background ( cmb ) carries unique informations on physical processes occurring during early cosmic epochs ( see e.g. danese & burigana 1993 and references therein ) . the comparison between models of cmb spectral distortions and cmb absolute temperature measures can constrain the physical parameters of the considered dissipation processes . we recently discussed ( salvaterra & burigana 2002 ) the implications of the current cmb spectrum data by jointly considering distortions generated in a wide range of early or intermediate cosmic epochs and at late cosmic epochs . various cmb spectrum experiments at long wavelengths , @xmath17 cm , are ongoing and planned for the future in order to improve the still quite poor accuracy of the data in this spectral region , where the maximum deviations from a pure blackbody spectrum are expected in the case of dissipation processes occurred at early and intermediate epochs . in this work we jointly consider the data from the firas instrument aboard the cobe satellite and simulated sets of cmb spectrum observations at wavelengths larger than 1 cm with the sensitivities expected from future experiments in order to discuss their impact for the recovery of the thermal history of the universe . in section 2 we briefly summarize the general properties of the cmb spectral distortions and the main physical informations that can be derived from the comparison with the observations . in section 3 we briefly discuss the performances of current and future cmb spectrum observations at long wavelengths and describe the generation of the simulated observations used in this work . the implications of observations with sensitivities typical of forthcoming and future ground and balloon experiments are presented in section 4 , while in section 5 we extensively discuss the implications of experiments at long wavelengths with a sensitivity comparable to that of cobe / firas , as foreseen for a space experiment , dimes , proposed to the nasa in the 1995 and designed to measure the cmb absolute temperature at @xmath18 cm with a sensitivity of @xmath3 mk ( kogut 1996 ) . in section 6 we present a detailed discussion of the capabilities of future cmb spectrum observations to discriminate between the firas calibration by the cobe / firas team ( referred as here `` standard '' calibration ; see fixsen et al . 1994 , 1996 , mather et al . 1999 , and references therein ) and that proposed by battistelli , fulcoli & macculi 2000 . the possibility to improve our knowledge of the free - free distortions is considered in section 7 , while section 8 is devoted to identify the experimental sensitivity requirements for an accurate baryon density evaluation through the detection of possible long wavelength distortions . finally , we draw our main conclusions in section 9 . the cmb spectrum emerges from the thermalization redshift , @xmath19 , with a shape very close to a planckian one , owing to the strict coupling between radiation and matter through compton scattering and photon production / absorption processes , radiative compton and bremsstrahlung , which were extremely efficient at early times and able to re - establish a blackbody ( bb ) spectrum from a perturbed one on timescales much shorter than the expansion time ( see e.g. danese & de zotti 1977 ) . the value of @xmath20 ( burigana , danese & de zotti 1991a ) depends on the baryon density ( in units of the critical density ) , @xmath21 , and the hubble constant , @xmath22 , through the product @xmath23 ( @xmath22 expressed in km / s / mpc ) . on the other hand , physical processes occurring at redshifts @xmath24 may lead imprints on the cmb spectrum . the timescale for the achievement of kinetic equilibrium between radiation and matter ( i.e. the relaxation time for the photon spectrum ) , @xmath25 , is @xmath26 where @xmath27 is the photon electron collision time , @xmath28 , @xmath29 being the electron temperature and @xmath30 ; @xmath31 is the mean fractional change of photon energy in a scattering of cool photons off hot electrons , i.e. @xmath32 ; @xmath33 is the present radiation temperature related to the present radiation energy density by @xmath34 ; a primordial helium abundance of 25% by mass is here assumed . it is useful to introduce the dimensionless time variable @xmath35 defined by @xmath36 where @xmath37 is the present time and @xmath38 is the expansion time given by @xmath39^{-1/2 } \sec \ , , \ ] ] @xmath40 being the redshift of equal non relativistic matter and photon energy densities ( @xmath41 is the density of non relativistic matter in units of critical density ) ; @xmath42 , @xmath43 being the number of relativistic , 2component , neutrino species ( for 3 species of massless neutrinos , @xmath44 ) , takes into account the contribution of relativistic neutrinos to the dynamics of the universe , is itself a function of the amount of energy dissipated . the effect , however , is never very important and is negligible for very small distortions . ] . burigana , de zotti & danese 1991b have reported on numerical solutions of the kompaneets equation ( kompaneets 1956 ) for a wide range of values of the relevant parameters and accurate analytical representations of these numerical solutions , suggested in part by the general properties of the kompaneets equation and by its well known asymptotic solutions , have been found ( burigana , de zotti & danese 1995 ) . the cmb distorted spectra depend on at least three main parameters : the fractional amount of energy exchanged between matter and radiation , @xmath45 , @xmath46 being the radiation energy density before the energy injection , the redshift @xmath47 at which the heating occurs , and the baryon density @xmath48 . the photon occupation number can be then expressed in the form @xmath49 where @xmath50 is the dimensionless frequency @xmath51 ( @xmath52 being the present frequency ) , and @xmath53 characterizes the epoch when the energy dissipation occurred , @xmath47 being the corresponding redshift ( we will refer to @xmath53 computed assuming @xmath54 , so that the epoch considered for the energy dissipation does not depend on the amount of released energy ) . the continuous behaviour of the distorted spectral shape with @xmath55 can be in principle used also to search for constraints on the epoch of the energy exchange . of course , by combining the approximations describing the distorted spectrum at early and intermediate epochs with the comptonization distortion expression describing late distortions , it is possible to jointly treat two heating processes ( see burigana et al . 1995 and salvaterra & burigana 2002 and references therein for a more exhaustive discussion ) . in this work the measures of the cmb absolute temperature are compared with the above models of distorted spectra for one or two heating processes by using a standard @xmath56 analysis . we determine the limits on the amount of energy possibly injected in the cosmic background at arbitrary primordial epochs corresponding to a redshift @xmath47 ( or equivalently to @xmath55 ) . this topic has been discussed in several works ( see e.g. burigana et al . 1991b , nordberg & smoot 1998 , salvaterra & burigana 2002 ) . as in salvaterra & burigana 2002 , we improve here the previous methods of analysis by investigating the possibility of properly combining firas data with longer wavelength measures with the sensitivities expected for forthcoming and future experiments and by refining the method of comparison with the theoretical models . we will consider the recent improvement in the calibration of the firas data , that sets the cmb scale temperature at @xmath57 k at 95 per cent confidence level ( cl ) ( mather et al . we do not consider the effect on the estimate of the amount of energy injected in the cmb at a given epoch introduced by the calibration uncertainty of firas scale temperature when firas data are treated jointly to longer wavelength measures , since the analysis of salvaterra & burigana 2002 shows that it introduces only minor effects . then , we study the combined effect of two different heating processes that may have distorted the cmb spectrum at different epochs . this hypothesis has been also taken into account by nordberg & smoot 1998 , who fit the observed data with a spectrum distorted by a single heating at @xmath58 , a second one at @xmath59 and by free - free emission , obtaining limits on the parameters that describe these processes . as in salvaterra & burigana 2002 , we extend their analysis by considering the full range of epochs for the early and intermediate energy injection process , by taking advantage of the analytical representation of spectral distortions at intermediate redshifts ( burigana et al . since the relationship between free - free distortion and comptonization distortion is highly model dependent , being related to the details of the thermal history at late epochs ( danese & burigana 1993 , burigana et al . 1995 ) , and can not be simply represented by integral parameters , we avoid a combined analysis of free - free distortions and other kinds of spectral distortions and separately discuss the implications of future , more accurate long wavelength measures on free - free distortions . it is also possible to extend the limits on @xmath5 for heatings occurred at @xmath60 , where @xmath61 is the redshift corresponding to @xmath62 , when the compton scattering was able to restore the kinetic equilibrium between matter and radiation on timescales much shorter than the expansion time and the evolution on the cmb spectrum can be easily studied by replacing the full kompaneets equation with the differential equations for the evolution of the electron temperature and the chemical potential . this study can be performed by using the simple analytical expressions by burigana et al . 1991b instead of numerical solutions . a recent analysis of the limits on the amount of the energy possibly injected in the cosmic background from the currently available data is reported in salvaterra & burigana 2002 . in particular , they found that the measures at @xmath17 cm do not significantly contribute to these constraints because of their poor sensitivity compared to that of firas . new and more accurate measurements are also needed in this range , which is particular sensitive to early energy injection processes . in fact , the current constraints on @xmath5 at @xmath63 are a factor @xmath64 less stringent than those at @xmath55 less than @xmath3 , because of the frequency coverage of firas , which mainly set the current constraints at the all cosmic epochs . thus , we are interested to investigate the role of future ground , balloon and space experiments at @xmath17 cm jointed to the firas measures at @xmath0 cm . to evaluate the scientific impact represented by the future experiment improvements , we create different data sets simulating the observation of a not distorted spectrum both from ground and balloon experiments and from a space experiment like dimes through the method described in section 3.1 . for a dimes - like experiment , we also explore the possibility of the observation of distorted spectra for different amounts of the energy injected in the radiation field and for different cosmic epochs . each data set will be then compared to models of distorted spectra by using the method described in salvaterra & burigana 2002 ( see also burigana & salvaterra 2000 for the details of the code ) to recover the value of @xmath5 or constraints on it , the heating epoch , @xmath55 , the free - free distortion parameter @xmath65 , and the combination , @xmath48 , of the baryon density and the hubble constant . for simplicity , we restrict to the case of a baryon density @xmath66 our analysis of the implications for the thermal history of the universe , but the method can be simply applied to different values of @xmath48 . in presence of an early distortion , @xmath48 could be in principle measured by cmb spectrum observations at long wavelengths provided that they have the required sensitivity about the minimum of the cmb absolute temperature ( see section 8) . the cmb spectrum experiments currently under study are dedicated to improve our knowledge at wavelengths longer than those covered by firas . at centimeter and decimeter wavelengths , the available measures typically show large error bars although some experiments are rather accurate ( i.e. , the measure of staggs et al . 1996 at @xmath67 cm ) . very accurate data at long wavelengths could give a significant improvement to our knowledge of physical processes in the primeval plasma , particularly at high redshifts . these projects regard measurements from ground , balloon and space . as representative cases , and without the ambition to cover the whole set of planned experiments , we briefly refer here to the ground experiment tris at very long wavelengths and to the dimes experiment from space ( kogut 1996 ) designed to reach an accuracy close to that of firas up to @xmath68 cm . tris is a set of total power radiometers designed to measure the absolute temperature of the cmb at three frequencies : 0.6 , 0.82 and 2.5 ghz . at these wavelengths ( @xmath69 cm ) the measurements are difficult because the cmb signal is comparable to other components of the antenna temperature : galactic background , unresolved extra - galactic sources , sidelobes pickup and atmospheric emission . to improve the experimental situation , tris will make absolute maps of large areas of the sky at the three frequencies , to disentangle the various components of the celestial signal ; all the lossy parts of the antenna front ends of the receivers will be cooled down liquid helium temperature , to reduce the thermal noise of these components ; the receiver temperatures will be very carefully stabilized to reduce drifts and gain variations . the tris expected sensitivity is of about 200 mk at the three frequencies . dimes ( diffuse microwave emission survey ) is a space mission submitted to the nasa in 1995 , designed to measure very accurately the cmb spectrum at wavelengths in the range @xmath18 cm ( kogut 1996 ) . dimes will compare the spectrum of each 10 degree pixel on the sky to a precisely known blackbody to precision of @xmath3 mk , close to that of firas ( @xmath70 mk ) . the set of receivers is given from cryogenic radiometers operating at six frequency bands about 2 , 4 , 6 , 10 e 90 ghz using a single external blackbody calibration target common to all channels . in each channel , a cryogenic radiometer switched for gain stability between an internal reference load and an antenna with 10 degree beam width , will measure the signal change as the antenna alternately views the sky and an external blackbody calibration target . the target temperature will be adjusted to match the sky signal in the lowest frequency band , allowing the absolute temperature to be read off from the target thermometry with minimal corrections for the instrumental signature . with its temperature held constant , the target will rapidly move over the higher - frequencies antenna apertures , effectively comparing the spectrum of diffuse emission from the sky to a precise blackbody . by comparing each channel to the same target , uncertainties in the target emission cancel so that deviations from a blackbody spectral shape may be determined much more precisely than the absolute temperature . the dimes design is driven by the need to reduce or eliminate systematic errors from instrumental artifacts . the instrument emission will be cooled to 2.7 k , whereas the calibration uncertainty will be minimized by using a single calibration target , common to all channels . the atmospheric emission will be observed from low earth orbit and the multiple channels measurements will minimize the foreground emission problems . the dimes sensitivity represents an improvement by a factor better than 300 with respect to previous measurements at cm wavelengths . we collect different data sets , simulating measurements of different cmb spectra , distorted or not , at the frequency ranges of the considered experiments . we add to these simulated data the firas data at higher frequencies according to the most recent calibration of the temperature scale at 2.725 k ( mather et al . 1999 ) . for the cases of distorted spectra we calculate the theoretical temperature of the cmb spectrum at the wavelengths of the new experiments as discussed in the previous section . of course , the thermodynamic temperature held obviously constant at all the frequencies for the case of a non distorted spectrum . the theoretical temperatures are then fouled to simulate real measurements affected by instrumental noise . the simulated temperature @xmath71 at the frequency @xmath52 is given by @xmath72 where @xmath73 is the theoretical temperature at the frequency @xmath52 and err(@xmath52 ) is the expected rms error ( at 1 @xmath74 ) of the experiment at this frequency . the numbers @xmath75 are a set of random numbers generated according to a gaussian distribution with null mean value and unit variance with the routine gasdev by press et al . 1992 ( 7 ) . we analyse here the impact of possible future observations from ground and balloon in the case of a not distorted spectrum at the temperature @xmath76 k. the results are thus comparable to those obtained with the firas data alone ( see e.g. salvaterra & burigana 2002 ) . to build the first simulated data set ( g&b1-bb ) , we split the region from 1 to 80 cm in three ranges and associate different values of sensitivity to each range according to the analysis of the main problems affecting the available observations in different spectral regions ( e.g. salvaterra & burigana 2000 ) . 1 . @xmath77 cm . in this range the measurements of staggs et al . 1996 show an uncertainty of @xmath78 mk . thus , quite accurate measures could be carried out in this range . we choose to associate to the future experiments at these wavelengths an improved typical sensitivity of 10 mk . 2 . @xmath79 cm . ground experiments in this range show error bars of about @xmath80 mk . progresses could be reached by improving the accuracy of the subtraction of the atmospheric contribution which dominates the final error at these wavelengths . thus , we choose to associate to the data in this region a typical sensitivity of 40 mk ; 3 . @xmath81 cm . observations in this range are still quite difficult , the typical sensitivities being between 200 mk for measures at 10 cm and 1.5 k for those at longer wavelengths . the expected sensitivity of the tris experiment ( see section 3 ) is of @xmath82 mk . thus , we choose to associate to future experiments in this range a typical sensitivity of 200 mk . the frequencies of the experiments at @xmath17 cm of the two last decades ( see e.g. table 1 of salvaterra & burigana 2002 ) , where suitable observation windows should exist and the presence of man made interferences should be not a concern , have been adopted in the generation of simulated observations . finally , we complete this data set by adding the firas measures calibrated at 2.725 k according to mather et al . 1999 to the above simulated data . a second data set ( g&b2-bb ) is built as before but by improving by a factor 10 the sensitivity associated to each of the above three frequency range in order to evaluate the impact of highly optimistic future progresses of ground and ballon experiments . the results of the fits to the simulated data g&b1-bb and g&b2-bb jointed to firas data are shown in fig . 1 [ for graphic purposes , we report in the plots the exact value of @xmath55 and the power - law approximation @xmath83 ( burigana et al . 1991b ) for the redshift ] . as evident , realistic improvements of future experiments from ground and balloon do not significantly change the firas limits . even under much more optimistic experimental conditions , able to decrease the errors by a factor 10 , the situation can not substantially improve , being the limits on @xmath84 obtained in this case only just more stringent than those based on firas data alone . we then conclude that , unfortunately , observations of the cmb absolute temperature with sensitivity levels typical of future ground and balloon experiments do not seem able to improve the limits on the amount of energy injected in the cosmic radiation field inferred on the basis of the currently available measures . we generate a set ( d - bb ) of simulated data in the case of a blackbody spectrum at a temperature of 2.725 k in order to evaluate the capability of an experiment with a sensitivity comparable to that expect for dimes to improve the constraints on the amount of the energy injected in the cosmic radiation field . the analysis of this case is in fact directly comparable with the results obtained from the fit to the firas data alone . then , we build up other data sets representing the observations of cmb spectra distorted by energy injections at different cosmic epochs in order to investigate the possibility of a dimes - like experiments to firmly determine the presence of spectral distortions . we consider processes occurring at a wide range of cosmic epochs , represented by the dimensionless time @xmath85 5 , 4 , 3 , 2 , 1 , 0.5 , 0.25 , 0.1 , 0.05 , 0.025 , 0.01 , and @xmath86 . we consider four representative values of fractional injected energy : @xmath87 , a value not much below the upper firas limits ; @xmath88 , well above the firas upper limit ( see section 6 ) ; @xmath89 and @xmath90 , two values well below the firas upper limit , to test the chances to detect very small distortions with a dimes - like experiment . as a further representative case , we simulate the observation of a spectrum distorted by two heating processes occurring at different epochs , the first at @xmath58 and the second at @xmath86 , both characterized by @xmath91 . all these distorted spectra are computed by setting @xmath92 and @xmath93 km / s / mpc . as a variance with respect the previous section , we choose here the frequencies of the simulated observations by adopting the five frequency channels of the dimes experiment . as in previous section , we complete these data set by adding the firas measures calibrated at 2.725 k. we fit the simulated data d - bb with a spectrum distorted by an energy injection at different values of @xmath55 in order to recover the value of @xmath5 , expected to be null , and the limits on it . the fit results are reported in fig . 2 . it is evident how future data at this sensitivity level will allow a strong improvement of the limits obtained with the firas data alone . the recovered best - fit value of @xmath5 is always compatible with the absence of distortions within the limits at 95 per cent cl . for heating processes at low @xmath94 ( @xmath95 ) the fit is substantially dominated by the firas data and the lower and the upper limits on @xmath5 are still @xmath96 . on the contrary , for early distortions ( @xmath6 ) the low frequency measures of a dimes - like experiment will allow to improve the firas constraints by a factor @xmath97 , the proper value increasing with the considered dissipation redshift . we conclude that measures from an instrument like dimes could represent a very good complement to the firas data . in the next sections we will analyse in detail the capability of a dimes - like experiment to determine the presence of spectral distortions possibly present in the cmb spectrum . the test reported in the previous section suggests that even small distortions could be determined provided that the dissipation would have occurred at relatively early epochs , @xmath6 . thus , we analyse the sensitivity of a dimes - like experiment in the recovery of the amount of energy possibly injected in the radiation field and explore also the possibility to determine the dissipation process epoch . firstly , we fit the data simulated as above under the hypothesis that the heating epoch is known ; more explicitly , we fit the data with a theoretical spectrum distorted by a process occurring at the considered @xmath55 by allowing to optimize @xmath5 ( and @xmath33 ) but by taking @xmath55 fixed . in this way we can see how accurately @xmath5 could be in principle recovered . on the other hand , unless we want to use the cmb spectrum data to constrain theoretical models with a well defined dissipation epoch , we are typically interested to set constraints on the value of @xmath5 possibly injected at a given unknown epoch occurring within a relatively wide cosmic period ; in addition , many classes of physical processes in the plasma epoch involve time parameters and it is important to understand how they can be possibly constrained by the comparison with cmb spectrum observations . thus , we focus on the cases of spectra distorted at high ( @xmath58 ) , medium ( @xmath98 ) and at low ( @xmath86 ) redshifts by fitting the simulated data by relaxing the a priori knowledge of the dissipation epoch . in this way we would be able to evaluate the possibility of determining also the epoch of the heating with a proper higher value of @xmath99 would give a distorted spectrum essentially indistinguishable by that generated in the case of a dissipation at @xmath62 with a lower value of @xmath99 , see section 5.4 . ] without a priori informations by jointly evaluating the impact of the unknowledge of the dissipation epoch on the recovery of injected energy . we will test also the possibility of deriving at the same time information on the baryon density . 5.2.2.1 @xmath100 energy injections at firas limits dissipation epoch : known _ 0.2 cm as a representative case we consider the simulated observation of a spectrum distorted at different values of @xmath55 by an energy injection with @xmath101 , a value below , but not much , the firas upper limit on @xmath5 . these data are then compared with the theoretical cmb spectrum distorted at the same @xmath55 ( assumed to be known ) by performing the fit only over @xmath5 and @xmath33 : this is appropriate to cases in which we have a quite well defined a priori information on the dissipation epoch but not on the amount of released energy . we find that for high redshift processes , @xmath102 , @xmath5 is precisely determined . for distortions at lower @xmath94 , @xmath103 , we obtain limits similar to those given from the currently available data , since the the fit result is mainly driven by firas data , more sensitive to these kinds of distortions , mainly located at high frequencies . 0.4 cm _ 5.2.2.2 @xmath100 energy injections at firas limits dissipation epoch : unknown _ 0.2 cm we relax here the assumption to know the dissipation epoch . we consider firstly the case of the fit to data simulated assuming a spectrum distorted at @xmath58 with @xmath101 with cmb theoretical spectra distorted at different values of @xmath55 . the best - fit to these data assuming @xmath58 gives a very accurate recovery of the input value of @xmath5 with a small quoted error ( we find an associated statistical error of @xmath104% at 95 per cent cl ) . the best - fit on @xmath5 assuming lower values of @xmath55 is far from the input value and the @xmath56 increases . thus , we search for a favourite value of @xmath55 by performing the fit over @xmath5 , @xmath33 , and @xmath55 . we obtain that the recovered best - fit value of @xmath55 is exactly the input one , 5.0 , and lower limit on @xmath55 at 95 per cent cl is 2.4 . by searching also for a favourite value of @xmath48 ( set to 0.05 in the data simulation ) , we obtain a 68 per cent cl range of @xmath105 . we repeated the same analysis in the case of a spectrum distorted at @xmath98 with @xmath101 . again , the recovered value of @xmath5 is close to the input one for fits with @xmath106 ( in this case we recover the input value of @xmath5 with an uncertainty of @xmath107% at 95 per cent cl ) and we are also able to determine a significative range ( @xmath6 at 95 per cent cl ) of favourite values of @xmath55 , although wider than in the previous case , while @xmath48 is found to be in the range @xmath108 at 68 per cent cl . similar results on @xmath55 and @xmath48 can not be obtained in the case of fit to the data simulating the observation of a spectrum distorted at @xmath109 . the fit result then is then compatible also with energy injections with smaller values of @xmath5 but at higher @xmath55 and with a non distorted spectrum . the @xmath56 value does not significantly change when @xmath55 varies . this is again the result of the main role of firas data for late dissipation processes . 5.2.2.3 @xmath100 energy injections below firas limits dissipation epoch : known _ 0.2 cm a dimes - like experiment should be able to detect also small spectral distortions . let consider here the case of a spectrum distorted from an energy injection with @xmath91 , about a factor 10 below the firas limits at 95 per cent cl . as shown in fig . 3 , if the dissipation epoch is known , we find that for processes at early and intermediate epochs the best - fit result is very close to the input value of the simulated data , although the limits on @xmath5 are not so stringent as in the case with a larger energy injection . for @xmath6 , a spectral distortion would be firmly detected at 95 per cent cl . 0.4 cm _ 5.2.2.4 @xmath100 energy injections below firas limits dissipation epoch : unknown _ 0.2 cm we relax here again the assumption to know the dissipation epoch . our results are summarized in fig . 4 : even for distortions well below the firas limits ( @xmath110 is assumed here ) an experiment like dimes would provide significative constraints both on the amount of dissipated energy and on the dissipation epoch in the case of and early processes . in this test the input dissipation epoch ( @xmath111 ) is again quite well recovered , the @xmath56 increasing of 4 when @xmath55 becomes close to unity . it would be also possible to provide an independent estimate of the baryon density : we find @xmath112 at 68 per cent cl . for a process occurring at intermediate epochs ( @xmath98 in this specific test ) we find that it is still possible to determine the amount of injected energy , but in this case of distortions significantly smaller than the firas limits the @xmath56 is no longer particularly sensitive to @xmath55 and the recovered range of dissipation epochs is wide . energy dissipations processes at intermediate epochs may then result still compatible with these simulated data and only energy injections at late epochs , could be excluded ( in this test we find that the @xmath56 increases of @xmath113 for @xmath114 ) . as already found for larger distortions , significant information on late processes can not be obtained from accurate long wavelength data because of the more relevant role of the firas data . 0.4 cm _ 5.2.2.5 @xmath100 very small energy injections _ 0.2 cm finally , we consider the possibility to detect very small energy injections , namely with @xmath115 . if the dissipation epoch is known , the result of the fit shows that is still possible to determine a no null distortion provided that the dissipation process occurs at high redshifts . for energy injections at epochs close to @xmath58 the recovered value of @xmath5 , @xmath116 in this test , is quite close to the input one and the associate statistical error gives a @xmath5 range of @xmath117 at 95 per cent cl . an indication of the dissipation epoch can be also derived ( in this case we find that the @xmath56 increases of @xmath113 for @xmath118 ) . unfortunately , no significant informations on @xmath48 can be obtained with the considered sensitivity and frequency coverage in the case of so small distortions . for dissipations at @xmath119 a significative distortion can be also determined , the recovered value of @xmath5 ranging between @xmath120 and @xmath121 at 95 per cent cl for @xmath122 where the @xmath123d.o.f . is about its minimum , but significative informations on the dissipation epoch can be no longer derived . finally , in the case of processes at late epochs , the fit results are compatible with an unperturbed spectrum , being these kind of distortions mainly detectable at firas frequencies . we discuss here the possibility to significantly improve the constraints on ( or to detect ) energy exchanges also in the more general case of a joint analysis of early / intermediate and late dissipation processes . by exploiting the data set d - bb presented in sect . 5.1 we consider the case of no significant deviations from a planckian spectrum . top panel of fig . 5 shows the limits on the energy exchange as function of @xmath55 by allowing for a later dissipation process possibly occurred at @xmath86 ; bottom panel of fig . 5 shows the constraints on the energy injected at low @xmath94 by allowing for a previous distortion occurred at any given @xmath55 . in fig . 5 we report also the comparison with the results based only on the firas data ( as shown in salvaterra & burigana 2002 , the current long wavelength measures do not change significantly these results ) . the conclusion is impressive : the constraints on @xmath5 for early and intermediate dissipation processes could be improved by a factor @xmath124 , depending on the considered dissipation epoch . in addition , the constraints on the energy dissipation at late epochs can be also improved , by a factor of about two , because of the reduction of the partial degeneracy introduced by the rough compensation ( salvaterra & burigana 2002 ) between the effect of early and late energy exchanges on the cmb spectrum when no accurate measures are available at long wavelengths . to complete the analysis of the impact of a possible dimes - like experiment , we consider the simulated observation of a spectrum distorted by a first energy dissipation at @xmath58 with @xmath91 and a second one at @xmath86 with @xmath91 . we then compare these data with theoretical spectra distorted by a process at @xmath86 and another at any given @xmath125 according to our grid of @xmath55 ( see section 5.1 ) . we find that a dimes - like experiment would allow to firmly determine the presence of the distortion at high @xmath94 ; in particular , at @xmath58 the recovered @xmath5 is very close to the input one , as already found for the tests described in sections 5.2.2.3 and 5.2.2.4 . on the contrary , for the dissipation process at @xmath86 the fit result is compatible with an unperturbed spectrum , since the firas data dominate the limits on the distortions at low redshifts . the limits on @xmath5 for processes at low redshifts are , however , again more stringent , by a factor 2 , than those obtained with the currently available data . we find that the @xmath56 significantly increases by assuming in the fit an earlier process at decreasing @xmath55 ( the @xmath56 increases of @xmath113 for @xmath126 ) : even in the case of a combination of an early and a late process a dimes - like experiment would be able to significantly constrain the epoch of the earlier energy exchange . we extend here at @xmath127 ( i.e. @xmath128 ) the constraints on @xmath5 that would be possible to derive at @xmath129 ( @xmath62 ) with a dimes - like experiment . we remember that at @xmath130 the compton scattering is able to restore , after an energy injection , the kinetic equilibrium between matter and radiation , yielding a bose - einstein ( be ) spectrum , and the combined effect of compton scattering and photon production processes tends to reduce the magnitude of spectral distortions , possibly leading to a blackbody spectrum . we firstly consider here the case of the simulated observation of a not distorted spectrum , that represents a good test of the possible improvements of an instrument like dimes , because the limits on @xmath5 at relevant redshifts can be directly compared to those obtained with firas data alone . for simplicity , we consider the case of a single energy injection possibly occurred in the cosmic thermal history . the comparison is shown in fig . 6 . as evident , the constraints on @xmath5 can be improved by a factor @xmath131 for processes possibly occurred in a wide range of cosmic epochs , corresponding to about a decade in redshift at @xmath94 about @xmath132 . of course , large energy injections are still possible at very early epochs close to the thermalization redshift , when primordial nucleosynthesis set the ultimately constraints on energy injections in the cosmic radiation field . for late dissipations , firas data mainly constrain @xmath5 . as a further example , we consider the constraints on the energy injections at @xmath133 ( @xmath134 ) in the case of a fit with a single energy injection to simulated observations of a spectrum distorted at @xmath62 with @xmath135 . as shown by the high redshift tails of the curves of fig . 3 , in this case the constraints on the thermal history of the universe would be completely different from those derived in the case in which distortions are not detected ( fig . . a firm detection of early energy injections would be clearly possible with the considered experimental performances and the constraints on the energy possibly injected at @xmath130 could be directly derived from such kind of future cmb spectrum data . in addition , fig . 6 shows that , contrariously to the case of the current observational status , the constraints on early energy exchanges based on future high accuracy long wavelength measures are no longer appreciably relaxed by assuming that a late process could be also occurred . in the previous sections , we considered simulated data with a planckian spectrum or with distortions compatible with the limits derived from firas data calibrated according to mather et al . 1999 . on the other hand , a recent analysis of firas calibration ( referred here as `` revised '' ) by battistelli et al . 2000 suggests a frequency dependence of the firas main calibrator emissivity . the firas data recalibrated according to their `` favourite '' calibration emissivity law ( r - firas data in what follows ) indicate the existence of deviations from a planckian shape or at least a significant relaxation of the constraints on them . salvaterra & burigana 2002 discussed the main implications of this analysis . although it seems quite difficult to fully explain from a physical point of view the r - firas data , two classes of phenomenological models may fit them : in the first one the main contribution derives from an intrinsic cmb spectral distortion with @xmath136 few @xmath137 occurring at early / intermediate epochs ; the second one involves a millimetric component possibly due to cold dust emission , described by a modified blackbody spectrum , added to a cmb blackbody spectrum at a temperature of @xmath113 mk below firas temperature scale of 2.725 k. fig . 7 shows as these models , very similar at millimeter wavelengths , predict significant differences at centimeter and decimeter wavelengths . in this section we carefully discuss the capabilities of forthcoming and future cmb spectrum measures at long wavelengths to discriminate between these two different firas calibrations and , in the case of the calibration by battistelli et al . 2000 , to distinguish between the two above scenarios . in the case in which the firas calibration by mather et al . 1999 is substantially correct , and therefore that revised by battistelli et al . 2000 is wrong , the cmb spectrum is expected to show an essentially planckian shape also at @xmath17 cm . to test the capabilities of forthcoming and future cmb spectrum experiments to rule out the `` revised '' calibration we fit simulated observations of a planckian spectrum at @xmath17 cm generated as described in section 3.1 added to the r - firas data set in terms of a single energy exchange process and of a combination of two processes at different cosmic times . we consider long wavelength data simulated assuming different values of @xmath33 , at steps of 0.1 mk within the firas temperature scale range , in order to reach the best compromise between them and the non - flat r - firas temperature data . the first five rows of table 1 summarize our results in the case of a dimes - like experiment assuming @xmath138 k , the case in which we find the best agreement . for comparison , we report in the last three rows of table 1 the results obtained by considering the r - firas data alone ( salvaterra & burigana 2002 ) . as evident from the significant increase of the @xmath123d.o.f . , long wavelength measures indicating a cmb planckian shape are not compatible with the firas data calibrated according to battistelli et al . 2000 . similar analyses carried out in the case of simulated long wavelength planckian data with the sensitivity of forthcoming , or improved , ground and balloon experiments ( data sets g&b1-bb and g&b2-bb ) show values of @xmath123d.o.f . close to @xmath139 ( similar to those obtained considering r - firas data alone ) , when the epoch ( or the epochs ) and the energy exchange ( exchanges ) of the dissipation process ( processes ) is ( are ) properly chosen . we then conclude that a cmb spectrum experiment at long wavelengths designed to rule out the firas calibration as revised by battistelli et al . 2002 should have a sensitivity comparable to that of a dimes - like experiment . .results of the fit to the r - firas data combined ( first five rows ) or not ( last three rows ) with long wavelength data with a sensitivity of a dimes - like experiment and simulated according to a planckian shape at @xmath140 k ( first 5 rows ) in terms of a single or two dissipation processes at different epochs ( fits to two or three parameters : @xmath33 and one or two values of @xmath5 ) . see also the text . [ cols= " < , < , < , < " , ] for distortions at relatively high redshifts ( @xmath6 ) , the value of @xmath48 can be simply determined by the knowledge of the frequency position of the minimum of the cmb absolute temperature : @xmath141 this opportunity is very powerful in principle , since the dependence of @xmath142 on @xmath48 is determined only by the well known physics of the radiation processes in an expanding universe during the radiation dominated era . for dissipations at @xmath143 , the amplitude of this temperature decrement is @xmath144 where @xmath145 is the chemical potential at the redshift @xmath61 corresponding to @xmath58 . ( 7 ) gives the range of wavelengths to observe for a firm evaluation of @xmath48 . as example , for @xmath146 we need to accurately measure the cmb absolute temperature up to wavelengths of about 50 cm , clearly out from the dimes range . ground and ballon experiments are currently planned to reach these wavelengths . moreover , the amplitude of the maximum dip of the brightness temperature for the energy dissipations at @xmath143 , see eq . ( 8) , turns to be at the mk level for @xmath147 and distortions within the firas limits ; burigana et al . 1991a shown that it is about 3 times smaller for energy injections at @xmath126 . experiments designed to estimate @xmath48 through the measure of @xmath142 should then have a sensitivity level of @xmath13 mk or better . for sake of illustration , we consider the simulated observation of a spectrum distorted at @xmath58 with @xmath101 ( or @xmath148 , as suggested by the `` revised '' firas data ) in a @xmath146 universe through a very precise experiment extended up to @xmath149 cm . more precisely , we consider the dimes channels combined to measures at 73.5 , 49.1 , 36.6 , 21.3 , 12 and 6.3 cm as those proposed for the space experiment lobo dedicated to measure the cmb spectrum at very low frequencies ( @xmath150 ghz ; see sironi et al . 1995 , pagana & villa 1996 ) , but we assume a much better sensitivity , @xmath151 mk , comparable to that of the dimes - like experiment , ( or @xmath152 mk ) . again , we generate the simulated data as described in section 2.3 . the fit to these simulated data by assuming to know the dissipation epoch shows that it would be possible to accurately determine both the amount of injected energy and the baryon density . we recover @xmath153 ( or @xmath154 ) and @xmath155 ( errors at 95 per cent cl ) . unfortunately , experiments at decimeter wavelengths with a sensitivity of @xmath13 mk or better , although very informative in principle , seem to be very far from current possibilities . we have studied the implications of possible future observations of the cmb absolute temperature at @xmath156 cm , where both ground , balloon and space experiments are currently under study to complement the accurate firas data at @xmath0 cm . our analysis shows that future measures from ground and balloon will not be able to significantly improve the constraints on energy exchanges in the primeval plasma already provided by the firas data . even observations with a sensitivity better by a factor 10 with respect to the realistic performances of the next experiments at different centimeter and decimeter wavelengths can not significantly improve this conclusion . thus , we have studied the impact of very high quality data , such those that could be in principle reached with a space experiment . for this analysis , we referred to the dimes experiment ( kogut 1996 ) , submitted to the nasa in 1995 , planned to measure the cmb absolute temperature at @xmath157 cm with a sensitivity of @xmath151 mk , close to that of firas . we have demonstrated that these data would represent a substantial improvement for our knowledge of energy dissipation processes at intermediate and high redshifts ( @xmath6 ) . dissipation processes at @xmath158 could be accurately constrained and possibly firmly detected even for very small amounts of the injected energy ( @xmath159 ) . for these early dissipation processes it would be possible to estimate also the energy injection epoch . distortions at intermediate redshifts ( @xmath119 ) could be also firmly detected , although in this case interesting information on the heating epoch can be derived only for energy injections , @xmath5 , larger than about @xmath160 . on the contrary , by considering the case of a single energy exchange in the thermal history of the universe , for late processes ( @xmath161 ) a such kind of experiment can not substantially improve the limits based on the firas data at @xmath0 cm , which would still set the constraints on @xmath5 at late epochs . by the jointed analysis of two dissipation processes occurring at different epochs , we demonstrated that the sensitivity and frequency coverage of a dimes - like experiment would allow to accurately recover the amount of energy exchanged in the primeval plasma at early and intermediate redshifts , and possibly the corresponding epoch , even in presence of a possible late distortion . even in this case , the constraints on @xmath5 can be improved by a factor @xmath162 for processes possibly occurred in a wide range of cosmic epochs , corresponding to about one two decades in redshift at @xmath94 about @xmath132 , while the constraints on the energy possibly dissipated at late epochs can be also improved by a factor @xmath7 , because the rough compensation between the distortion effects at millimetric wavelengths from an early and a late process with opposite signs becomes much less relevant in presence of very accurate long wavelength data . in addition , accurate long wavelength measures can provide an independent cross - check of the firas calibration : a dimes - like experiment could accurately distinguish between the firas calibrations by mather et al . 1999 and by battistelli et al . 2000 and in this second case could discriminate between different scenarios to account for it . interesting , although not fully exhaustive , indications on this aspect could be also obtained by improving the sensitivity of the next ground and balloon experiments by a factor @xmath1 . further , we have shown that a possible accurate observation of spectral distortions at @xmath163 cm compatible with relatively large energy injections , compared to the `` standard '' firas limits , can not be consistently reconciled with the firas data , at least for the class of distortion considered here . in this observational scenario , `` exotic '' models for spectral distortions should be carefully considered . we have shown that future long wavelength measures can significantly improve the current observational status of the free - free distortion : constraints on ( or detection of ) @xmath65 at the ( accuracy ) level of @xmath164 can be reached by forthcoming experiments , while improving the sensitivity up to that of a dimes - like experiment will allow to measure @xmath165 with an accuracy up to about @xmath166 ( errors at 95 per cent cl ) . of course , not only a very good sensitivity , but also an extreme control of the all systematical effects and , in particular , of the frequency calibration is crucial to reach these goals . finally , a dimes - like experiment will be able to provide indicative independent estimates of the baryon density : the product @xmath8 can be recovered within a factor @xmath9 even in the case of ( very small ) early distortions with @xmath10 . on the other hand , for @xmath11 , an independent baryon density determination with an accuracy at @xmath12 per cent level , comparable to that achievable with cmb anisotropy experiments , would require an accuracy of @xmath13 mk or better in the measure of possible early distortions but up to a wavelength from @xmath12 few @xmath14 dm to @xmath15 dm , according to the baryon density value . it is a pleasure to thank m. bersanelli , n. mandolesi , c. macculi , g. palumbo , and g. sironi for useful discussions on cmb spectrum observations . warmly thank l. danese and g. de zotti for numberless conversations on theoretical aspects of cmb spectral distortions . battistelli e.s . , fulcoli v. , macculi c. 2000 , new astronomy , 5 , 77 burigana c. , danese l. , de zotti g. 1991a , a&a , 246 , 59 burigana c. , de zotti g. , danese l. 1991b , apj , 379 , 1 burigana c. , de zotti g. , danese l. 1995 , a&a , 303 , 323 burigana c. & salvaterra r. 2000 , int . itesre / cnr 291/2000 , august danese l. & burigana c. 1993 , in : `` present and future of the cosmic microwave background '' , lecture in physics , vol . 429 , eds . sanz , e. martinez - gonzales , l. cayon , springer verlag , heidelberg ( frg ) , p. 28 danese l. & de zotti g. 1977 , riv . nuovo cimento , 7 , 277 fixsen d.j . 1994 , apj , 420 , 457 fixsen d.j . 1996 , apj , 473 , 576 kogut a. 1996 , `` diffuse microwave emission survey '' , in the proceedings from xvi moriond astrophysics meeting held march march 16 - 23 in les arcs , france , astro - ph/9607100 kompaneets a.s . 1956 , zh . , 31 , 876 [ sov . jept , 4 , 730 , ( 1957 ) ] mather j.c . , fixsen d.j . , shafer r.a . , mosier c. , wilkinson , d.t . 1999 , apj , 512 , 511 nordberg h.p . & smoot g.f . 1998 , astro - ph/9805123 pagana e. , villa f. , 1996 , `` the lobo satellite mission : feasibility study and preliminary cost evaluation '' , int . rep . c.i.f.s . - 1996 press w.h . , teukolsky s.a . , vetterling w.t . , flannery b.p . 1992 , `` numerical recipes in fortran '' , second edition , cambridge university press , usa salvaterra r. & burigana c. 2000 , int . itesre / cnr 270/2000 , march , astro - ph/0206350 salvaterra r. & burigana c. 2002 , mnras , 336 , 592 sironi g. , bonelli g. , dalloglio g. , pagana e. , de angeli s. , perelli m. , 1995 , astroph . comm . , 32 , 31 staggs s.t . , jarosik n.c . , meyer s.s . , wilkinson d.t . 1996 , apj , 473 , l1 [ lastpage ]

we analyse the implications of future observations of the cmb absolute temperature at centimeter and decimeter wavelengths , where both ground , balloon and space experiments are currently under study to complement the accurate cobe / firas data available at @xmath0 cm . our analysis shows that forthcoming ground and balloon measures will allow a better understanding of free - free distortions but will not be able to significantly improve the constraints already provided by the firas data on the possible energy exchanges in the primeval plasma . the same holds even for observations with sensitivities up to @xmath1 times better than those of forthcoming experiments . thus , we have studied the impact of very high quality data , such those in principle achievable with a space experiment like dimes planned to measure the cmb absolute temperature at @xmath2 cm with a sensitivity of @xmath3 mk , close to that of firas . we have demonstrated that such high quality data would improve by a factor @xmath4 the firas results on the fractional energy exchanges , @xmath5 , associated to dissipation processes possibly occurred in a wide range of cosmic epochs , at intermediate and high redshifts ( @xmath6 ) , and that the energy dissipation epoch could be also significantly constrained . by jointly considering two dissipation processes occurring at different epochs , we demonstrated that the sensitivity and frequency coverage of a dimes - like experiment would allow to accurately recover the epoch and the amount of energy possibly injected in the radiation field at early and intermediate epochs even in presence of a possible late distortion , while the constraints on the energy possibly dissipated at late epochs can be improved by a factor @xmath7 . in addition , such measures can provide an independent and very accurate cross - check of firas calibration . finally , a dimes - like experiment will be able to provide indicative independent estimates of the baryon density : the product @xmath8 can be recovered within a factor @xmath9 even in the case of ( very small ) early distortions with @xmath10 . on the other hand , for @xmath11 , an independent baryon density determination with an accuracy at @xmath12 per cent level , comparable to that achievable with cmb anisotropy experiments , would require an accuracy of @xmath13 mk or better in the measure of possible early distortions but up to a wavelength from @xmath12 few @xmath14 dm to @xmath15 dm , according to the baryon density value . # 1 # 1 2truept2truept 2truept2truept @xmath16 # 1#2 = cmbx12 et al.et al . [ firstpage ] cosmology : cosmic microwave background cosmological parameters - cosmology : theory

questions such as whether the universe will expand forever or eventually re - collapse and end with a big crunch , and what its shape and size may be , are among the most fundamental challenges in cosmology . regarding the former question , it is well known that the ultimate fate of the universe is intrinsically associated with the nature of its dominant components . in the friedmann - lematre - robertson - walker ( flrw ) class of models , for instance , a universe that is dominated by a pressureless fluid ( as , e.g. , baryons and/or dark matter ) or any kind of fluid with positive pressure ( as radiation , for example ) will expand forever if its spatial geometry is euclidean or hyperbolic , or will eventually re - collapse if it is spherical . this predictable destiny for the universe , however , may be completely modified if it is currently dominated by some sort of negative - pressure dark component , as indicated by a number of independent observational results ( see , e.g. , ref . @xcite ) . in this case , not only the dynamic but also the thermodynamic fate of the universe may be completely different , with the possibility of an eternally expanding closed model @xcite , an increasingly hot expanding universe @xcite or even a progressive rip - off of the large and small scale structure of matter ending with the occurrence of a curvature singularity , the so - called big smash @xcite . the remaining questions , concerning the shape and size of our @xmath2dimensional world , go in turn beyond the scope of general relativity ( gr ) , since they have an intrinsically topological nature . in this way , approaches or answers to these questions are ultimately associated with measurements of the _ global _ structure ( topology ) of the universe and , as a _ metric theory , gr can not say much about it , leaving the global topology of the universe undetermined . over the past few years , several aspects of the cosmic topology have become topical ( see , e.g. , the review articles ref . @xcite ) , given the wealth of increasingly accurate cosmological observations , especially the recent results from the wilkinson microwave anisotropy probe ( wmap ) experiment @xcite , which have heightened the interest in the possibility of a universe with a nontrivial spatial topology . a pertinent question the reader may ask at this point is whether the current values of cosmological density parameters , which help us to answer the above first question ( associated with the ultimate fate of the universe ) , can be constrained by a possible detection of the spatial topology of the universe . our primary objective here is to address this question by focusing our attention on possible topological constraints on the density parameters associated with the baryonic / dark matter ( @xmath0 ) and dark energy ( @xmath1 ) . motivated by the best fit value for the total energy density @xmath3 ( @xmath4 level ) reported by wmap team @xcite , which includes a positively curved universe as a realistic possibility , we shall consider globally homogeneous spherical manifolds , some of which account for the suppression of power at large scales observed by wmap @xcite , and also fits the wmap temperature two - point correlation function @xcite . to this end , in the next section we present our basic context and prerequisites , while in the last section we discuss our main results and present some concluding remarks . within the framework of standard cosmology , the universe is described by a space - time manifold @xmath5 with a locally homogeneous and isotropic robertson walker ( rw ) metric @xmath6 \;,\ ] ] where @xmath7 , @xmath8 , or @xmath9 depends on the sign of the constant spatial curvature ( @xmath10 , respectively ) . the @xmath2space @xmath11 is usually taken to be one of the following simply - connected spaces : euclidean @xmath12 , spherical @xmath13 , or hyperbolic @xmath14 . however , given that the simple - connectedness of our space @xmath11 has not been established , our @xmath2space may equally well be any one of the possible quotient manifolds @xmath15 , where @xmath16 is a fixed point - free group of isometries of the covering space @xmath17 . thus , for example , in a universe whose geometry of the spatial section is euclidean ( @xmath18 ) , besides @xmath19 there are 6 classes of topologically distinct compact orientable @xmath2spaces @xmath11 that admits this geometry , while for universes with either spherical ( @xmath20 ) and hyperbolic ( @xmath21 ) spatial geometries there is an infinite number of topologically non - homeomorphic ( inequivalent ) manifolds with nontrivial topology that can be endowed with these geometries . quotient manifolds are compact in three independent directions , or compact in two or at least one independent direction . in compact manifolds , any two given points may be joined by more than one geodesic . since the radiation emitted by cosmic sources follows geodesics , the immediate observational consequence of a nontrivial detectable spatial topology of @xmath11 is that the sky may show multiple images of radiating sources : cosmic objects or specific correlated spots of the cosmic microwave background radiation ( cmbr ) . at very large scales , the existence of these multiple images ( or pattern repetitions ) is a physical effect that can be used to probe the @xmath2-space topology . in this work , we use the so - called circles - in - the - sky " method ( for cosmic crystallographic methods see , e.g. , refs . @xcite ) , which relies on multiple copies of correlated circles in the cmbr maps @xcite , whose existence is clear from the following reasoning : in a space with a detectable nontrivial topology , the last scattering sphere ( lss ) intersects some of its topological images along pairs of circles of equal radii , centered at different points on the lss , with the same distribution of temperature fluctuations , @xmath22 . since the mapping from the lss to the night sky sphere preserves circles @xcite , these pairs of matching circles will be inprinted on the cmbr temperature fluctuations sky maps regardless of the background geometry and detectable topology . as a consequence , to observationally probe a nontrivial topology on the available largest scale , one should scrutinize the full - sky cmb maps in order to extract the correlated circles , whose angular radii and relative position of their centers can be used to determine the topology of the universe . thus , a nontrivial topology of the space section of the universe may be observed , and can be probed through the circles - in - the - sky for all locally homogeneous and isotropic universes with no assumption on the cosmological density parameters . let us now state our basic cosmological assumptions and fix some notation . in addition to the rw metric ( [ rwmetric ] ) , we assume that the current matter content of the universe is well approximated by cold dark matter ( cdm ) of density @xmath23 plus a cosmological constant @xmath24 . in this standard @xmath24cdm context , for nonflat spaces the scale factor @xmath25 can be identified with the curvature radius of the spatial section of the universe at time @xmath26 , which is given by @xmath27 where here and in what follows the subscript @xmath28 denotes evaluation at present time @xmath29 , @xmath30 is the hubble constant , and @xmath31 is the total density at @xmath26 . in this way , for nonflat spaces the distance @xmath32 of any point with coordinates @xmath33 to the origin ( in the covering space ) _ in units of the curvature radius _ , @xmath34 , reduces to @xmath35 where @xmath36 is an integration variable , and @xmath37 . throughout this paper we shall measure the lengths in unit of curvature radius @xmath38 . a typical characteristic length of nonflat manifolds @xmath11 , which we shall use in this paper , is the so - called injectivity radius @xmath39 , which is defined as the radius of the smallest sphere ` inscribable ' in @xmath11 . an important mathematical result is that @xmath39 , expressed in terms of the curvature radius , is a constant ( topological invariant ) for any given spherical and hyperbolic manifolds . in this work we shall focus our attention in globally homogeneous spherical manifolds , as presented in table [ singleaction ] ( see also its caption for more details ) . these manifolds satisfy a topological principle of homogeneity , in the sense that all points in @xmath11 are topologically equivalent . .the globally homogeneous spherical manifolds are of the form @xmath40 . the first column gives the name we use for the manifolds . the second column displays the covering groups @xmath16 . finally , the remaining columns present the order of the group @xmath16 and the injectivity radius @xmath39 . the cyclic and binary dihedral cases actually constitute families of manifolds , whose members are given by the different values of the integers @xmath41 and @xmath42 . the order of @xmath16 gives the number of fundamental polyhedra needed to fulfill the whole covering space @xmath13 . thus , for example , for the manifold @xmath43 which is the the well - known poincar dodecahedral space , the fundamental polyhedron is a regular spherical dodecahedron , @xmath44 of which tile the @xmath2sphere into identical cells that are copies of the fp . [ cols="^,^,^,^",options="header " , ] to investigate the extent to which a possible detection of a nontrival topology may place constraints on the cosmological density parameters , we consider here the globally homogeneous spherical manifolds . in these @xmath2spaces the number of pairs of matching circles depends on the ratio of the injectivity radius @xmath39 to the radius @xmath45 of lss , which in turn depends on the density parameters ( see ref . @xcite for examples of specific estimates of this number regarding @xmath46 , @xmath47 and @xmath43 ) . nevertheless , if the topology of a globally homogeneous spherical manifold is detectable the correlated pairs will be antipodal , i.e. the centers of correlated circles are separated by @xmath48 , as shown in figure [ cinthesky1 ] . clearly the distance between the centers of each pair of the _ first _ correlated circles is twice the injectivity radius @xmath39 . now , a straightforward use of known trigonometric rules to the right - angled spherical triangle shown in figure [ cinthesky1 ] yields a relation between the angular radius @xmath49 and the angular sides @xmath39 and radius @xmath45 of the last scattering sphere , namely @xmath50 where @xmath39 is a topological invariant , whose values are given in table [ singleaction ] , and the distance @xmath45 of the last scattering surface to the origin in units of the curvature radius is given by ( [ redshift - dist ] ) with @xmath51 @xcite . 0.1 in equations ( [ cosalpha ] ) along with ( [ redshift - dist ] ) give the relations between the angular radius @xmath49 and the cosmological density parameters @xmath1 and @xmath52 , and thus can be used to set bounds on these parameters . to quantify this we proceed in the following way . firstly , as an example , we assume the angular radius @xmath53 . secondly , since the measurements of the radius @xmath49 unavoidably involve observational uncertainties , in order to obtain very conservative results we take @xmath54 . and its uncertainty . ] in order to study the effect of the cosmic topology on the density parameters @xmath55 and @xmath56 , we consider the binary tetrahedral @xmath46 and the binary octahedral @xmath47 spatial topologies ( see table [ singleaction ] ) , to reanalyze with these two topological priors the constraints on these parameters that arise from the so - called _ gold _ sample of 157 sne ia , as compiled by riess _ et al . _ @xcite , along with the latest chandra measurements of the x - ray gas mass fraction in 26 x - ray luminous , dynamically relaxed galaxy clusters ( spanning the redshift range @xmath57 ) as provided by allen _ et al . _ @xcite ( see also @xcite for details on sne ia and x - ray statistics ) . the @xmath46 and @xmath47 spatial topology is added to the conventional sne ia plus clusters data analysis as a gaussian prior on the value of @xmath45 , which can be easily obtained from an elementary combination of ( [ cosalpha ] ) and ( [ redshift - dist ] ) . in other words , the contribution of the topology to @xmath58 is a term of the form @xmath59 . -0.2 cm -0.2 cm figures 2b and 2c ( central and right panels ) show the results of our statistical analysis . confidence regions 68.3% and 95.4% confidence limits ( c.l . ) in the parametric space @xmath52@xmath56 are displayed for the above described combination of observational data . for the sake of comparison , we also show in fig . 2a the @xmath52@xmath56 plane for the conventional sne ia plus galaxy clusters analysis , i.e. , the one without the above cosmic topology assumption . by comparing both analyses , it is clear that a nontrivial space topology reduces considerably the parametric space region allowed by the current observational data , and also breaks some degeneracies arising from the current sne ia and x - ray gas mass fraction measurements . at 95.4% c.l . our sne ia+x - ray+topology analysis provides @xmath60 and @xmath61 ( binary octahedral @xmath62 ) and @xmath63 and @xmath64 ( binary tetrahedral @xmath65 ) . concerning the above analysis it is worth emphasizing three important aspects . first , that the best - fit values depend weakly on the value used for radius @xmath49 of the circle . second , the uncertainty @xmath66 alters predominantly the area corresponding to the confidence regions , without having a significant effect on the best - fit values . third , we also notice that there is a topological degeneracy in that the same best fits and confidence regions for , e.g. , the @xmath46 topology , would equally arise from either @xmath67 or @xmath68 spatial topology . similarly , @xmath47 , @xmath69 and @xmath68 give rise to identical bounds on the density parameters . this kind of topological degeneracy passed unnoticed in refs . @xcite . finally , we emphasize that given the wealth of increasingly accurate cosmological observations , especially the recent results from the wmap , and the development of methods and strategies in the search for cosmic topology , it is reasonable to expect that we should be able to detect it . besides it importance as a major scientific achievement , we have shown through concrete examples that the knowledge of the spatial topology allows to place constraints on the density parameters associated to dark matter ( @xmath0 ) and dark energy ( @xmath1 ) . we thank cnpq for the grants under which this work was carried out . we also thank a.f.f . teixeira for the reading of the manuscript and indication of relevant misprints and omissions . v. sahni and a. starobinsky , int . j. mod d * 9 * , 373 ( 2000 ) ; j.e . peebles and b. ratra , rev . mod . phys . * 75 * , 559 ( 2003 ) ; t. padmanabhan , phys . rep . * 380 * , 235 ( 2003 ) ; j.a.s . lima , braz . j. phys . * 34 * , 194 ( 2004 ) . m. lachize - rey and j .- p . luminet , phys . rep . * 254 * , 135 ( 1995 ) ; g.d . starkman , class . quantum grav . * 15 * , 2529 ( 1998 ) ; j. levin , phys . rep . * 365 * , 251 ( 2002 ) ; m.j . rebouas and g.i . gomero , braz . j. phys . * 34 * , 1358 ( 2004 ) . astro - ph/0402324 ; m.j . rebouas , a brief introduction to cosmic topology , in _ proc . xith brazilian school of cosmology and gravitation _ , eds . m. novello and s. e. perez bergliaffa ( americal institute of physics , melville , new york , 2005 ) aip conference proceedings vol . * 782 * , p 188 ( 2005 ) . e. komatsu et al . , * 148 * , 119 ( 2003 ) ; h.k . eriksen , f.k . hansen , a.j . banday , k.m . gorski , and p.b . lilje , astrophys . j. * 605 * , 14 ( 2004 ) ; c.j . copi , d. huterer , and g.d . starkman , phys . d * 70 * , 043515 ( 2004 ) . spergel et al . , astrophys . j.suppl . * 148 * , 175 ( 2003 ) . m. tegmark , a. de oliveira - costa , and a.j.s . hamilton , phys . d * 68 * , 123523 ( 2003 ) ; a. de oliveira - costa , m. tegmark , m. zaldarriaga , and a. hamilton , phys . d * 69 * , 063516 ( 2004 ) ; j.r . weeks , astro - ph/0412231 ; p. bielewicz , h.k . eriksen , a.j . banday , and k.m . gorski , and p.b . lilje , astro - ph/0507186 ; k. land and j. magueijo , phys . * 95 * , 071301 ( 2005 ) . a. bernui , b. mota , m.j . rebouas , and r. tavakol , astro - ph/0511666 ; k. land and j. magueijo , mon . not . astron . 357 * , 994 ( 2005 ) . luminet , j. weeks , a. riazuelo , r. lehoucq and j .- p . uzan , nature * 425 * , 593 ( 2003 ) ; n.j . cornish , d.n . spergel , g.d . starkman , and e. komatsu , phys . * 92 * , 201302 ( 2004 ) ; j. gundermann , astro - ph/0503014 ; b.f . roukema , b. lew , m. cechowska , a. marecki , and s. bajtlik , astron . astrophys . * 423 * , 821 ( 2004 ) . r. lehoucq , m. lachize - rey , and j .- p luminet , astron . astrophys . * 313 * , 339 ( 1996 ) ; b.f . roukema and a. edge , _ mon . not . soc . _ * 292 * , 105 ( 1997 ) ; b.f . roukema , class . quantum grav . * 15 * , 2645 ( 1998 ) ; r. lehoucq , j .- p luminet , and j .- uzan , astron . astrophys . * 344 * , 735 ( 1999 ) ; h.v . fagundes and e. gausmann , phys . a * 238 * , 235 ( 1998 ) ; h.v . fagundes and e. gausmann , phys . a * 261 * , 235 ( 1999 ) ; j .- uzan , r. lehoucq and j .- p . luminet , astron . astrophys . * 351 * , 766 ( 1999 ) ; g.i . gomero , m.j . rebouas , and a.f.f . teixeira , int . d * 9 * , 687 ( 2000 ) ; r. lehoucq , j .- uzan , and j .- p luminet , astron . astrophys . * 363 * , 1 ( 2000 ) ; g.i . gomero , m.j . rebouas , and a.f.f . teixeira , phys . a * 275 * , 355 ( 2000 ) ; g.i . gomero , m.j . rebouas , and a.f.f . teixeira , class . quantum grav . * 18 * , 1885 ( 2001 ) ; g.i . gomero , a.f.f . teixeira , m.j . rebouas and a. bernui , int d * 11 * , 869 ( 2002 ) ; a. marecki , b. roukema , and s. bajtlik , astron . astrophys . * 435 * , 427 ( 2005 ) . gomero , m.j . rebouas and r. tavakol , class . quantum grav . * 18 * , 4461 ( 2001 ) ; g.i . gomero , m.j . rebouas , and r. tavakol , int . j. mod . a * 17 * , 4261 ( 2002 ) ; j.r . weeks , r. lehoucq , and j .- uzan , class . quantum grav . * 20 * , 1529 ( 2003 ) ; j.r . weeks , mod . phys . lett . a * 18 * , 2099 ( 2003 ) ; g.i . gomero and m.j . rebouas , phys . lett . a * 311 * , 319 ( 2003 ) ; b. mota , m.j . rebouas , and r. tavakol , class . quantum grav . * 20 * , 4837 ( 2003 ) ; b. mota , g.i . gomero , m. j. rebouas and r. tavakol , class . quantum grav . * 21 * , 3361 ( 2004 ) . s.w . allen , r.w . schmidt , h. ebeling , a.c . fabian , and l. van speybroeck , mon . not . soc . * 353 * , 457 ( 2004 ) . j.a.s . lima , j.v . cunha and j.s . alcaniz , phys . d * 68 * , 023510 ( 2003 ) ; j.s . alcaniz and z .- h . zhu , phys . d * 71 * , 083513 ( 2005 ) ; d. rapetti , s.w . allen and j. weller , mon . not . . soc . * 360 * , 546 ( 2005 ) .

given the wealth of increasingly accurate cosmological observations , especially the recent results from the wmap , and the development of methods and strategies in the search for cosmic topology , it is reasonable to expect that we should be able to detect the spatial topology of the universe in the near future . motivated by this , we examine to what extent a possible detection of a nontrivial topology of positively curved universe may be used to place constraints on the matter content of the universe . we show through concrete examples that the knowledge of the spatial topology allows to place constraints on the density parameters associated to dark matter ( @xmath0 ) and dark energy ( @xmath1 ) .

a series of rare elementary processes involving more than two particles in the final state are going to be measured with increasing precision . the multiplicity of the final state makes it difficult to extract predictions by the standard gauge theories even if semplifications arise when either partecipants are all massless or only some of the external particles are massive . however more accurate rate measurements of processes with heavy quark hadrons in the final state will soon be available as is the case of the chorus experiment where direct evidence for the associate charm production in charged current neutrino nucleon scattering has been shown @xcite . in the one loop calculation of such processes we encounter pentagon integrals with a massive line as skeched in figure 1 where massive particles are bold , massless ones thin and dashed ones can be either massive or not . in general the inclusion of masses makes things more involved , although the calculation simplifies when either external masses are equal to each other or they are equal to the internal masses or both eventualities occur as it is often the case in normal gauge theories . recently a lot of progress has been made in the technics for perturbative calculations with different approaches . a non - comprehensive list is given in @xcite and reference therein and in @xcite , @xcite , @xcite , @xcite and @xcite . in particular adopting the dimensional regularization approach for feynman parametrized integrand the authors of ref.@xcite derived simplifications and recursion formulas by the implementation of algebraic technic . using these methods the problem of the evaluation of a one loop @xmath0 points scalar integral is translated to the evaluation of a combination of @xmath1 points scalar integrals and the original @xmath0 points integral in @xmath2 dimensions ; moreover the original @xmath0 points one loop integral can be represented as the solution of a partial differential equation system . in the present paper we use this approach to perform the calculation of the pentagon integral represented in figure 1 . other massive pentagon integrals have been recently evaluated in next to leading order calculations of processes in which an higgs particle can be generated at hadron colliders . in particular two independent groups report the nlo corrections for the process in which an higgs particle is generated together with a @xmath3 pair , @xcite and @xcite . another nlo calculation involving massive pentagon integrals is given in @xcite in which the final state considered consists of an higgs particle plus two jets . the general methods employed here do not concern with the specific processes and the results must be analitically continued to describe a specific process . finally only the most simple tensor integral is given while we postpone other cases to a dedicated paper @xcite . the paper is organized as follow : in section ii relevant formulas from ref.@xcite are collected , in section iii they are applied to the scalar massive pentagon represented in figure 1 transforming it in a combination of four points integrals ; section iv is devoted to four point integrals evaluation and in section v more simple tensor integral ( vector ) is given with the conclusions . the initial condition for the differential equations originated in the four points evaluation are calculated in the appendix . the starting point is the integral in @xmath4 dimensions @xmath5 with the momenta @xmath6 taken to be outgoing , @xmath7 and @xmath8 applying feynman parametrization , wick rotating and integrating over loop momentum this integral can be cast in the form @xmath9 @xmath10 with @xmath11 and the matrix @xmath12 given by @xmath13 with @xmath14 and @xmath15 for @xmath16 . we will not repeat the derivations obtained in ref.@xcite but , to introduce notation and to be self - consistent , in the rest of this section we just collect relevant formulas that will be used in section iii and iv . performing a projective transformation @xcite with parameters @xmath17 in such a way that the denominator in eq.([aaa ] ) has no @xmath17 dependence the definition of a new matrix follows ( indices are not summed ) @xmath18 using the following definitions @xmath19 the authors of ref.@xcite find @xmath20\\ \frac{1}{n-4 + 2\varepsilon}\,\frac{\partial\hat{i}_n}{\partial\alpha_i}= \frac{1}{2n_n}\left[\sum_{j=1}^n\eta_{ij}\,\hat{i}_{n-1}^{(j)}+ ( n-5 + 2\varepsilon)\,\gamma_i\,\hat{i}_n^{d=6 - 2\varepsilon}\right]\end{aligned}\ ] ] where @xmath21 stands for the @xmath1 integral with the denominator obtained from an @xmath22 integral eliminating the propagator between legs @xmath23 and @xmath24 ; once feynman parameter has been introduced in the usual way for @xmath22 the denominator in @xmath21 is obtained putting @xmath25 . by the observation that @xmath26 and @xmath27 are finite in @xmath28 dimensions , performing one - loop calculation one can limit to evaluate @xmath29 taking only the divergent part from the @xmath30 integrals in eq.([bbb ] ) . to write down the integral in figure 1 we set @xmath31 , @xmath32 and @xmath33 giving @xmath34 and @xmath35 with @xmath36 given in eq.([ccc ] ) and the matrix @xmath12 given by @xmath37 with @xmath38 . we define @xmath39 and in the following we will assume @xmath40 . performing the projective transformation with @xmath41 we get for the @xmath42 matrix in eq.([rrhhoo ] ) @xmath43 with @xmath44 @xmath45 if @xmath46 we only have to take @xmath47 in eq.([eee ] ) . the coefficient relevant for the evaluation of the pentagon by eq.([ppp ] ) are given in the table 1 , keeping apart the case @xmath48 . due to the presence of masses we have not cyclic relations between the coefficients but only the relations @xmath49 table 1 . coefficients to be used in eq.([ppp ] ) [ cols="^,^,^ " , ] in terms of new kinematical variables @xmath17 , @xmath50 and @xmath51 the denominator in the @xmath27 integral represented in eq.([ccc ] ) is given by @xmath52 and the four points denominators in the @xmath53 integrals in eq.([ppp ] ) can be obtained putting @xmath54 to zero in the expression above . it is easy to verify the relations @xmath55 in the next section we proceed to the evaluation of @xmath56 , @xmath57 and @xmath58 using the set of partial differential eqs.([bbb ] ) . here we evaluate the integrals @xmath56 and @xmath57 , corrsponding to massive boxes with an internal massive line , in the variables defined in eqs.([c1 ] , [ c2 ] , and [ c3 ] ) and translate the integrals @xmath58 that are well known and correspond to massive boxes with massles internal lines . after putting @xmath59 in eq.([ggg ] ) we have for the denominator in @xmath56 @xmath60 before solving the integral we perform the following kinematic transformation : @xmath61 in terms of the new variables we get for the denominator : @xmath62 and @xmath63 the only divergent three points functions extracted by the expression above are @xmath64 and @xmath65 obtained putting @xmath66 and @xmath67 respectively ; these correspond to two two - mass triangles , while the other two obtained putting @xmath68 and @xmath69 respectively are three - mass triangles checked to be finite . at the @xmath70 we have @xmath71 the system of partial differential equations in eq.([bbb ] ) is then given by @xmath72 \nonumber \\ \frac{\partial\hat{i}_4^{(1)}}{\partial c_5}&= & \frac{2\,\gamma(1+\varepsilon)}{c_4-c_5 } \ , \log \left ( \frac{c_5}{c_4 } \right)\end{aligned}\ ] ] with the solution @xmath73.\end{aligned}\ ] ] the integration constant @xmath74 is evaluated in the appendix and its value is @xmath75 where @xmath76 is the dilogarithm function and @xmath77 . afetr some manipulation we have @xmath78 reintroducing the original variables inverting eq.([tr1 ] ) we get for @xmath56 : @xmath79 being @xmath56 independent from @xmath50 its value does not change in the limit @xmath80 . here and in the following subsection we proceed performing the same steps as in the derivation of @xmath56 . the limit @xmath80 now gives a different situation ; in fact in this limit there will be three divergent three - point integrals extracted by @xmath57 so as explained in @xcite the limit procedure is not smooth and the two case have to be taken separately . in this case @xmath57 is a three external mass box but , differently from @xmath56 , it has all external masses different from each other and so it needs evaluation . after putting @xmath66 in eq.([ggg ] ) we have for the denominator in @xmath57 @xmath82 rescaling the variables with @xmath83 we get @xmath84 giving @xmath85 and @xmath86 the only divergent three points functions extracted by the expression above are @xmath64 and @xmath65 obtained putting @xmath59 and @xmath67 respectively ; these correspond to two two - mass triangle , while the other two obtained putting @xmath68 and @xmath69 respectively are three - mass triangle checked to be finite . at the @xmath70 we have @xmath87 the system in eq.([bbb ] ) is then given by @xmath88 \nonumber \\ \frac{\partial\hat{i}_4^{(2)}}{\partial c_5}&= & \frac{2\,\gamma(1+\varepsilon)}{c_5-\delta \,c_4 } \log \left ( \frac{\delta \,c_4}{c_5 } \right)\end{aligned}\ ] ] with the solution @xmath89.\end{aligned}\ ] ] the integration constant @xmath90 is evaluated in the appendix and its value is @xmath91 after some maipulation we have @xmath92\ ] ] reintroducing the original variables inverting eq.([tr2 ] ) we get for @xmath57 : @xmath93 \\\end{aligned}\ ] ] in this case @xmath57 is a two external mass box . putting @xmath47 the denominator in eq.([den2 ] ) became @xmath95 rescaling the variables as in eq.([tr2 ] ) we get @xmath96 the divergent three point functions @xmath64 , @xmath65 and @xmath97 are obtained putting @xmath98 , @xmath99 and @xmath67 respectively . @xmath100 while @xmath101 is a three - mass triangle checked to be finite . the partial differential equation system is given by @xmath102 with the solution @xmath103\end{aligned}\ ] ] the integration constant @xmath104 is evaluated in the appendix and its value is @xmath105 after some manipulation we find @xmath106\ ] ] reintroducing the original variables we have @xmath107\ ] ] putting @xmath67 in eq.([ggg ] ) we eliminate the massive propagator and obtain the easy ( opposite ) two mass box @xcite or the one external massive box if we take respectively @xmath108 or @xmath94 . these integrals are well - known and are reported also in @xcite . here we just put these integrals in the kinematics specified in section 3 . after putting @xmath67 in eq.([ggg ] ) the denominator is given by @xmath109 using eq.(4.44 ) from the third paper in ref.@xcite the integral in the kinematics of section 3 reads @xmath110\end{aligned}\ ] ] putting @xmath47 in eq.([den3 ] ) the denominator of this integrals is given by @xmath111 using eqs.(4.27 , 4.40 ) from the third paper in ref.@xcite the integral in the kinematics of section 3 reads @xmath112\end{aligned}\ ] ] an expression for the scalar pentagon integral shown in figure 1 can be built via eqs.([ppp ] ) , ( [ rel ] ) , the four points integrals evaluated in the last section and the coefficients in table 1 . the expressions for @xmath27 are very long and are not reported . more familiar kinematics is realized by replacing the variables @xmath17 , @xmath50 and @xmath113 with their definitions in terms of @xmath114 , @xmath115 and @xmath116 . tensor integrals will be considered in a separate paper @xcite , however the simplest one of them , the vector integral , is related to the scalar integrals with one feynman parameter in the numerator by the following relation @xmath117\rightarrow i_n^d[\mathcal{p}^\mu]\ ] ] in which the arrow means integration over loop momentum @xmath118 , the integrand numerator is in the square brackets and @xmath119 with @xmath120 given in eqs.([pi ] ) . the integrals @xmath121 $ ] can be evaluated by @xcite @xmath122=\frac{1}{2n_5}\sum_{i=1}^5\eta_{ji}\hat{i}_{4}^{(i)}+ { \mathcal o}(\varepsilon)\ ] ] where @xmath123 defined in eq.([fff ] ) is deduced by @xmath124 given in eq.([eee ] ) @xmath125 higher tensor integrals can be evaluated considering that they are linked to scalar integrals with more powers of feynman parameters in the numerator @xcite . such a decompositioncan can also be organized in a way that drastically reduces numerical instabilities genarated by the presence of inverse powers of gram determinants @xcite . besides the deep inelastic case mentioned in the introduction , the results obtained in the present paper with @xmath81 can be useful in the evaluation at one loop of the decay amplitude of a real @xmath126 boson or a virtual photon in a heavy quark - antiquark pair and two light quarks . let us consider the case in which all massless particles and @xmath127 are gluons , then the pentagon studied with @xmath94 can be identified with one of the four pentagon in the perturbative evaluation of the one - loop associated production of heavy quark in the gluon - gluon - fusion with a gluon in the final state ( @xmath128 ) ; in this case , indeed , pentagons are found in which the propagators form chains with @xmath129 , @xmath130 , @xmath131 and @xmath132 equal mass fermions the first of which is calculated in the present paper while the other ones can be calculated analogously . the author gratefully acknowledges prof . p. strolin who supported the present research , profs . g. cosenza and a. della selva for discussions , dr . g. de lellis for suggesting the topic and for many comments on the manuscript , dr . d. falcone for suggestions on the manuscript , drs . r. mertig and f. orellana for help with @xmath133 @xcite , dr . f. di capua and dr . l. scotto lavina for a quick help with x - fig and dr . g. celentano for help with latex . in this appendix we report the calculation of the integration constants for the four points integral of section four systematically neglecting @xmath134 terms . instead of reporting all length passages , we give the steps that can be followed by programs of function manipulation like the used mathematica . the point chosen to evaluate @xmath56 is @xmath135 where the expression in eq.([sol1 ] ) gives @xmath136 the expression for the integral at the point selected deduced using eq.([quat ] ) and the first of eqs.([fff ] ) is @xmath137 the factor @xmath130 is given by @xmath138 . renaming @xmath99 with @xmath139 , @xmath140 with @xmath141 and @xmath142 with @xmath143 , and performing the transformation @xmath144 , @xmath145 and @xmath146 we arrive at the expression @xmath147 putting apart the gamma function for the moment , the @xmath143 integration gives @xmath148 the two integrals can be evaluated by shifting both in @xmath141 @xmath149 simplifying the @xmath139 integral @xmath150 and inverting the integration order @xcite . after some manipulation and expanding some hypergeometric and generalized hypergeometric functions the result is @xmath151 finally , making the substitution @xmath152 in eq.([n1 ] ) and taking into account eq.([ap1 ] ) we find @xmath153 in eq.([c01 ] ) . [ [ integration - constant - for - hati_42-integral - with - q2-neq-0 ] ] integration constant for @xmath57 integral with @xmath154 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ to evaluate the integration constant we evaluate the integral in the point : @xmath155 with @xmath156 the integrations are trivial but the expression is very long . the @xmath157 chosen can not be simultaneously positive so we checked the result in the point @xmath158 where the expression in eq.([sol2 ] ) gives @xmath159 the expression for the integral at the point selected is @xmath160 renaming @xmath98 with @xmath139 , @xmath140 with @xmath141 and @xmath142 with @xmath143 , and performing the transformation @xmath144 , @xmath145 and @xmath146 we arrive at the expression @xmath161 putting apart the gamma function for the moment , the @xmath143 integration gives @xmath162 performing the @xmath139 integration before and adding and subtracting terms we find eq.([ss2 ] ) . [ [ integration - constant - for - hati_42-integral - with - q2 - 0 ] ] integration constant for @xmath57 integral with @xmath94 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ the point chosen to evaluate @xmath57 is given by @xmath163 in which the expression in eq.([sol3 ] ) gives @xmath164 the expression for the integral at the point selected is @xmath165 renaming @xmath140 with @xmath139 , @xmath142 with @xmath141 and @xmath166 with @xmath143 , and performing the transformation @xmath144 , @xmath145 and @xmath146 we arrive at the expression @xmath167 putting apart the gamma function for the moment , the @xmath143 integration gives @xmath168 performing the shift @xmath149 gives @xmath169 finally performing the @xmath139 integration before and adding and subtracting terms before expanding in @xmath170 the result is @xmath171 substituting @xmath172 in eq.([n3 ] ) and taking into account eq.([ap3 ] ) we find @xmath104 in eq.([c03 ] ) .

* abstract * in this paper we present the calculation of a scalar pentagon integral with two consecutive massive external legs having an equal mass propagator embedded between them . we also deal with the two situations where the farest external leg is either massive or not . the relevance of the calculation comes from its application in many perturbative qcd calculations as well as in qcd corrections for weak precesses . = 0.5 cm = 0.0 cm = 0.75 cm dsf-22/2002 * pentagon integrals for heavy quark physics * dipartimento di scienze fisiche , universit di napoli , complesso di monte s. angelo , via cintia , napoli , italy

over 200 extrasolar giant planets have now been discovered by radial velocity surveys . fourteen of these planets have been observed transiting their parent star , allowing an accurate determination of their radius and mean density . for half of these objects , notably the first ever discovered transit hd209458b , the predicted theoretical radius lies several @xmath0 s ( up to @xmath1 ) below the observed mean value @xcite . the various scenarios proposed so far to solve this discrepancy either have been rejected on observational or theoretical arguments @xcite or lack an identified robust mechanism to convert surface kinetic energy into thermal energy at depth @xcite . a fair conclusion of these studies is that an important physical mechanism is probably missing in our present description of at least some short period planets , for which we have a radius determination , but possibly of all extrasolar or even solar giant planets . according to the conventional core - accretion model for planet formation @xcite , planets are believed to have a substantial enrichment in heavy elements compared with their parent star , with a total of @xmath2 for a jupiter - mass object @xcite . observational constraints of jupiter and saturn show that these planets do have a significantly enhanced z - abundance compared with the sun , with a global mean mass fraction @xmath3=@xmath4 10 - 20% @xcite . in all these calculations , the big planetesimals are generally supposed to drown to the core during the early phase of solid accretion , while the smaller ones are distributed _ uniformly _ throughout the envelope , leading to a uniform heavy element abundance . this is a rather simplistic description of the planet internal structure , which implies ( i ) well defined interfaces between the central core and the ( highly diffusive ) h / he rich envelope and ( ii ) very efficient large - scale thermal convection throughout the entire gaseous envelope . the observed atmospheric abundances of jupiter and saturn , however , seem to require the redistribution of a subsequent fraction of heavy elements in the interior of these planets @xcite . in this letter , we explore the consequences of the presence of an initial compositional gradient in the envelope , as a result of either early planetesimal accretion or subsequent core erosion , and of the resulting less efficient heat transport and compositional mixing on the fate of gaseous planets . we show that layered convection , if it occurs as a result of this compositional gradient , might be the lacking physical mechanism to explain the transiting planet abnormally large radii . the presence of a positive compositional gradient , i.e. a gradient of mean molecular weight @xmath5=@xmath6 , tends to stabilize the fluid against convective instability according to the ledoux stability condition : @xmath7 where @xmath8 and @xmath9 denote the usual temperature and adiabatic gradients , respectively , and @xmath10=@xmath11 , @xmath12=@xmath13 . in most of the giant planet interiors , superadiabaticity is extremely small , with @xmath14 , so a small molecular weight gradient over a typical mixing length size region can affect significantly and even damp out convection . in convective systems where buoyancy effects of ( destabilizing ) heat and ( stabilizing ) composition are opposed , the process leads generally to quasi - static uniformly mixed convective layers separated by small diffusive interfaces with steep gradients , @xmath15 . such stable layered convection is indeed observed in some areas of the earth s oceans , due to the presence of the stabilizing salt gradient ( thermohaline convection ) leading to a stratified step - like temperature profile with stable boundary layers @xcite . laboratory experiments have also confirmed this layering @xcite . once formed , the stratification is stable provided the compositional gradient remains large enough at the interfaces to satisfy eq.([ledoux ] ) . would such a stratification occur in planetary interiors , the layered part of the interior can be considered as a _ semiconvection _ zone with a reduced efficiency to transport the internal heat and composition flux compared with large - scale convection . one may argue that the conditions in planetary interiors differ from the ones in the oceans or in the experiments . characteristic thermal diffusivity in h / he planetary interiors , dominated by electronic transport in the central , ionized parts , and by molecular motions in the outer envelope , lies in the range @xmath16-@xmath17 @xmath18 , while the kinematic viscosity is @xmath19-@xmath17 @xmath18 @xcite . the characteristic prandtl number thus ranges from @xmath20 @xmath21 to 1 . these values do not differ by large factors from the ones characteristic in the oceans or in laboratory experiments , @xmath22 - 10 , in contrast to the ones characteristic of stellar conditions ( @xmath23 ) . therefore , a layering process in giant planet interiors can not be excluded and it is worth exploring the consequences on the planet evolution . such layered convection may occur near the discontinuity in composition at the boundary of the central rocky - icy core or in chemically inhomogeneous regions in the interior , reminiscent of the early planetesimal accretion episodes . there is presently no widely accepted description of semiconvection . water - salt experiments @xcite show that a series of quasi - static convective layers separated by diffusive interfaces develop when a balance is reached between the variation of potential energy ( i.e. of buoyancy ) due to mixing at the interface and the kinetic energy of the eddies available at the interface . this translates into a critical richardson number @xmath24=@xmath25 of order 1 to 10 , where @xmath26 is the density contrast between the diffusive and convective layers , @xmath27 is the gravity and @xmath28 is the kinetic energy of the convective flow , of characteristic average length scale @xmath29 and rms velocity @xmath30 . guided by experimental results @xcite and energetic arguments , stevenson ( 1979 ) , in a wave description of semiconvection , showed that , should layers form as a result of small - scale wave breaking whereupon the compositional gradient is redistributed , they would be stable if @xmath31 where the inverse lewis number @xmath32=@xmath33=@xmath34 is the ratio of the solute microscopic diffusivity to the thermal diffusivity . spruit s ( 1992 ) stability condition is less restrictive , since layered formation is supposed to always occur , and is given essentially by eq.(1 ) . under jovian planet conditions , typical values are @xmath35-@xmath36 @xmath18 @xcite , then @xmath37 , so that , according to this criterion , diffusive layers could be at least marginally stable in giant planet interiors . it is worth noting that the molecular to thermal diffusivity ratio is the same for a h / he mixture under jovian interior conditions as for salty water , @xmath38 , so that the extent of the solute versus thermal layer is about the same . since , according to experiments and condition ( 2 ) , this ratio is the relevant criterion for stability of the layers , this adds some support to the planetary case . in a layered convection stratification , heat is carried away from the interfaces by descending and ascending plumes in the overturning regions while transport across the interface occurs by diffusion . because of the boundary layers , only a part of the fluid transports heat efficiently . on average , one has `` convective - like '' motions having a much shorter length scale than for ordinary convection . the _ thermal _ thickness of the diffusive layers , @xmath39 , is determined by a balance between the thickening due to diffusion and the entrainment due to convective motions so the convective time , @xmath40 , in the mixed layer of size @xmath29 must be comparable to the thermal diffusion time , @xmath41 , across the boundary layer @xcite . this yields @xmath42 and @xmath43 for the thickness of the heat and compositional interfacial layers , respectively . the number @xmath44 of layers is of course very uncertain . it can be crudely estimated as follows . the heat flux @xmath45 transported by convection in each mixed layer is the mass flux carried by the plumes fed from the diffusive layers , and thus of width comparable to these layer thickness , times the energy variation across the convective layer , @xmath46 : @xmath47 for a semiconvective region extending over a planet - size region , the total number of layers is thus given by : @xmath48 using characteristic numbers for jovian planet conditions ( with @xmath49 ) , one gets @xmath50-@xmath51 as a rough estimate . if convection is inhibited , however , the smaller heat flux and larger superadiabaticity require less layers . we have conducted calculations following the evolution of a template jupiter - mass planet , representative of hd209458b and similar short - period planets , with a global metal content @xmath52 , including a 6 @xmath53 core , i.e. @xmath54 ( @xmath55 ) , in agreement with previously mentioned planet formation models and jupiter and saturn s observational constraints . the amount of heavy elements is distributed initially throughout the planet following a gradient , distributed within a certain number of boundary layers @xmath44 where condition ( 1 ) is fulfilled . the layers are located within the inner @xmath56 by mass ( 60% in radius ) of the planet , where h and he are fully ionized , to ensure high enough thermal conductivity . the present calculations have been done with @xmath44=50 and @xmath44=100 ; the width of each boundary layer corresponds to @xmath57 cm @xmath58 . a larger number of layers would be computationally too difficult to resolve correctly . these boundary layers are separated by larger convective , mixed layers , with a uniform composition ( @xmath59 ) , where the usual mixing - length formalism applies . the sizes of the boundary and mixed layers ( @xmath60 ) obey the aforementioned relationships . the heat flux @xmath61 and solute flux @xmath62 in the boundary layers are calculated with the appropriate diffusion equations : @xmath61=@xmath63 , where @xmath64=@xmath65 and @xmath66 denote the conductive @xcite and radiative @xcite mean opacities , respectively , and @xmath62=@xmath67 , where @xmath68=@xmath69 , with @xmath70 , is the mean concentration gradient across the layer . conduction remains efficient enough in the thin boundary layers to fulfill condition ( 1 ) . because diffusion limits the heat transport , the internal heat flow of the planet is significantly reduced compared with that of a fully convective object . the signatures of double - diffusive convection in a planetary interior are thus a reduced heat output and a larger radius compared with an object where heat is transported efficiently by large - scale convection . this is illustrated in fig . 1 , which compares the evolution of the radius and thermal intrinsic luminosity of the planet in both cases . the excellent agreement with the otherwise unexplained observed radii of hd209458b and similar irradiated planets suggests that diffusive convection might be taking place in the interior of at least certain giant planets . as seen , the expected luminosity at young ages is more than one order of magnitude fainter than that of a fully convective planet evolving from a comparable initial state . the observational confirmation of the present scenario would be either the determination of an exoplanet temperature or luminosity at young ages , is smaller than the absorbed and reflected contributions of the incident stellar flux , @xmath71 . for long - period planets or for planed telescopes like the lbt or the jwst , dedicated to infrared planet searches , however , the planet intrinsic luminosity can be determined . ] or the observation of an inflated radius for a transiting planet at large enough orbital distance , @xmath72 au for a solar - type parent star , for stellar irradiation not to affect the planet s internal structure . figure 1 illustrates also the dependence of the evolution upon the number of layers . less boundary layers implies larger convective layers and thus more efficient heat transport , as illustrated by the more rapidly decreasing radius in the 50-layer calculations . a key question is to know if diffusive interfaces can persist on time scales comparable to the characteristic time for the evolution of the planet . according to the aforementioned critical richardson number criterion , supported by experiments @xcite , a quantitative argument is that if the average kinetic energy in the convective layers is smaller than ( a fraction of ) the potential energy wall of the interface , convection can not penetrate deeply into this latter and significant entrainment across the interface can not occur . this implies that the molecular diffusion time scale be long enough . this latter can be estimated for the entire stack of layers , distributed over a region of size @xmath73 in the planet ( presently @xmath74 cm ) . the flux of element across an interface is @xmath75 , where @xmath76 is the jump in the element mass fraction at each interface while @xmath77 is the total variation over the entire semiconvective region . the time scale to redistribute the entire gradient over the entire region is then @xmath78 10 gyr . this admitedly crude estimate shows that the stable diffusive convection configuration might last long enough to affect substantially the evolution . with the typical value @xmath79 , about 10% of the initial gradient @xmath77 has been transported by diffusion over a gyr , as confirmed by our numerical calculations . in principle , the compositional gradient thus remains large enough during the evolution for the ledoux criterion to remain valid in a majority of layers . in other words , the temperature jump at interfaces is too small to offset the molecular weight stabilization of interfaces ( @xmath80 ) . the composition and temperature profiles in our calculations at 5 gyr are portrayed in fig . 2 , with @xmath81 and @xmath82 k at each diffusive interface . note that , if layers form in sequence through turbulent entrainment or from sporadically breaking internal waves generated by oscillatory instabilities , interfaces may be dynamically renewed with time , if some compositional gradient or stirring effects remain present . such a process occurs in laboratory systems and oceans . different reasons can be advocated for the cause of the initial compositional gradient . this latter can be inherited from the formation process . large incoming planetesimals could disseminate part of their constituents , iron , silicates , ices , by ablation and break - up as they penetrate the building gaseous envelope @xcite . note also that accretion will not proceed homogeneously as capture mechanisms differ for the gas ( h , he ) , ice ( essentially c , o , n ) and rock ( silicates and iron ) components . this will increase substantially the compositional gradients before the core is reached . as mentioned earlier , even modest gradients can easily offset superadiabatic excess over planet - size regions , preventing large - scale convective motions . a compositional gradient might also result from disruption and redistribution of the core due to a giant impact or erosion at the core - envelope interface because of metallic hydrogen high diffusivity , leading to a core diluted into a fraction of the planet @xcite . the redistribution of these elements might be partially inhibited by diffusive processes , forming diffusive interfaces because of the opposite buoyancy effects of heat and composition . furthermore , when accreting the envelope , part of the outermost regions of the protoplanet might be nearly isothermal @xcite , which favors the stability of a compositional gradient . interestingly , when distributing the layers in the _ outer _ 10% by mass ( @xmath83 by radius ) of the planet , where _ radiative _ thermal diffusivity starts to dominate over conductive diffusivity , we get an effect similar to the one portrayed in fig . the situation for the formation of diffusive convection is particularly favorable for short - period exoplanets for several reasons . first of all , a substantial fraction of the gaseous envelope has been eliminated by evaporation @xcite , leading to a larger metal fraction . second of all , for short - period exoplanets , the numerous collisions tend to eject the gas , leading to a larger enrichment in planetesimals than for the other planets . third of all , the higher internal temperatures for short - period , irradiated planets , favor ( i ) ionization of the various elements and thus the thermal conductivity , ( ii ) solubility of the core material into the envelope . at last , because of the stellar irradiation , the outer layers of short - period planets are isothermal and not adiabatic . the aim of the present letter is to suggest an alternative , possibly important energy transport mechanism in giant planet interiors and to explore its effects on the evolution . these calculations provide a consistent description of the evolution of giant planets , _ with a metal enrichment in agreement with observational constraints _ , in case heat is transported by layered convection . assuming an initial compositional stratification within a certain number of double - diffusive interfaces , diffusive and convective transport in the respective layers are calculated consistently during the evolution . only the outermost 40% in radius ( 70% in mass ) of the planet can convect freely . these calculations , however , can not be expected to give an accurate description of the onset and stability of layered convection . there is presently no accurate treatment of this mechanism under conditions characteristic of giant planets . the only attempt to study the onset of double - diffusive layer formation at low prandtl numbers ( @xmath84 ) @xcite remains inconclusive . indeed , insufficient numerical resolution and artificially enhanced viscous and molecular diffusivities in the simulations might suppress small - scale motions / instabilities and the formation of a statistically steady state of intermitent diffusive layers . simulations of vertical salinity in water , @xmath85=@xmath86 , on the other hand , well reproduce the experiments and confirm the formation of quasi - static convective layers separated by diffusive interfaces above a critical richardson number @xcite . even though the present calculations rely on some uncertain ground , the agreement with the puzzling and otherwise unexplained observed radius of hd209458b and other abnormally large exoplanets leads to the conclusion that layered convection might be taking place in at least some planets and could explain their particular properties . this transport mechanism yields a much reduced heat escaping rate compared with a homogeneous adiabatic structure . even if a stable layered configuration does not occur , overstable modes of convection ( fullfilling eq.(1 ) but not eq.(2 ) ) , due to the presence of opposite diffusive processes ( composition and heat ) of different efficiencies , can lead to the growth of small - scale fluid oscillations @xcite . overstability , however , is more similar to an enhanced diffusion process than to a convective mixing process , with a much smaller energy transport efficiency , as shown by experiments @xcite . the onset or persistence of layered or overstable ( oscillatory ) convection might require optimal conditions , inherited from particularly favorable formation or evolution histories ( e.g. late accretion of large unmixed planetesimals or giant impacts stirring up completely the planet interior ) . the present paper suggests that , under such appropriate conditions , the heat transport mechanism in giant planet interiors can be severely affected , decreasing the efficiency of or even inhibiting large - scale convection . this should motivate 3d investigations of convection in the presence of a stabilizing compositional gradient under conditions suitable to giant planets and the search for transits at larger orbital distances . alibert , y. , mordasini , c. , benz , w. & winisdoerffer , c. , 2005 , , 434 , 343 baraffe , i. et al . , 2005 , , 436 , 47 baraffe , i. , alibert , y. , chabrier , g. & benz , w. , 2006 , , 450 , 25 ferguson et al . , j. , 2005 , , 623 , 585 fernando , h. , 1989 , , 209 , 1 guillot , t. , stevenson , d. , hubbard , w.b . , saumon , d. , 2004 , jupiter : the planet , satellites and magnetosphere , cambridge university press iaroslavitz , e. & podolak , m. , 2007 , icarus , 187 , 600 levrard , b. et al . , 2007 , , 462 , l5 linden , p.f . & shirtcliffe , t. , 1978 , , 87 , 417 merryfield , w. , 1995 , , 444 , 318 mizuno , h. , 1980 , prog . physics , 64 , 544 molemaker , m. & dijkstra , h. , 1997 , , 331 , 199 pollack , j. et al . , 1996 , icarus , 124 , 62 potekhin , a. , 1999 , , 351 , 787 rogers , f. , swenson , f. & iglesias , c. , 1996 , , 456 , 902 saumon , d. & guillot , t. , 2004 , , 609 , 1170 showman , a. , & guillot , t. , 2002 , , 385 , 156 schmitt , r. , 1994 , ann . rev . fluid mech . , 26 , 255 spruit , h. , 1992 , , 253 , 131 stevenson , d. , 1979 , , 187 , 129 stevenson , d. , 1982 , planet . space sci . , 755 stevenson , d. , salpeter , e. , 1977a , , 35 , 221 stevenson , d. , salpeter , e. , 1977b , , 35 , 239

we explore the possibility that large - scale convection be inhibited over some regions of giant planet interiors , as a consequence of a gradient of composition inherited either from their formation history or from particular events like giant impacts or core erosion during their evolution . under appropriate circumstances , the redistribution of the gradient of molecular weight can lead to double diffusive layered or overstable convection . this leads to much less efficient heat transport and compositional mixing than large - scale adiabatic convection . we show that this process can explain the abnormally large radius of the transit planet hd209458b and similar objects , and may be at play in some giant planets , with short - period planets offering the most favorable conditions . observational signatures of this transport mechanism are a large radius and a reduced heat flux output compared with uniformly mixed objects . if our suggestion is correct , it bears major consequences on our understanding of giant planet formation , structure and evolution , including possibly our own jovian planets .

the mission has recently made public a catalog of all transiting planet candidates identified during the first four months of observation by the spacecraft @xcite . included in this list of 1235 objects are nearly 300 in the category of super - earths ( defined here as having radii in the range @xmath5 ) , and several dozen of earth size ( @xmath6 ) . the wealth of new information promises to revolutionize our knowledge of extrasolar planets . although strictly speaking these are still only _ candidates _ since confirmation by spectroscopic or other means is not yet in hand , expectations are high that the rate of false positives in this list is relatively small ( see * ? ? ? * ; * ? ? ? consequently , results from this sample concerning the general properties of exoplanets have already begun to emerge , including studies of the architecture and dynamics of multiple transiting systems @xcite , an investigation of the statistical distribution of eccentricities @xcite , and first estimates of the rate of occurrence of planets larger than 2@xmath7 with orbital periods up to 50 days @xcite , among others . for good reasons the confirmation or `` validation '' of small transiting planets ( earth - size or super - earth - size ) has attracted considerable attention , but has proven to be non - trivial in many cases because of the difficulty of detecting the tiny radial - velocity ( rv ) signatures that these objects cause on their parent stars , as exemplified by the cases of corot-7b @xcite , kepler-9d @xcite , and kepler-11 g @xcite . in fact , such spectroscopic signals are often too small to detect with current instrumentation , and the planetary nature of the candidate must be established statistically , as in the latter two cases . the smallest planet discovered to date , kepler-10b , was announced recently by @xcite , and is the mission s first rocky planet . it has a measured radius of @xmath8 and a mass of @xmath9 , leading to a mean density of @xmath10 that implies a significant iron mass fraction @xcite . its parent star , kepler-10 ( kic11904151 , 2mass119024305 + 5014286 ) , is relatively bright among the targets ( magnitude @xmath11 ) and displays _ two _ periodic signals with periods of 0.84 days and 45.3 days , and flux decrements ( ignoring limb darkening ) of @xmath12 ppm and @xmath1 ppm , respectively @xcite . the extensive observations that followed the detection of these signals are documented in detail by those authors , and include the difficult measurement of the reflex radial - velocity motion of the star with a semi - amplitude of only @xmath13 and a period that is consistent with the shorter signal . as is customary also in ground - based searches for transiting planets , the shapes of the spectral lines were examined carefully to rule out changes of similar amplitude correlating with orbital phase that might indicate a false positive , such as a background eclipsing binary ( eb ) blended with the target , or an eb physically associated with it . however the precision of the measurements ( bisector spans ) compared to the small rv amplitude did not allow such changes to be ruled out unambiguously . false positive scenarios were explored with the aid of , a technique that models the transit light curves to test a wide range of blend configurations @xcite , and it was found that the overwhelming majority of them can be rejected . this and other evidence presented by @xcite allowed the planetary nature of kepler-10b to be established with very high confidence . this was not the case , however , for the 45-day period signal referred to as koi-072.02 ( object of interest 72.02 ) , which is the subject of this paper . no significant rv signal was detected at this period , and only an upper limit on its amplitude could be placed . using , @xcite were able to rule out a large fraction of the blend scenarios involving circular orbits ( including hierarchical triples ) , but eccentric orbits were not explored because of the increased complexity of the problem and the much larger space of parameters for false positives . while circular orbits are a reasonable assumption for kepler-10b because of the strong effects of tidal forces at close range , this is not true for koi-072.02 on account of its much longer orbital period ( see , e.g. , * ? ? ? * ) ; eccentric orbits can not be ruled out . this provides the motivation for the present work , in which we set out to examine all viable astrophysical false positive scenarios for koi-072.02with the goal of validating it as a bona - fide planet . in addition to improvements in the modeling , we bring to bear new near - infrared observations obtained with the _ spitzer _ space telescope in which the transits are clearly detected , as well as the complete arsenal of follow - up observations gathered by the team , including high - resolution adaptive optics imaging and speckle interferometry , high - resolution spectroscopy , and an analysis based on the observations themselves of the difference images in and out of transit for positional displacements ( centroid motion ) . all of these observations combined with the strong constraints provided by significantly limit the kinds of blends that remain possible , and as we describe below they allow us to claim with very high confidence that koi-072.02 is indeed a planet . its estimated radius is approximately 60% of that of neptune . with this , kepler-10 becomes the mission s third confirmed multi - planet system ( after kepler-9 and kepler-11 ; * ? ? ? * ; * ? ? ? * ) containing a transiting super - earth - size planet and at least one larger planet that also transits . we begin with a brief recapitulation of the technique , including recent improvements . we then present the _ warm spitzer_observations at 4.5 that help rule out many blends , and we summarize additional constraints available from other observations . this is followed by the application of to koi-072.02 in order to identify all blends scenarios that can mimic the transit light curve . next we combine this information with the other constraints and carry out a statistical assessment of the false alarm rate for the planet hypothesis , leading to the validation of the candidate as kepler-10c . we conclude with a discussion of the possible constitution of the new planet in the light of current models , and the significance of this type of validation . the detailed morphology of a transit light curve ( length of ingress / egress , total duration ) contains important information that can be used to reject many false positive scenarios producing brightness variations that do not quite have the right shape , even though they may well match the observed transit depth ( see , e.g. , * ? ? ? @xcite takes advantage of this to explore a very large range of scenarios , including background or foreground eclipsing binaries blended with the target , as well as eclipsing binaries physically associated with the target in a hierarchical triple configuration . following the notation introduced by @xcite , the objects composing the binary are referred to as the `` secondary '' and `` tertiary '' , and the candidate is the `` primary '' . the tertiary can be either a star ( including a white dwarf ) or a planet , and the secondary can be a main - sequence star or a ( background ) giant . with the help of model isochrones to set the stellar properties , simulates blend light curves resulting from the flux of the eclipsing pair diluted by the brighter target ( and any additional stars that may fall within the photometric aperture ) . each simulated light curve is compared with the observations in a @xmath14 sense to identify which of them result in acceptable fits ( to be defined later ) . the parameters varied during the simulations are the mass of the secondary star ( @xmath15 ) , the mass of the tertiary ( @xmath16 , or its radius @xmath17 if a planet ) , the impact parameter ( @xmath18 ) , the relative linear distance ( @xmath19 ) between the eclipsing pair and the target , and the relative duration ( @xmath20 ) of the transit compared to the duration for a circular orbit ( see below ) . for convenience the relative linear distance is parametrized in terms of the difference in distance modulus , @xmath21 , where @xmath22 . in the case of hierarchical triple configurations the isochrone for the binary is assumed to be the same as for the primary ( metallicity of @xmath23 } = -0.15 $ ] and a nominal age of 11.9gyr ; see * ? ? ? * ) , whereas for background blends we have adopted for the binary a representative 3gyr isochrone of solar metallicity , although these parameters have a minimal impact on the results . for full details of the technique we refer the reader to the references above . three recent changes and improvements that are especially relevant to the application to koi-072.02 are described next : ( _ i _ ) the relatively long orbital period of koi-072.02 ( 45.3 days ) precludes us from assuming that the eccentricity ( @xmath24 ) is zero , as we were able to suppose in previous applications of to kepler-9d and kepler-10b , which have periods of 1.59 and 0.84 days , respectively . the reason this matters is that the duration of the transit is set , among other factors , by the size of the secondary star . eccentricity can alter the speed of the tertiary around the secondary , making it slower or faster than in the circular case depending on the orientation of the orbit ( longitude of periastron , @xmath25 ) . given a fixed ( measured ) duration , blends with smaller or larger secondary stars than in the circular case may still provide satisfactory fits to the light curve , effectively increasing the pool of potential false positives . now takes this into account , although rather than using as parameters @xmath24 and @xmath25 , which are the natural variables employed in the binary light - curve generating routine at the core of ( see * ? ? ? * ) , a more convenient variable that captures the effects of both is the duration relative to a circular orbit . following @xcite , this may be expressed as @xmath26 . operationally , then , we vary @xmath20 over wide ranges as we explore different blend scenarios , and for each value we infer the corresponding values of @xmath24 and @xmath25 . in practice , in order to solve for \{@xmath24 , @xmath25 } from @xmath20 it is only necessary to consider the limiting cases with @xmath27 and 270 , corresponding to transits occurring at periastron and apastron , respectively , since these are the orientations resulting in the minimum and maximum durations for a given eccentricity . other combinations of @xmath24 and @xmath25 will lead to intermediate relative durations that are already sampled in our @xmath20 grid . it is worth noting that use of only these two values of @xmath25 leads to predicted secondary eclipses in the simulated light curves that are always located at phase 0.5 , whereas secondary eclipses in the real data might be present at any phase . for our purposes this is of no consequence , as koi-072.02 has already had its light curve screened for secondary eclipses at any phase that might betray a false positive , as part of the vetting process . no such features are present down to the 100 ppm level . thus , any simulated light curves from that display a significant secondary eclipse will yield poor fits no matter where the secondary eclipse happens to be , and will lead to the rejection of that particular blend scenario . ( _ ii _ ) for each false positive configuration can predict the overall photometric color of the blend , for comparison with the measured color index of the candidate as reported in the input catalog ( kic ; * ? ? ? * ) . a color index such as @xmath28 , where @xmath29 is the magnitude and @xmath30 derives from the 2mass catalog , provides a reasonable compromise between wavelength leverage and the precision of the index . the latter varies typically between 0.015 and 0.030 mag , depending on the passband and the brightness of the star ( see * ? ? ? we consider a particular blend to be rejected when its predicted color deviates from the kic value by more than three times the error of the latter . as it turns out , color is a particularly effective way of rejecting blends that include secondary stars of a different spectral type than the primary , such as those that become possible when allowing for eccentric orbits . ( _ iii _ ) recent refinements in the resolution of the simulations to better explore parameter space , in addition to the inclusion of eccentricity ( or @xmath20 ) as an extra variable , have increased the complexity of the problem as well as the computing time ( by nearly two orders of magnitude ) compared to the relatively simple case of circular orbits . the number of different parameter combinations examined with ( and corresponding light - curve fits ) can approach @xmath31 in some cases . consequently the simulations are now performed on the pleiades cluster at the nasa advanced supercomputing division , located at the ames research center ( california ) , typically on 1024 processors running in parallel . for convenience hierarchical triple configurations ( 4 parameters ) and background / foreground blends ( 5 parameters ) are studied separately , each for the two separate cases of stellar and planetary tertiaries ( for a total of four grids ) . one additional fit is carried out using a true transiting planet model to provide a reference for the quality of the false positive fits in the other grids . the discriminating value of the shape information contained in the light curves , mentioned at the beginning of this section , is highlighted by our results for kepler-10b , as described by @xcite . in that study it was found that _ all _ background eclipsing binary configurations with stellar tertiaries yield very poor fits to the light curve , and are easily rejected . the underlying reason is that all such blend models predict obvious brightness changes out of eclipse ( ellipsoidal variations ) with an amplitude that is not seen in the data , and that are a consequence of the very short orbital period . ) artificially suppresses out - of - eclipse variations to some extent , typically by median filtering , so that the light curves for periods as short as that of kepler-10b ( 0.84 days ) are rendered essentially flat except for the transits themselves . in this sense the situation is similar to that mentioned earlier regarding the presence of secondary eclipses : obvious ellipsoidal variability in the raw data would normally trigger a false positive warning during the vetting process , preventing the target from becoming an object of interest . but if it reaches koi status , we assume that out - of - eclipse modulations are insignificant so that the comparison with any model in which those variations are present is meaningful and would yield a poor fit , sufficient in most cases to reject the blend . ] hierarchical triple scenarios were also excluded based on joint constraints from and other follow - up observations . the only configurations providing suitable alternatives to the true planet scenario involved stars in the foreground or background of the target that are orbited by a larger transiting planet . the considerable reduction in the blend frequency from the exclusion of all background eclipsing binaries led to a false alarm probability low enough to validate kepler-10b with a very high level of confidence , _ independently _ of any spectroscopic evidence . this remarkable result speaks to the power of when combined with all other observational constraints . it also assumes considerable significance for kepler-10b , given that it was not possible to provide separate proof of the planetary nature of this signal in the @xcite study from an examination of the bisector spans . the scatter of the bisector span measurements ( 10.5 ) was three times larger than the rv semi - amplitude ( 3.3 ) , rendering them inconclusive . the situation regarding the analysis of the koi-072.02 signal in the @xcite study was very different : the orbital period is much longer , and ellipsoidal variations are predicted to be negligible , so that background eclipsing binaries with stellar tertiaries remain viable blends . this , and the added complication from eccentric orbits , hindered the efforts of those authors to validate this candidate . with the benefit of the enhancements in described above , we are now in a better position to approach this problem anew . as follow - up observations provide important constraints that are complementary to those supplied by , and play an important role in determining the false alarm rate for the planetary nature of koi-072.02 ( sect . [ sec : statistics ] ) , we describe those first below , beginning with our new near - infrared _ spitzer _ observations . koi-072.02 was observed during two transits with the irac instrument on the _ spitzer _ space telescope @xcite at 4.5(program i d 60028 ) . the observations were obtained on ut 2010 august 30 and november 15 , with each visit lasting approximately 15hr 10min . the data were gathered in full - frame mode ( @xmath32 pixels ) with an exposure time of 6.0s per image , which resulted in approximately a 7.1s cadence and yielded 7700 images per visit . the method we used to produce photometric time series from the images is described by @xcite . it consists of finding the centroid position of the stellar point spread function ( psf ) and performing aperture photometry using a circular aperture on individual exposures . the images used are the basic calibrated data ( bcd ) delivered by the _ spitzer _ archive . these files are corrected for dark current , flat - fielding , and detector non - linearity , and are converted to flux units . we converted the pixel intensities to electrons using the information on the detector gain and exposure time provided in the fits headers . this facilitates the evaluation of the photometric errors . we extracted the utc - based julian date for each image from the fits header ( keyword date_obs ) and corrected to mid - exposure . we converted to tdb - based barycentric julian dates using the ` utc2bjd ` procedure developed by @xcite . this program uses the jpl horizons ephemeris to estimate the position of the _ spitzer _ spacecraft during the observations . we then corrected for transient pixels in each individual image using a 20-point sliding median filter of the pixel intensity versus time . to do so , we compared each pixel s intensity to the median of the 10 preceding and 10 following exposures at the same pixel position , and we replaced outliers greater than @xmath33 with their median value . the fraction of all pixels we corrected is 0.02% for the first visit and 0.06% for the second . -10pt the centroid position of the stellar psf was determined using the daophot - related procedures ` gcntrd ` , from the idl astronomy library . we applied the ` aper ` routine to perform aperture photometry with a circular aperture of variable radius , using a range of radii between 1.5 and 8 pixels in steps of 0.5 . the propagated uncertainties were derived as a function of the aperture radius , and we adopted the aperture providing the smallest errors . we found that the transit depths and errors varied only weakly with aperture radius for all light - curves analyzed in this project . the optimal aperture was found to have a radius of 4.0 pixels . we estimated the background by examining a histogram of counts from the full array . we fit a gaussian curve to the central region of this distribution ( ignoring bins with high counts , which correspond to pixels containing stars ) , and we adopted the center of this gaussian as the value of the residual background intensity . as seen already in previous _ warm spitzer _ observations @xcite , we found that the background varies by 20% between three distinct levels from image to image , and displays a ramp - like behavior as function of time . the contribution of the background to the total flux from the stars is low for both observations , from 0.1% to 0.55% depending on the image . therefore , photometric errors are not dominated by fluctuations in the background . we used a sliding median filter to select and trim outliers in flux and position greater than 5@xmath34 , representing 1.6% and 1.3% of the data for the first and second visits , respectively . we also discarded the first half - hour s worth of observations , which is affected by significant telescope jitter before stabilization . the final number of photometric measurements used is 7277 and 7362 . the raw time series are presented in the top panel of figure [ fig : spitzerlightcurves ] . we find that the point - to - point scatter in the photometry gives a typical signal - to - noise ratio ( s / n ) of 280 per image , which corresponds to 90% of the theoretical signal - to - noise . therefore , the noise is dominated by poisson statistics . in order to determine the transit parameters and associated uncertainties from the _ spitzer _ time series we used a transit light curve model multiplied by instrumental decorrelation functions , as described by @xcite the transit light curves were computed with the idl transit routine ` occultsmall ` from @xcite . for the present case we allowed for a single free parameter in the model , which is the planet - to - star radius ratio @xmath35 ( or equivalently , the depth , in the absence of limb darkening ) . the normalized orbital semi - major axis ( system scale ) @xmath36 , the impact parameter @xmath18 , the period @xmath37 , and the time of mid transit @xmath38 were held fixed at the values derived from the light curve , as reported by @xcite and summarized below in sect . [ sec : discussion ] . limb darkening is small at 4.5 , but was nevertheless included in our modeling using the 4-parameter law by @xcite and theoretical coefficients published by @xcite . spitzer_/irac photometry is known to be systematically affected by the so - called `` pixel - phase effect '' ( see , e.g. , * ? ? ? * ; * ? ? ? this effect is seen as oscillations in the measured fluxes with a period corresponding to that of the telescope pointing jitter . for the first visit this period was 70 min , and the amplitude of the oscillations was approximately 2% peak - to - peak ; for the second visit the period was 35 min , and the amplitude about 1% . we decorrelated our signal in each channel using a linear function of time for the baseline ( two parameters ) and a quadratic function of the psf position ( four parameters ) to correct the data for each channel . we performed a simultaneous levenberg - marquardt least - squares fit to the data @xcite to determine the transit and instrumental model parameters ( 7 in total ) . the errors on each photometric point were assumed to be identical , and were set to the rms residual of the initial best fit . to obtain an estimate of the correlated and systematic errors in our measurements @xcite we used the residual permutation bootstrap technique , or `` prayer bead '' method , as described by @xcite . in this method the residuals of the initial fit are shifted systematically and sequentially by one frame , and then added to the transit light curve model before fitting again . we considered asymmetric error bars spanning 34% of the points above and below the median of the distributions to derive the @xmath39 uncertainties for each parameter , as described by @xcite . the bottom panel of figure [ fig : spitzerlightcurves ] shows the best - fit model superimposed on the observations from the two visits combined , with the data binned in 36min bins for clarity ( 295 points per bin ) . the transit depths at 4.5 ( after removing limb - darkening effects ) are @xmath40 ppm for the first visit and @xmath41 for the second , which are in good agreement with each other . the weighted average depth of @xmath0 is consistent with the non - limb - darkened value of @xmath1 ppm derived from the light curve @xcite well within the 1@xmath34 errors , strongly suggesting the transit is achromatic , as expected for a planet . the above _ spitzer _ observations provide a useful constraint on the kinds of false positives ( blends ) that may be mimicking the koi-072.02signal . for example , if kepler-10 were blended with a faint unresolved background eclipsing binary of much later spectral type that manages to reproduce the transit depth in the passband , the predicted depth at 4.5 may be expected to be larger because of the higher flux of the contaminating binary at longer wavelengths compared to kepler-10 . since the transit depth we measure in the near infrared is about the same as in the optical , this argues against blends composed of stars of much later spectral type . based on model isochrones and the properties of the target star ( see below ) , we determine an upper limit to the secondary masses of 0.77@xmath42 . spitzer _ constraint is used in sect . [ sec : blender_app ] to eliminate many blends . further constraints of a different kind are provided by high - resolution imaging as described in more detail by @xcite . briefly , these consist of speckle observations obtained on ut 2010 june 18 with a two - color ( approximately @xmath43 and @xmath44 ) speckle camera on the wiyn 3.5 m telescope on kitt peak ( see * ? ? ? * ) , and near - infrared ( @xmath45-band ) adaptive optics ( ao ) observations conducted on ut 2009 september 8 with the pharo camera on the 5 m palomar telescope . no companions were detected around kepler-10 within 15 ( for speckle ) or 125 ( ao ) , and more generally these observations place strong limits on the presence of other stars as a function of angular separation ( down to 005 in the case of speckle ) and relative brightness ( companions as faint as @xmath46 for ao ) . these sensitivity curves are shown in fig . 9 of @xcite , and we make use of that information below . high - resolution spectra described also by @xcite and obtained with the hires instrument on the 10 m keck i telescope place additional limits on the presence of close companions falling within the spectrograph slit ( 087 ) , such that stars within about 2 magnitudes of the target would generally have been seen . a small chance remains that these companions could escape detection if their radial velocity happens to be within a few of that of the target ( which is a narrow - lined , slowly rotating star with @xmath47 ; * ? ? ? * ) , so that the spectral lines are completely blended . this would be extremely unlikely for a chance alignment with a background / foreground star , but not necessarily for physically associated companions in wide orbits , i.e. , with slow orbital motions . we explored this through monte carlo simulations . the results indicate that the probability of having a physical companion within a conservative range of @xmath4810 of the rv of the target that would also go unnoticed in our speckle observations , and that additionally would not induce a rv drift on the target large enough to have been detected in the high - precision measurements of @xcite , is only about 0.1% . finally , an analysis of the image centroids measured from the observations rules out background objects of any brightness beyond about 2 of the target . this exclusion limit ( equivalent to half a pixel ) is considerably more conservative than the 06 reported by @xcite , and accounts for saturation effects not considered earlier ( given that at @xmath11 the star is very bright by standards ) as well as quarter - to - quarter variations ( where `` quarters '' usually represent 3-month observing blocks interrupted by spacecraft rolls required to maintain the proper illumination of the solar panels ) . the photometry used here is the same as employed in the work of @xcite , and was collected between 2009 may 2 and 2010 january 9 . these dates correspond to quarter 0 ( first nine days of commissioning data ) through the first month of quarter 4 . for this study we used only the long - cadence observations ( 10,870 measurements ) obtained by the spacecraft at regular intervals of about 29.4 min . all blend models generated with were integrated over this time interval for comparison with the measurements . the original data have been de - trended for this work by removing a first - order polynomial , and then applying median filtering with a 2-day wide sliding window . observations that occur during transits were masked and did not contribute to the median calculation . because this sliding window is considerably shorter than the 45.3-day orbital period , any ellipsoidal variations present in the original data should be largely preserved , although in any case they are expected to be very small for binaries with periods as long as this . we adopted also the ephemeris of mid - transit for koi-072.02 as reported by @xcite , which is @xmath49 } = 2,\!454,\!971.6761 + n \times 45.29485 $ ] days , where @xmath50 is the number of cycles from the reference epoch . because it is relatively bright ( @xmath11 ) , kepler-10 was also observed by the mission with a shorter cadence of approximately 1 min for a period of several months to allow an asteroseismic characterization of the star . a total of 19 oscillation frequencies were detected , and enabled a very precise determination of the mean stellar density . when combined with stellar evolution models and a spectroscopic determination of the effective temperature and chemical composition , the resulting parameters for the star are very well determined . kepler-10 is relatively old ( @xmath51gyr ) but is otherwise quite similar to the sun , with a temperature of @xmath52k , a mass and radius of @xmath53@xmath42 and @xmath54@xmath55 , and a composition [ fe / h ] @xmath56 slightly below solar @xcite . as indicated earlier we considered four general scenarios for false positives : chance alignments ( a pair of background / foreground eclipsing objects ) and hierarchical triple systems , each with tertiaries that can be either stars or planets . the free parameters were varied over the following ranges : secondary mass @xmath15 between 0.10 and 1.40@xmath42 , in steps of 0.02@xmath42 ; tertiary mass @xmath16 between 0.10 and @xmath15 , also in steps of 0.02@xmath42 ; tertiary radius @xmath17 between 0.06 and 2.00@xmath57 in steps of 0.02@xmath57 ; impact parameter @xmath18 between 0.00 and 1.00 in steps of 0.05 ; relative duration @xmath20 between 0.2 and 4.6 in steps of 0.2 , corresponding to eccentricities up to 0.92 and values of @xmath25 of 90 and 270 ( see sect . [ sec : blender ] ) ; and relative distance @xmath21 ( distance modulus difference ) between @xmath58 and @xmath59 in steps of 0.5 mag , except for hierarchical triple configurations , for which @xmath60 . the goodness of the fit of each of the large number of synthetic light curves generated by is quantified here by computing the @xmath14 statistic and comparing it with that of the best planet model fit . the difference can be assigned a significance level ( or false alarm rate ) that depends on the number of free parameters of the problem . for example , for a blend scenario corresponding to a hierarchical triple system ( 4 degrees of freedom ) , a trial model giving a worse fit than the planet solution by @xmath61 4.72 is statistically different at the 1@xmath34 level , assuming gaussian errors ( see , e.g. , * ? ? ? a fit that is worse by @xmath61 16.3 is different at the 3@xmath34 level . hierarchical triple blends giving poorer fits than this are considered here to be ruled out by the photometry . for background / foreground scenarios ( 5 degrees of freedom ) the 3@xmath34 blend rejection level is @xmath61 18.2 . in this section we describe the simulations carried out for the four general blend configurations mentioned above . although the secondaries for the background scenarios can in principle also be evolved stars ( giants ) , as opposed to main - sequence stars , we consistently found that the transit light curves generated by such systems give a very poor match to the observations because they do not have the right shape ( the ingress / egress phases are too long ) . therefore , we restricted our exploration of parameter space to main - sequence stars only . an additional possibility for a false positive may stem from an error in the determination of the orbital period . if the true period were twice the nominal value , alternating transit events would correspond to primary and secondary eclipses , implicating a blended eclipsing binary . the primary and secondary eclipses would often ( but not always ) be of different depth . as part of the vetting process for each candidate , the team examines the even - numbered and odd - numbered events to look for differences in depth that may indicate a false positive of this kind . as described by @xcite , no significant differences were found for koi-072.02 beyond the 2@xmath34 level , where @xmath34 represents the uncertainty in the transit depth ( 9 ppm ) . nevertheless , as the possibility still exists that the components of the eclipsing binary are identical , experiments were run with to examine the transit shape produced by such scenarios , and it was found that the ingress and egress phases are always much too long compared to the observations , as expected for two equal - size stars eclipsing each other . thus , these scenarios are easily ruled out as well . the simulations with indicate that few background blend scenarios with stellar tertiaries are able to mimic the transit features in the light curve at an acceptable level , and they all correspond to somewhat eccentric orbits . in figure [ fig : backstar ] we show the goodness of fit of these scenarios , with the small closed 3@xmath34 contour representing the region of parameter space within which the fits are satisfactory , according to the criteria given above . only blends with secondary masses @xmath15 larger than about 1.3@xmath42 are allowed , and the eclipsing binary can only be within a small range of distances behind the target ( @xmath62 ) for the dilution effect to be just right , such that the corresponding apparent brightness difference @xmath63 is between 2.5 and 3.5 mag ( see figure ) . the best among these blend models ( located near the center of the contour ) provides a fit that is about 2.1@xmath34 worse than a planet model ( but still acceptable ) , and is shown in the top panel of figure [ fig : fits ] compared against the planet model . the tertiary stars in these blends are constrained to be very small , between 0.10 and 0.16@xmath42 . that most blends involving background eclipsing binaries can be ruled out may appear somewhat surprising , and is worth investigating . indeed , for a given measured transit depth @xmath64 , a blend can only reproduce the light curve if it contributes at least a fraction @xmath64 of the total flux collected in the aperture . thus , one would expect that binaries as faint as @xmath65 mag relative to the target should be able to match that amount of dimming if they were totally eclipsed ( see , e.g. , * ? ? ? * ) , and furthermore , that the measured duration could also be reproduced by a large range of secondary sizes with an appropriate combination of orbital eccentricity and @xmath25 . yet we find that no blends fainter than @xmath66 give tolerable fits to the light curve ( see figure [ fig : backstar ] ) . a visual understanding of the underlying reason for this may be seen in the bottom panel of figure [ fig : fits ] , in which we show a blend model that one would naively expect should be able to match the observations , according to the crude recipe described above . this particular blend scenario is marked with a cross in figure [ fig : backstar ] , and corresponds to @xmath67 and @xmath68@xmath42 , resulting in a magnitude difference of @xmath69 for the eb relative to the target . while this model does yield a good match to the measured depth , and even the total duration , it does nt perform nearly as well in the ingress / egress phases , which are too long when compared against the observations . the quality of this fit relative to the best planet fit , which can also be seen in the figure , corresponds to a 10.1@xmath34 difference , and therefore rejects it . thus , the reason most blends of this class can be ruled out is ultimately the high precision of the light curves , which provides a very strong constraint on the shape of the transit light curve , and in particular on the size ratio between the secondary and tertiary , which sets the duration of the ingress and egress phases . there is a very broad range of blends consisting of a background or foreground star transited by a planet ( as opposed to a star ) that are found by to give satisfactory fits to the data , as shown in figure [ fig : back_plan ] . these viable blends occupy the area below the 3@xmath34 contour represented with a thick white line . secondary stars of all spectral types ( masses ) are permitted , in principle , although in practice other constraints described below eliminate a substantial fraction of them . all of these blends involve secondary+tertiary pairs that are within 4 magnitudes of the target in the passband ( diagonal dashed line in the figure ) . the tertiary sizes in these blends range from 0.42@xmath57 to 1.84@xmath57 . our _ warm spitzer _ observations set a lower limit of about 0.77@xmath42 for the secondary masses of these blends , as described earlier ; scenarios involving redder stars would result in transits at 4.5 significantly deeper than we observe ( i.e. , deeper than the measured depth + 3@xmath34 ) . this exclusion region is indicated by the shaded area . additionally , blends that are much brighter than @xmath70 would most likely have been detected spectroscopically ( see * ? ? ? * ) , so we consider those to be ruled out as well . we indicate this with the green hatched region in the lower right - hand side of the figure finally , the colors of the background / foreground configurations simulated with provide a further constraint which is represented by the blue hatched area on the lower left of the figure . this swath of parameter space is excluded because the blends are significantly redder than the color index measured for kepler-10 ( @xmath71 ) , by more than three times the uncertainty in the observed index . as a result of these complementary constraints , the only section of parameter space remaining for viable blends involving star+planet pairs is the area under the 3@xmath34 contour and limited from below and on the left by the hatched areas ( color and brightness conditions ) and shaded area ( _ spitzer _ constraint ) , respectively . all of these blends have the eclipsing pair behind the target ( foreground scenarios are all ruled out ) . we note that in this star+planet blend scenario white dwarfs can also act as tertiaries , as long as they are cooler than the secondaries so that they do not lead to deep occultation events that would have been seen in the light curve of koi-072.02 . the above range of tertiary radii ( 0.42@xmath57 to 1.84@xmath57 ) excludes essentially all cool carbon - oxygen and oxygen - neon white dwarfs more massive than about 0.4@xmath42 , as these are smaller than the lower limit set by , which corresponds to 4.7@xmath7 ( see , e.g. , * ? ? ? low - mass helium - core or oxygen - core white dwarfs that are the product of common - envelope evolution in binary stars can be considerably larger in size , although they appear to be very rare . the mission itself has uncovered only three examples to date @xcite . however , all of them are very hot ( @xmath72k ) , and produce deep and unmistakable flat - bottomed occultation signals . model calculations such as those of @xcite show that as these helium - core white dwarfs cool , their radii quickly become earth - size or smaller . therefore , we do not consider white dwarfs to be a significant source of blends for koi-072.02 . eclipsing binaries composed of two stars physically associated with the target are clearly ruled out by , as they produce very poor fits to the light curves . for cases in which the tertiaries are planets , viable scenarios identified by span a range of secondary masses and tertiary radii within the 3@xmath34 contour shown in figure [ fig : htp_r3 ] . most of these configurations turn out to involve eccentric orbits , with transit durations longer than those corresponding to circular orbits along with secondary stars that are smaller than the primary ( see figure [ fig : htp_dur ] ) . once again other observational constraints are very complementary , and in this case they are sufficient to exclude all of these blends . for example , the shaded area of parameter space to the left of 0.77@xmath42 is eliminated by the _ spitzer _ observations , as described earlier . the constraint on the @xmath73 color ( hatched area on the left ) is partly redundant with the nir observations , but extends to slightly larger secondary masses . and finally , the spectroscopic constraint removes the remaining scenarios corresponding to higher - mass ( brighter ) secondaries . we conclude that of all the hierarchical triple blend scenarios that are capable of precisely reproducing the detailed shape of the transit light curve , _ none _ would have escaped detection by one or more of our follow - up efforts , including nir _ spitzer_observations , high - resolution spectroscopy , or absolute photometry ( colors ) . ) , not only here but in all previously discovered transiting planets . for the present purposes we do not consider this `` twin star '' scenario as a false positive in the strict sense ( see also * ? ? ? * ) , as the transiting object would still be a planet , only that it would be larger than we thought by about a factor of @xmath74 because of the extra dilution from the companion.[twin ] ] this highlights the importance of these types of constraints for validating candidates , given that blends involving physically associated stars would generally be spatially unresolved by our high - resolution imaging with adaptive optics or speckle interferometry , and they would typically also be below the sensitivity limits of our centroid motion analysis , so that they would not be detected by those means . therefore , the only blends we need to be concerned about for koi-072.02 are those consisting of stars in the background of the target that are orbited by other stars or by transiting planets . in order to estimate the frequency of the blend scenarios ( i.e. , background configurations ) that remain possible after applying and all other observational constraints , we follow a procedure similar to that described by @xcite for kepler-9d . we appeal to the besanon galactic structure models of @xcite to predict the number density of background stars of each spectral type ( mass ) and brightness around kepler-10 , in half - magnitude bins , and we make use of estimates of the frequencies of transiting planets and of eclipsing binaries from recent studies by the team to infer the number density of blends . using constraints from our high - resolution imaging ( specifically , the sensitivity curves presented by * ? ? ? * their fig . 9 ) we calculate the area around the target within which blends would go undetected , and with this the expected number of blends . ) , is significantly less constraining than the high - resolution imaging , so is not as useful here as it was for kepler-9d . ] the recent release by @xcite of a list of 1235 candidate transiting planets ( kois ) from provides a means to estimate planet frequencies needed for our calculations , with significant advantages over the calculations of @xcite for kepler-9d , which were based on the earlier list of candidates published by @xcite . not only is the sample now much larger , but the knowledge of the rate of false positives for is also much improved , and that rate is believed to be relatively small ( 2040% depending on the level of vetting of the candidate , according to @xcite ; less than 10% according to the recent study by @xcite ) . thus , our results will not be significantly affected by the assumption that all of the candidates are planets ( see also below ) . an additional assumption we make is that this census is largely complete . among these candidates we count a total of 267 having radii in the range allowed by for the tertiaries of viable blends ( i.e. , between 0.42 and 1.84@xmath57 ) . with the total number of targets being 156,453 @xcite , the relevant frequency of transiting planets for our blend calculation is @xmath75 . @xcite have recently published a catalog of the 2165 eclipsing binaries found in the field , from the first four months of observation . only the 1225 detached systems among these are considered here , since binaries in the category of semi - detached , over - contact , or ellipsoidal variables would not produce light curves with a shape consistent with a transit . the frequency of eclipsing binaries for our purposes is then @xmath76 . table [ tab : new_stats ] presents the results of our calculation of the frequency of blends , separately for background blends with stellar tertiaries ( eclipsing binaries ) and with planetary tertiaries . columns 1 and 2 give the @xmath29 magnitude range of each bin and the magnitude difference @xmath77 relative to the target , calculated at the upper edge of each bin . column 3 reports the mean number density of stars per square degree obtained from the besanon models , for stars in the mass range allowed by as shown in figure [ fig : backstar ] . in column 4 we list the maximum angular separation @xmath78 at which stars in the corresponding magnitude bin would go undetected in our imaging observations , taken from the information in the work of @xcite . the product of the area implied by this radius and the stellar densities in the previous column give the number of stars in the appropriate mass range , listed in column 6 in units of @xmath79 . multiplying these figures by the frequency of eclipsing binaries @xmath80 then gives the number of background star+star blends in column 7 . a similar calculation for the background star+planet blends , making use of @xmath81 , is presented in columns 710 . we sum up the contributions from each magnitude bin at the bottom of columns 6 and 10 . the total number of blends we expect _ a priori _ ( blend frequency ) is given in the last line of the table by adding these two values together , and is @xmath82 . the calculations show that background blends consisting of star+planet pairs contribute to this frequency about three times more than background eclipsing binaries . while we have assumed up to now that any companions to koi-072.02 within @xmath70 mag of the target would have been seen spectroscopically , we note that relaxing this condition to a much more conservative @xmath83 has no effect at all on the contribution from eclipsing binaries , and a negligible effect on the contribution of star+planet scenarios . to obtain a bayesian estimate of the probability that koi-072.02 is indeed a planet as opposed to a false positive ( or equivalently , the `` false alarm rate '' , far ) we follow the general methodology of @xcite and compare the _ a priori _ likelihoods of blends and of planets : far = bf / pf . if the _ a priori _ blend frequency is sufficiently small compared the planet frequency ( pf ) , we consider the planet validated . our _ a priori _ blend frequencies above correspond to false positive scenarios giving fits to the light curve that are within 3@xmath34 of the best planet fit . we use a similar criterion to estimate the _ a priori _ planet frequency by counting the kois in the @xcite sample that have radii within 3@xmath34 of the best fit from a planet model ( @xmath84 ; see table [ tab : systemparams ] below ) . we find that 157 among the 1235 kois are in this radius range ( 2.062.38@xmath7 ) , giving pf @xmath85 . this results in a false alarm rate for koi-072.02 of @xmath86 , which is so small that it allows us to validate the candidate with a very high level of confidence . the planet is designated kepler-10c . this result rests heavily on the _ a priori _ frequency of planets from the mission , derived from the assumption that all 1235 candidates reported by @xcite are indeed planets rather than false positives . if we were to be as pessimistic as to assume that as many as 90% of the small - size candidates are actually false positives ( a similar rate of false positives as is typically found in ground - based surveys for transiting planets ) , and at the same time that all of the larger - size candidates that come into the blend frequency calculation are real planets ( thereby maximizing bf and minimizing pf ) , the false alarm rate would be 10 times larger than before , or @xmath87 . this is still a very small number , and our conclusion regarding validation is unchanged . we note that a rate of false positives as high as 90% yields a planet frequency that is strongly inconsistent not only with the expectations of @xcite and @xcite , but also with the independent results of ground based doppler surveys as reported by @xcite . in the above calculations we have implicitly assumed similar period distributions for planets of all sizes and for eclipsing binaries . however , it is conceivable that the results could change if the period distribution of planets such as kepler-10c were significantly different from the one for larger planets that go into the blend frequency calculations , or from the one for ebs ( which have a smaller contribution to bf ; see table [ tab : new_stats ] ) . therefore , as a further test we considered the impact of restricting the periods to be within an arbitrary factor of two of the kepler-10c period of 45.3 days , both in our blend frequency calculations and for the _ a priori _ estimate of the planet frequency , pf . we find that the planet frequencies are reduced by a factor of 4.5 , and the eclipsing binary frequency by a factor of 10.4 , and as a result the false alarm rate for koi-072.02 is @xmath88 , which is about the same as before . thus , our conclusions are robust against assumptions about the period distributions . finally , our false alarm rate is conservative in the sense that we have not accounted for the flatness ( coplanarity ) of the kepler-10 system . only a small fraction of single transiting planets with periods as long as 45 days orbiting background stars ( i.e. , those acting as blends ) are likely to transit , _ a priori _ , whereas a planet of this period such as kepler-10c is much more likely to transit if it is coplanar with kepler-10b . taking this into account would boost the planet frequency ( pf ) and decrease the far by as much as an order of magnitude ( see , e.g. , * ? ? ? coplanarity in multiple systems is in fact supported by the large number of multiple transiting system candidates found by @xcite , and their mutual inclinations seem to be small ( 15 ; * ? ? ? therefore , we consider our estimate of the far for kepler-10c to be conservative . the stellar , orbital , and planetary parameters inferred for the system as determined by @xcite are summarized in table [ tab : systemparams ] , to which we add the transit duration . the small formal uncertainty in the planetary radius ( @xmath892.4% ) derives from the relatively high precision of the stellar radius , which is based on asteroseismic constraints on the mean density of the star . with its radius of about 2.2@xmath7 , kepler-10c is among the smallest exoplanets discovered to date . the mass is undetermined as the doppler signature has not been detected . nevertheless , @xcite placed a constraint on it based on the distribution of masses resulting from the markov chain monte carlo fitting procedure they applied to the existing radial - velocity measurements of kepler-10 . their conservative 3-@xmath34 upper limit for the mass is 20@xmath4 . the corresponding maximum mean density is 10 g @xmath90 . given a precise radius measurement and mass upper limit of 20@xmath4 , some minimal constraints can be placed on the composition of kepler-10c . using the models of @xcite , we find that an earth - like rock - iron composition is only possible at @xmath89 20 @xmath4 . lower masses would require a depletion in iron compared to rock , or more likely an enrichment in low - density volatiles such as water and/or h@xmath91/he gas . a 50/50 rock / water composition yields 2.23@xmath7 at 7@xmath4 . still lower masses are possible with a h@xmath91/he gas envelope . using models presented in @xcite , a planet with a rock / iron core and a 5% h@xmath91/he atmosphere ( by mass ) matches the measured radius of kepler-10c at only 3@xmath4 . a massive 20@xmath4 core should have attained a h@xmath91/he envelope , and it would appear to be stable at kepler-10c s relatively modest irradiation level , which would lead to a planetary radius dramatically larger than 2.23@xmath7 . this would tend to favor a scenario where kepler-10c is more akin to gj 1214b @xcite and kepler-11b and kepler-11f , which are all below 7@xmath4 and enriched in volatiles . the well measured inclinations of both kepler-10b and kepler-10c allow us to put a weak constraint on the true mutual inclination ( @xmath92 ) between the orbital planes of the two planets . although the relative orientation in the plane of the sky ( i.e. , the mutual nodal angle ) is unknown , the different impact parameters and resulting apparent inclinations place a lower limit on @xmath92 . as discussed by @xcite , the geometric limits to the mutual inclination are given by @xmath93 , where @xmath94 @xcite and @xmath95 ( table [ tab : systemparams ] ) are the usual inclinations with respect to the line of sight . assuming a random orientation of the lines of nodes ( which does not account for the _ a priori _ knowledge that both planets are transiting ) , the mutual inclination is constrained to be in the interval @xmath96 , with the most likely values being at the extremes of this distribution . making the reasonable supposition of non - retrograde orbits , a mutual inclination close to the lower limit of about 5 is most likely for these planets . a more detailed probabilistic argument requires making assumptions about the number of planets in the kepler-10 system . this mutual inclination is on the high end of the distribution inferred for other multiple candidate systems ( 15 ) by @xcite . if this mutual inclination is typical for planets in this system , then it is relatively likely ( depending on the orbital period ) that other planets , if present , are not transiting . when considering the set of candidates in multiple systems that have periods less than 125 days , the ratio of periods between kepler-10c and kepler-10b ( which is 54.1 ) is by far the highest of all period ratios of neighboring pairs of candidates ( the next highest being 23.4 ) , and is even higher than the period ratios between non - neighboring planets . clearly , there is room for multiple additional planets between kepler-10b and kepler-10c . the preponderance of tightly - packed multiple candidate systems suggests that additional planets may exist , and these may be revealed in the future with more detailed transit timing variation measurements . kepler-10c is the first target observed with _ warm spitzer_with the aim of testing the wavelength dependence of the transit depth . this is currently the only facility available that has the capability of detecting such shallow transits at wavelengths that are sufficiently separated from the passband to be helpful . in this case the observations were successful , and the transit at 4.5 is shown to have virtually the same depth as in the optical . this places a very strong constraint on the color of potential blends , which are restricted to have secondaries of similar spectral type as the primary star . the detailed analysis of the photometry with combined with constraints from other observations eliminates the vast majority of possible blend scenarios . this includes most background eclipsing binaries ( leaving only a small range of possible spectral types and relative fluxes for the secondaries ) , most of the scenarios involving chance alignments with a star transited by a larger planet , and all possible hierarchical triple configurations . the latter are among the most difficult to detect observationally since they are typically spatially unresolved . the key factors that have allowed this , and made possible the validation of the planet , are the high - precision of the photometry , the relatively short ingress and egress phases ( which places strong constraints on the size ratio between the secondary and tertiary ) , and the near equatorial orientation , resulting in a relatively flat transit that leaves less freedom for the parameters of the eclipsing binaries . we expect to be similarly effective for other candidates that show similar features in their light curves . kepler-10c along with kepler-9d and kepler-11 g are examples of transiting planets that have not received the usual confirmation by dynamical means that previous discoveries have enjoyed ( including essentially all ground - based discoveries ) , in which either the doppler signature is detected unambiguously ( and verified by the lack of bisector span variations ) , or transit timing variations in a multiple system are directly measured ( as in kepler-9b and c as well as the five inner planets of the kepler-11 system ) . instead , the planets in those three cases have been _ validated _ statistically , with a bayesian approach to estimate the probability that the transit signals are due to a planet rather than a false positive . this probability has been computed by first estimating the _ a priori _ likelihood of a false positive , and then comparing it with the _ a priori _ chance of having observed a true planet . in the three cases mentioned above the ratio of the false positive to planet likelihoods is small enough that the planetary nature of the signal is established with a very high degree of confidence . for kepler-10c the false alarm rate is @xmath2 . the recent work of @xcite has provided a means of assessing a rough false alarm rate for candidates as a function of the depth of the transit signal and the brightness of the object . as noted also by those authors , while these estimates are extremely valuable for statistical studies , the validation of candidates on an individual basis with a sufficiently high degree of confidence will usually require a much more detailed analysis of false positives , such as we have performed here . masses for these objects ( other than upper limits ) may of course be difficult or impractical to determine in many cases , but it is worth keeping in mind that some of the most exciting candidates to be discovered by will be in this category , namely , earth - size planets in the habitable zones of their parent stars . except for stars of late spectral type , the rv signals will generally be very challenging to detect with the sensitivity of current instrumentation . thus , statistical validation of planets is likely to play an important role for in the years to come . funding for this discovery mission is provided by nasa s science mission directorate . this research has made use of the facilities at the nasa advanced supercomputing division ( nasa ames research center ) , and is based also on observations made with the spitzer space telescope which is operated by the jet propulsion laboratory , california institute of technology under a contract with nasa . support for this work was provided by nasa through an award issued by jpl / caltech . we thank mukremin kilic and rosanne di stefano for helpful discussions about white dwarfs , and the anonymous referee for constructive comments . ccccccccccc 11.011.5 & 0.5 & & & & & & & & & + 11.512.0 & 1.0 & & & & & & & & & + 12.013.0 & 1.5 & & & & & & & & & + 12.513.0 & 2.0 & & & & & & & & & + 13.013.5 & 2.5 & & & & & & 139 & 0.12 & 0.485 & 0.0008 + 13.514.0 & 3.0 & 32 & 0.15 & 0.175 & 0.0014 & & 197 & 0.15 & 1.074 & 0.0018 + 14.014.5 & 3.5 & 44 & 0.18 & 0.346 & 0.0027 & & 278 & 0.18 & 2.183 & 0.0037 + 14.515.0 & 4.0 & & & & & & 351 & 0.20 & 3.403 & 0.0058 + 15.015.5 & 4.5 & & & & & & & & & + 15.516.0 & 5.0 & & & & & & & & & + 16.016.5 & 5.5 & & & & & & & & & + 16.517.0 & 6.0 & & & & & & & & & + 17.017.5 & 6.5 & & & & & & & & & + 17.518.0 & 7.0 & & & & & & & & & + 18.018.5 & 7.5 & & & & & & & & & + 18.519.0 & 8.0 & & & & & & & & & + & 76 & & 0.521 & * 0.0041 * & & 965 & & 7.145 & * 0.0121 * + + [ -1.5ex ] effective temperature , @xmath97 ( k ) & @xmath98 & a + surface gravity , @xmath99 ( cgs ) & @xmath100 & a + metallicity , [ fe / h ] & @xmath101 & a + projected rotation , @xmath102 ( ) & @xmath103 & a + mass , @xmath104 ( @xmath42 ) & @xmath105 & b + radius , @xmath106 ( @xmath55 ) & @xmath107 & b + surface gravity , @xmath108 ( cgs ) & @xmath109 & b + luminosity , @xmath110 ( @xmath111 ) & @xmath112 & b + absolute @xmath43 magnitude , @xmath113 ( mag ) & @xmath114 & b + age ( gyr ) & @xmath115 & b + distance ( pc ) & @xmath116 & b + orbital period , @xmath37 ( days ) & @xmath117 & c + mid - transit time , @xmath38 ( hjd ) & @xmath118 & c + scaled semimajor axis , @xmath119 & @xmath120 & c + scaled planet radius , @xmath121 & @xmath122 & c + impact parameter , @xmath18 & @xmath123 & c + orbital inclination , @xmath124 ( deg ) & @xmath125 & c + transit duration , @xmath126 ( hours ) & @xmath127 & c + radius , @xmath128 ( @xmath7 ) & @xmath129 & b , c + mass , @xmath130 ( @xmath4 ) & @xmath131 & d + mean density , @xmath132 ( g @xmath90 ) & @xmath133 & d + orbital semimajor axis , @xmath134 ( au ) & @xmath135 & e + equilibrium temperature , @xmath136 ( k ) & 485 & f + [ -1.5ex ]

the mission has recently announced the discovery of kepler-10b , the smallest exoplanet discovered to date and the first rocky planet found by the spacecraft . a second , 45-day period transit - like signal present in the photometry from the first eight months of data could not be confirmed as being caused by a planet at the time of that announcement . here we apply the light - curve modeling technique known as to explore the possibility that the signal might be due to an astrophysical false positive ( blend ) . to aid in this analysis we report the observation of two transits with the _ spitzer _ space telescope at 4.5 . when combined they yield a transit depth of @xmath0 ppm that is consistent with the depth in the passband ( @xmath1 ppm , ignoring limb darkening ) , which rules out blends with an eclipsing binary of a significantly different color than the target . using these observations along with other constraints from high - resolution imaging and spectroscopy we are able to exclude the vast majority of possible false positives . we assess the likelihood of the remaining blends , and arrive conservatively at a false alarm rate of @xmath2 that is small enough to validate the candidate as a planet ( designated kepler-10c ) with a very high level of confidence . the radius of this object is measured to be @xmath3 ( in which the error includes the uncertainty in the stellar properties ) , but currently available radial - velocity measurements only place an upper limit on its mass of about 20@xmath4 . kepler-10c represents another example ( with kepler-9d and kepler-11 g ) of statistical `` validation '' of a transiting exoplanet , as opposed to the usual `` confirmation '' that can take place when the doppler signal is detected or transit timing variations are measured . it is anticipated that many of s smaller candidates will receive a similar treatment since dynamical confirmation may be difficult or impractical with the sensitivity of current instrumentation .

the measurement of masses of galaxies has been , over a long period of time , an interesting and difficult problem , which has elicited the application of various and diverse techniques @xcite . since the determination of rotation curves for a large number of spiral galaxies @xcite and the suggestion that these rotation curves are flat because of the presence of an unseen amount of mass which has been called ` dark matter ' , the determination of the mass of all types of galaxies has become a pressing concern of modern astronomy . it is fair to say at this point that there is no direct evidence of the existence of dark matter and that there are other explanations which , although not as currently popular as dark matter , may explain the observations quite reasonably . the total mass of a galaxy is composed of two elements ; luminous matter and dark matter . if we assume that both luminous and dark matter respond to the newtonian gravitational law in the same way , then the difference between the dynamical mass and the luminous mass of a galaxy provides us with an estimation of the amount of dark matter present in the galactic system in question . from such a determination we would be able to study if a dependence of the amount of dark matter with dynamical mass and/or redshift exists . measuring the amount of radiation from a particular galaxy , combined with typical mass to light ratios ( @xmath3/@xmath4 ) that have been calibrated using different stellar samples in our own galaxy , allows us to estimate its stellar , gas and dust content . moreover , rotation curves for spiral galaxies permit the calculation of dynamical mass inside any radius for which a value of rotation velocity is known , allowing us , in principle , to calculate from these two determinations the amount of dark matter present in the galaxy under study . as is well known , rotation velocity curves are used for studying the kinematics of galaxies , determining the amount and distribution of mass interior to a given radius , to derive an insight into galactic evolutionary histories and the possible role that interactions with other systems may have played . since rotation curves may be obtained at different wavelengths they provide information as to the kinematics of different constituents of a galaxy . they may be observed in the infrared as well as in the optical , which may be used to trace ionised gas and the stellar motions , also in the radio and microwave regimes which trace the neutral and molecular gas components of a galaxy . recently , stellar population synthesis models have been used to calculate galactic masses . these models also give us an idea of the total stellar content of a galaxy as well as the distribution of stars of all the different spectral types and luminosity classes @xcite . dynamical theoretical models can also be used to calculate masses for early - type galaxies ( etgs ) , such as those which @xcite constructed for 37 bright elliptical galaxies . from these models he found an average ( m/@xmath5 . discrepancies of the observed velocities in the outer parts with those predicted by the models may be explained by the inclusion of massive dark haloes . @xcite performed dynamical studies of the shapes of line - profiles for 21 elliptical galaxies ; they used them to investigate the dark halo properties and dynamical family relations of these galaxies . they appear to have minimal haloes implied from the fact that the ratio m/@xmath6 turned out maximal . some of these galaxies showed no dark matter within @xmath7 . @xcite investigated the correlations between the mass - to - light m/@xmath4 ratios of 25 elliptical and lenticular galaxies . field and cluster galaxies presented no difference , and their dark matter content within an effective radius @xmath8 was @xmath9 of the total mass contained there . it appeared that the amount of dark matter correlates with galactic rotation velocity ; in the sense that more massive slow - rotating galaxies contain less dark matter that the fast - rotating galaxies . there have been many papers in which dynamical arguments are used to calculate the dynamical mass of galaxies , and hence , by comparison with the amount of luminous mass , they calculate the amount of dark matter present , see for example : @xcite , @xcite , @xcite , @xcite to mention a few . also check the detailed introduction to this topic published in @xcite . the gravitational lens phenomenon provides direct and precise measurements of masses of galaxies at different scales , and allows us to establish the nature and presence of dark matter in a galactic system . elliptical galaxies have been considered to have extended dark - matter massive haloes @xcite that follow the @xcite density profiles . @xcite and @xcite have studied the kinematics of different components in nearby elliptical systems and have concluded that dark matter haloes are required to explain the dynamics of massive elliptical galaxies , provided that newtonian gravity be valid at these scales . galactic mass determinations have also been made using weak and strong lensing observations @xcite . the fraction of total mass in the form of dark matter in etgs , @xmath10 , appears to increase with growing radius reaching values of @xmath11 at five effective radii @xcite . furthermore , @xmath10 within a fixed radius seems to grow with galaxy stellar mass and with velocity dispersion @xcite . @xmath10 varies from small values as in the case of bright giant elliptical galaxies @xcite to very large values , as has been found for dwarf spheroidal galaxies by @xcite . studies of the virgo giant elliptical galaxy ngc 4949 ( m60 ) by @xcite reveal that the kinematics of planetary nebulae in this object is consistent with the presence of a dark matter halo with @xmath12 for @xmath13 . @xcite presented three - integral axisymmetric models for ngc 4649 and ngc 7097 and concluded that the kinematic data for ngc 4649 only require a small amount of dark matter , however @xcite determine @xmath14 at @xmath15 for ngc 4649 . using gravitational lensing experiments , @xcite find a projected dark matter fraction of @xmath16 for 15 etgs , while @xcite studying sixteen early - type lens galaxies determine the lower limit for dark matter @xmath10 inside the effective radius . the median value for this fraction is @xmath17 with variations from almost 0 to up to @xmath18 . as mentioned above , direct detection of dark matter has not been achieved yet . its presence requires the validity of newtonian gravity . if we were to assume that at these very low acceleration regimes newtonian gravity is not valid or may be slightly modified @xcite then further developments have explained several phenomena without the need of dark matter e.g. spiral galaxies , flat - rotation curves @xcite , projected surface density profiles and observational parameters of the local dwarf spheroidal galaxies @xcite , the relative velocity of wide binaries in the solar neighbourhood @xcite , fully self - consistent equilibrium models for ngc 4649 @xcite and references within among others . in this paper we present a study of luminous and dynamical mass inside the effective radius of etgs considering newtonian dynamics . we search for differences between these masses and assume that any difference is due to dark matter or a non - universal imf or a combination of both . the structure of this study is as follows ; in 2 we present the sample of etgs used in this work , in 3 we discuss the calculation of the stellar and virial masses for the galaxies in the sample , in 4 we discuss the distribution of stellar mass as a function of virial mass , in 5 and 6 we outline the difference between virial and stellar mass as a function of mass and redshift , in 7 we discuss our results in the fundamental plane context and finally in 8 we present the conclusions . we use a sample of etgs from the ninth data release ( dr9 ) of the sloan digital sky survey ( sdss ) @xcite and two subsamples of it , all of them in the @xmath19 and @xmath20 filters . these samples were compiled by and described in great detail in @xcite . here we shall describe briefly the selection criteria used . \1 ) the brightness profile of the galaxy must be well adjusted by a de vaucouleurs profile , in both the @xmath19 and @xmath20 filters ( fracdevg = 1 and fracdevr = 1 according to the sdss nomenclature ) . \2 ) the de vaucouleurs magnitude of the galaxies must be contained in the interval @xmath21 and its equivalent in the g filter . \3 ) the quotient of the semi axes ( b / a ) for the galaxies must be larger than 0.6 in both filters @xmath19 and @xmath20 . \4 ) the galaxies must have a velocity dispersion of @xmath22 60 @xmath23 and a signal - to - noise ratio ( s / n ) @xmath24 10 . the main sample is called total - sdss - sample " . it contains approximately 98000 galaxies , is distributed in a redshift interval @xmath1 and within a magnitude range @xmath25 @xmath26 @xmath27 ( @xmath28 ) . the first subsample is named the - morphological - sample " . the main characteristic of the - morphological - sample is that the selection criteria for the morphological classification are more rigorous than in the total - sdss - sample , due to the fact that @xcite use the morphological classification from the galaxy zoo project ( see @xcite ) . with these added criteria they obtain approximately 27,000 etgs . the last subsample is named the - homogeneous - sdss - sample " . in this case @xcite consider a volume limited sample ( 0.04 @xmath29 0.08 ) with the objective of obtaining a complete sample in the bright end of the magnitude range . in this volume they obtain approximately 19 000 etgs . this subsample covers a magnitude range @xmath25 @xmath30 @xmath27 ( @xmath31 ) and is approximately complete for @xmath32 ( see nigoche - netro et al . 2015 for details ) . the photometry and spectroscopy of the samples of galaxies were corrected due to different biases . below we list these corrections : * seeing correction : the seeing - corrected parameters were obtained from the sdss pipeline . * extinction correction : the extinction correction values were obtained from the sdss pipeline . * k correction : the k correction was obtained from @xcite . * cosmological dimming correction : the cosmological dimming correction was obtained from @xcite . * evolution correction : the evolution correction was obtained from @xcite . * effective radius correction : the effective radius correction to the rest reference frame was obtained from @xcite . * aperture correction to the velocity dispersion : the velocity dispersion inside the radius subtended by the sdss fibre was corrected using the aperture correction from @xcite . we use the stellar and virial masses obtained in @xcite . here we shall describe briefly the procedure used to calculate those masses and some terms that are important for the present work . the total stellar mass was obtained by @xcite considering different stellar population synthesis models , using a universal imf ( salpeter or kroupa ) and different brightness profiles ( de vaucouleurs or srsic ) . the combination of these ingredients results in three mass estimations , as follows : \i ) de vaucouleurs salpeter - imf stellar mass . \ii ) srsic salpeter - imf stellar mass . \iii ) kroupa - imf stellar mass . according to @xcite , within a sphere of radius equal to @xmath0 , 42% of the total stellar mass is contained . the stellar masses described before assume a universal imf . however , some papers in the astronomical literature claim that the imf is not universal but rather it depends on the stellar mass @xcite . we do not correct the stellar mass for the behaviour of the imf as a function of mass because there is no accurate equation describing this effect . since our results have to take into account this effect , we will discussed them in the subsequent sections . the total virial mass was obtained by @xcite using an equation from @xcite . this method assumes newtonian mechanics and virial equilibrium for the galaxies in question . the equation is as follows : @xmath33 where the variables @xmath34 , @xmath0 and , @xmath35 represent respectively the total virial mass , the effective radius and the velocity dispersion inside @xmath0 . @xmath36 stands for the gravitational constant and @xmath37 is a scale factor . for the de vaucouleurs profile case @xmath38 @xcite . the amount of mass within an effective radius corresponds to 0.42 times the value calculated from equation ( 1 ) this mass may or may not be luminous . the errors calculated for the different parameters reported in this paper are obtained using the rules of error propagation and considering possible systematics on the photometric and spectroscopic parameters as discussed in detail by @xcite . in the following sections , and taking into consideration only the region internal to @xmath0 , we will carry out an analysis of the behaviour of the virial vs. stellar mass . in @xcite we have made a complete analysis of the distribution of the stellar mass with respect to the virial mass for etgs samples . in this section we present an extract of that analysis only with the relevant information for the goals of the present work . figure 1 is the most important part of the extract because it shows the comparison of viral and stellar mass for each galaxy in our samples . in this figure , column 1 represents the total sample , column 2 the morphologic sample and column 3 the homogeneous sample . the rows correspond to different profiles and imfs , being the first one associated with the de vaucouleurs salpeter - imf stellar mass , the second with the sersic salpeter - imf stellar mass and the third with the kroupa - imf stellar mass . the solid line is the one - to - one line . @xcite discuss different procedures to analyse the distribution of masses shown in figure 1 considering that the mentioned distribution may depend on observational biases , on physical properties of the galaxies , and on arbitrary cuts performed in the observed samples ( see also * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? . those procedures may be helpful in investigating whether there is dark matter inside etgs . particularly they found that the application of the weighted bisector fit ( @xmath39 fit ) to the mean value of the distribution at quasi - constant mass , results in a reduction of the possible biases which may creep in the process ( for details see section 7.3 of appendix a from * ? ? ? * ) . this method is only a first approximation to the study of dark matter inside etgs because the distribution of masses seen in figure 1 have a high intrinsic dispersion and the physical causes of the intrinsic dispersion are at present yet unknown . refinements to this method must be sought for in the physical causes of the intrinsic dispersion seen in the mass distributions . in the previous section we have mentioned that there are different procedures to analyse the properties of our etgs samples . in particular , we can study the difference between the virial and stellar mass using the mass distribution of the samples and the @xmath39 fit . to obtain the parameters of the linear regression ( @xmath39 fit ) for the different etgs samples we use the nigoche - netro et al . ( 2015 ) procedure as follows : * we calculate the mean value of the logarithm of stellar mass at quasi - constant logarithm of virial mass . * we calculate the mean value of the logarithm of virial mass at quasi - constant logarithm of stellar mass . * we perform a linear regression ( @xmath39 fit ) to the previously mentioned mean values . the term quasi - constant mass in this context means mass intervals in the logarithm of width equal to 0.1 . in figure 2 we show a mosaic of the behaviour of the stellar mass with respect to the virial mass for the etgs total ( column 1 ) , morphologic ( column 2 ) and homogeneous ( column 3 ) samples . the first row corresponds to the de vaucouleurs salpeter - imf stellar mass , the second one corresponds to the srsic salpeter - imf stellar mass and the third one corresponds to the kroupa - imf stellar mass . each graph shows the mean values of the luminous mass distribution at quasi - constant virial mass ( black dots ) , the mean values of the virial mass distribution at quasi - constant stellar mass ( blue squares ) and the @xmath39 fit ( dashed line ) to both point distributions . the solid line is the one - to - one line . in table 1 we show the results of the @xmath39 fit to the different samples of etgs . the difference between virial and stellar mass shown has been obtained considering the slope and the zero point of the fit for each sample . for each value of virial mass we calculated the stellar mass and the average , maximum and minimum differences among these masses . the mean error in the average , maximum and minimum values of the difference between log@xmath40 and log@xmath41 is approximately 0.12 . from table 1 we find the following : * the average difference between log@xmath40 and log@xmath41 considering the samples with salpeter - imf stellar mass is 0.380 , whereas if we consider kroupa - imf stellar mass profiles the average difference is 0.281 . this seems to indicate that the estimated difference between masses is affected by the imf used in the calculation of luminous mass . however the mentioned difference seems to be due to the zero point because the slopes of the samples are similar . * considering only those samples where the masses were obtained using de vaucouleurs profiles the average difference is 0.376 , whereas if we consider srsic profiles the average difference is 0.384 . this seems to indicate that the estimated difference between masses is not affected by the profile used in the calculation of luminous and virial mass . this result is confirmed by the slopes of the samples which are similar . * if we consider only samples in large intervals of redshift ( total and morphological samples ) the average difference is 0.348 , whereas if we consider only the sample restricted in redshift ( homogeneous sample ) the average difference is 0.344 . this seems to indicate that the average difference between masses is similar when we move from a wide to a narrow redshift interval . however the slopes of the samples seems to refute this result . the previous results are in agreement with the results found by @xcite . however , the comparison of the mean values of the difference in masses could mask the real behaviour of the samples because the midpoints of two straight lines could be similar even if the slopes of those straight lines are different . so it is necessary to compare the masses in a different way considering that the difference in mass could depend on other variables such as mass and/or redshift . from table 1 we can see that , if we consider all the samples , the average difference between the maximum and minimum values for log(@xmath42 - log@xmath43 ) is 0.401 . this relatively large difference as well as the equations corresponding to table 1 suggest that there is a dependence of the log(@xmath42 - log@xmath43 ) on virial and stellar mass . this behaviour can be easily seen in figure 3 where , for all samples , the difference between virial and stellar mass depends on virial mass . from figure 3 , it is interesting to note that the behaviour of the samples with different imf - stellar mass is similar , although with different zero points . a similar behaviour also occurs when we consider different profiles . also from figure 3 , it is important to note that the slope for the restricted in redshift sample ( homogeneous sample ) is steeper than the slope for the samples in large intervals of redshift . that is to say , the difference between virial and stellar mass seems to depend on redshift . the behaviour of the differences between virial and stellar mass as a function of mass and redshift requires a deeper analysis which we will address in the following section . in the previous section we have found that the difference between virial and stellar mass depends on mass and seems to depend on redshift . to investigate these dependences in a deeper way we can analyse the difference in masses considering quasi - constant mass ( virial and stellar ) and quasi - constant redshift . the term quasi - constant mass means mass interval in the logarithm of width equal to 0.1 . the term quasi - constant redshift means redshift interval equal to 0.01 . in figure 4 we can see the behaviour of the stellar mass as a function of virial mass considering quasi - constant redshift for the kroupa - imf stellar mass sample . each colour and symbol represents quasi - constant redshift . the redshift value goes from approximately zero ( lower - left part of the graph ) to approximately 0.3 ( upper - right part of the graph ) . in this figure we can see that for the same value of the virial mass , the stellar mass grows with redshift , that is to say , the difference between virial and stellar mass diminishes with redshfit . we can also see that for the high - redshift and high - mass regime the dispersion of the distribution is lower than in the low - mass regime of the samples . from this figure we can see that the redshift plays an important role in the intrinsic dispersion seen in figure 1 . in figure 5 we show the difference between virial and stellar mass as a function of virial mass considering quasi - constant redshift for the kroupa - imf stellar mass sample . the redshift value goes from approximately zero ( upper - left part of the graph ) to approximately 0.3 ( lower - right part of the graph ) . in this graph we can see that for the same value of the virial mass the difference between virial and stellar mass diminishes with redshift . we can also see that the more massive galaxies have a greater difference between virial and stellar mass and that for the high - redshfit and high - mass regime the dispersion of the distribution is lower than in the low - mass regime of the samples . the difference between virial and stellar mass at different redshift is directly related with the dispersion seen in figures 1 and 4 . in figure 6 we show the behaviour of the stellar mass as function of redshift considering quasi - constant virial mass for the kroupa - imf stellar mass sample . the logarithm of virial mass value goes from approximately 10 ( lower - left part of the graph ) to approximately 12 ( upper - right part of the graph ) . in this graph we can see that for a constant value of virial mass the stellar mass increases as function of redshift . we can also see that for the high - mass regime the behaviour of the stellar mass as a function of redshift is less steep than in the low - mass regime of the samples . the slope of the stellar mass as function of redshift at quasi - constant virial mass is related with the intrinsic dispersion seen in figures 1 and 4 . from the previous results we can conclude that the redshift plays an important role in the intrinsic dispersion of the distribution of log(@xmath42 vs. log@xmath43 ) . we also can conclude that the difference between virial and stellar mass , in the redshift range @xmath1 and in the dynamical mass range @xmath44 , depends on mass and redshift . the difference between dynamical and stellar mass increases as a function of dynamical mass and decreases as a function of redshift . this last result is in agreement with recent works from the literature where it is shown that the amount of dark matter could depend on mass @xcite and redshift @xcite . if we convert the data shown in figure 5 to percentages , we find that the difference between masses goes from almost zero to approximately 70% of the virial mass . this difference could be due to the dark matter and/or a non universal imf . therefore , the amount of dark matter , in the redshift range @xmath1 and in the dynamical mass range @xmath44 , goes from almost zero to 70% of the virial mass depending on mass and redshift and on the impact of the imf on the stellar mass estimation . it is important to note that we have found similar results for the de vaucouleurs salpeter - imf sample and srsic salpeter - imf sample . the results previously described can be analysed in the fundamental plane ( fp ) context . during the last 30 years a lot of scientific papers about the fp have been published @xcite . the fp is a relation among the variables ; effective radii ( @xmath45 ) , the effective mean surface brightness ( @xmath46 ) and the central velocity dispersion ( log @xmath47 ) , as follows : @xmath48 where @xmath49 , @xmath50 and @xmath51 represent scale factors . this relation seems to be due to the virial equilibrium of etgs , however the theoretical and observational results do not agree . the difference between the theoretical and observational results is known as the tilt of the fp . there are different explanations for the fp tilt , for example , it could be due to : the increase of m / l with l @xcite , the variation of the fp parameters with redshift @xcite , the variation in the homology of etgs ( non - constant k in the virial relation -see equation 1- ) @xcite , the variation of the shape of the light profile and the content and concentration of dark matter relative to luminous matter @xcite among others . some of these works have found that the tilt could be due to a combination of several of the mentioned effects @xcite which seems to be the most plausible explanation . given that the fp relates dynamic variables and stellar formation processes as log(@xmath42 vs. log@xmath43 ) does , we can extrapolate our findings to the fp . in this sense , the mass and redshift dependence of log(@xmath42 - log@xmath43 ) found in this work go in the same direction as the increase of the content of dark matter relative to luminous matter along fp and the variation of the fp parameters with redshift . that is to say , our findings go in the same direction of the ` hybrid ' explanation to the tilt of the fp . however we have to take into account that in this work we consider that the dark matter follows the light and that the scale factor k in the virial relation ( see equation 1 ) depends only on the light profile which , according to some authors @xcite is not necessarily appropriate . in a forthcoming paper we will analyse these variables and their relation with the difference between log(@xmath42 and log@xmath43 ) . the analysis of the distribution of stellar mass with respect to virial mass on several samples of etgs from the sdss dr9 in the redshift range @xmath1 and in the dynamical mass range @xmath44 has yielded the following results : * a significant part of the intrinsic dispersion of the distribution of log(@xmath42 vs. log@xmath43 ) is due to redshift ( see fig . the difference between dynamical and stellar mass depends on mass and redshift . * the difference between dynamical and stellar mass increases as a function of dynamical mass and decreases as a function of redshift . * the difference between dynamical and stellar mass goes from almost zero to approximately 70% of the virial mass depending on mass and redshift ( see figure 5 ) . this difference is due to dark matter or a non - universal imf or a combination of both . * the amount of dark matter inside etgs would be equal to or less than the difference between dynamical and stellar mass depending on the impact of the imf on the stellar mass estimation . the previous results have been analysed in the fp context and we have found that they go in the same direction as some fp results found in the literature in the sense that they could be interpreted as an increase of dark matter along the fp and a dependence of the fp on redshift . however in this work we have considered that the dark matter follows the same density profile as the stellar component and that the scale factor k in the virial relation ( see equation 1 ) depends only on the light profile which , according to some authors @xcite , is not appropriate for massive and compact galaxies respectively . in a forthcoming paper we will analyse these variables and their possible relation with the log(@xmath42-log@xmath43 ) difference . to the memory of mrs . eutiquia netro castillo , an extraordinary woman . we thank and acknowledge the comments made by an anonymous referee , they improved greatly the presentation of this paper . we also thank to instituto de astronoma y meteorologa ( udg , mxico ) and instituto de astronoma ( unam , mxico ) for all the facilities provided for the realisation of this project . a. nigoche - netro and g. ramos - larios acknowledge support from conacyt and prodep ( mxico ) . patricio lagos is supported by a postdoctoral grant sfrh / bpd/72308/2010 , funded by fct ( portugal ) and fundao para a cincia e a tecnologia ( fct ) under project fcomp-01 - 0124-feder-029170 ( reference fct ptdc / fis - ast/3214/2012 ) , funded by the feder program . a. ruelas - mayorga thanks direccin general de asuntos del personal acadmico , dgapa at unam for financial support under project number papiit in103813 . a. m. hidalgo - gmez thanks instituto politcnico nacional sip20161416 for financial support under project number sip20161416 . cappellari , m. , bacon , r. , bureau , r. , damen , m.c . , davies , r.l . , de zeeuw , p.t . , emsellem , e. , falcn - barroso , j. , krajnovi , d. , kuntscher , h. , mcdermid , r.m . , peletier , r.f . , sarzi , m. , van den bosch , r.c.e . & van de ven , g. 2006 , mnras , 366 , 1126 donofrio , m. , fasano , g. , moretti , a. , marziani , p. , bindoni , d. , j. fritz , j. , varela , j. , bettoni , d. , cava , a. , poggianti , b. gullieuszik , m. , kjrgaard , p. , moles , m. , vulcani , b. , omizzolo , a. , w.j . couch , w. j. & dressler , a 2013 , mnras , 435 , 45 peralta de arriba , l. , balcells , m. , trujillo , i. , falcon - barroso , j. , tapia , t. , cardiel , n. , gallego , j. , guzman , r. , hempel , a. , martn - navarro , i. , perez - gonzalez , p. g. & sanchez - blaquez , p. mnras , 453 , 704

we study the behaviour of the dynamical and stellar mass inside the effective radius ( @xmath0 ) of early - type galaxies ( etgs ) . we use several samples of etgs -ranging from 19 000 to 98 000 objects- from the ninth data release of the sloan digital sky survey . we consider newtonian dynamics , different light profiles and different initial mass functions ( imf ) to calculate the dynamical and stellar mass . we assume that any difference between these two masses is due to dark matter and/or a non universal imf . the main results for galaxies in the redshift range @xmath1 and in the dynamical mass range 9.5 @xmath2 log(m ) @xmath2 12.5 are : i ) a significant part of the intrinsic dispersion of the distribution of dynamical vs. stellar mass is due to redshift . ii ) the difference between dynamical and stellar mass increases as a function of dynamical mass and decreases as a function of redshift . iii ) the difference between dynamical and stellar mass goes from approximately 0% to 70% of the dynamical mass depending on mass and redshift . iv ) these differences could be due to dark matter or a non universal imf or a combination of both . v ) the amount of dark matter inside etgs would be equal to or less than the difference between dynamical and stellar mass depending on the impact of the imf on the stellar mass estimation . vi ) the previous results go in the same direction of some results of the fundamental plane ( fp ) found in the literature in the sense that they could be interpreted as an increase of dark matter along the fp and a dependence of the fp on redshift . [ firstpage ] galaxies : fundamental parameters , photometry , distances and redshifts . cosmology : dark matter .

the ua5 collaboration noticed for the first time that charged - particle multiplicity distributions measured in high energy proton-(anti)proton collisions in limited intervals of pseudo - rapidity have the negative binomial form @xcite . in the present paper this observation will be verified for the collisions at @xmath0 = 0.9 and 2.36 tev performed by ua5 @xcite and alice collaborations @xcite . only non - single diffractive ( nsd ) events will be considered because such a case was analyzed with this respect by both collaborations . in fact , the author investigated alice inelastic events also ( including the case of @xmath0 = 7 tev @xcite ) , but all fits were entirely unacceptable . the negative binomial distribution ( nbd ) is defined as @xmath1 where @xmath2 , @xmath3 and @xmath4 is a positive real number . in the application to high energy physics @xmath5 has the meaning of the number of charged particles detected in an event . the expected value @xmath6 and variance @xmath7 and the variance @xmath8 . [ przyp1 ] ] are expressed as : @xmath9 in this analysis the hypothesis that the charged - particle multiplicities measured in high energy @xmath10 collisions are distributed according to the nbd is verified with the use of the maximum likelihood method ( ml ) and the likelihood ratio test . more details of this approach can be found in refs . . there are two crucial reasons for this approach : the fitted quantity is a probability distribution function ( p.d.f . ) , so the most natural way is to use the ml method , where the likelihood function is constructed directly from the tested p.d.f .. but more important is that because of wilks s theorem ( see appendix b ) one can easily define a statistic , the distribution of which converges to a @xmath11 distribution as the number of measurements goes to infinity . thus for the large sample the goodness - of - fit can be expressed as a @xmath12-value computed with the corresponding @xmath11 distribution . the most commonly used method , the least - squares method ( ls ) ( called also @xmath11 minimization ) , has the disadvantage of providing only the qualitative measure of the significance of the fit , in general . only if observables are represented by gaussian random variables with known variances , the conclusion about the goodness - of - fit equivalent to that mentioned in the point ( i ) can be derived @xcite . it is worth noting that the ml method with binned data and poisson fluctuations within a bin was already applied to fitting multiplicity distributions to the nbd but at much lower energies ( e-802 collaboration @xcite ) . the number of charged particles @xmath13 is assumed to be a random variable with the p.d.f . given by eq . ( [ nbdist ] ) . each event is treated as an independent observation of @xmath13 and a set of a given class of events is a sample . for @xmath14 events in the class there are @xmath14 measurements of @xmath13 , say @xmath15 . some of these measurements can be equal , _ i.e. _ @xmath16 for @xmath17 can happen . the whole population consists of all possible events with the measurements of 0 , 1 , 2 , ... charged particles and by definition is infinite . ] for the class of events one can defined the likelihood function @xmath18 where @xmath19 is the nbd , eq . ( [ nbdist ] ) . the values @xmath20 and @xmath21 for which @xmath22 has its maximum are the maximum likelihood estimates of parameters @xmath12 and @xmath4 . this is equivalent to the maximization of the log - likelihood function @xmath23 thus the values @xmath20 and @xmath21 are the solutions of the equations : @xmath24 @xmath25 it can be proven that one of the necessary conditions for the existence of the maximum is ( see appendix a for details ) : @xmath26 _ i.e. _ the distribution average has to be equal to the experimental average . let divide the sample defined in sect . [ likmeth ] into @xmath27 bins characterized by @xmath28 - the number of measured charged particles for @xmath17 and @xmath29 . [ przyp3 ] ] and @xmath30 - the number of entries in the @xmath31th bin , @xmath32 ( details of the theoretical framework of this section can be found in refs . ) . then the expectation value of the number of events in the @xmath31th bin can be written as @xmath33 where @xmath34 is the expected number of all events in the sample , @xmath35 . this is because one can treat the number of events in the sample @xmath14 also as a random variable with its own distribution - poisson one . generally , the whole histogram can be treated as one measurement of @xmath27-dimensional random vector @xmath36 which has a multinomial distribution , so the joint p.d.f . for the measurement of @xmath14 and @xmath37 can be converted to the form @xcite : @xmath38 since now @xmath39 is the p.d.f . for one measurement , @xmath40 is also the likelihood function @xmath41 with the use of eq . ( [ neventi ] ) the corresponding likelihood function can be written as @xmath42 then the likelihood ratio is defined as @xmath43 where @xmath44 , @xmath20 and @xmath21 are the ml estimates of @xmath34 , @xmath12 and @xmath4 with the likelihood function given by eq . ( [ liksubset ] ) and @xmath45 , @xmath46 are the ml estimates of @xmath47 treated as free parameters . note that since the denominator in eq . ( [ likeliratio ] ) does not depend on parameters , the log - ratio defined as @xmath48 where @xmath47 are expressed by eq . ( [ neventi ] ) , can be used to find the ml estimates of @xmath34 , @xmath12 and @xmath4 . further , the statistic given by @xmath49 approaches the @xmath11 distribution asymptotically , _ i.e. _ as the number of measurements , here the number of events @xmath14 , goes to infinity ( the consequence of the wilks s theorem , see appendix b ) . the values @xmath50 are the estimates of @xmath47 given by @xmath51 and if one assumes that @xmath34 does not depend on @xmath12 and @xmath4 then @xmath52 . for such a case @xmath53 and eq . ( [ poissonchi ] ) becomes @xmath54 also then one can just put @xmath55 and eq . ( [ logratio ] ) can be rewritten as @xmath56 with the term depending on @xmath12 and @xmath4 the same as eq . ( [ loglikfubi ] ) and @xmath57 . therefore conclusions of appendix a holds here , _ i.e. _ the necessary conditions for the existence of the maximum is @xmath58 , eq . ( [ dloglikfk ] ) is the equation which determines @xmath21 and @xmath20 is obtained with the help of eq . ( [ oneoverp ] ) . note that the maximum of @xmath59 is the minimum of @xmath60 , so from eqs . ( [ multinomchi ] ) and ( [ logratfreq ] ) one arrives at @xmath61 in fact , the method just described assumes that the sum of @xmath62 over all bins equals 1 . but only the infinite sum of @xmath63 is 1 . however the measured values of @xmath64 are big enough ( of the order of 20 at least for all considered cases ) so in the vicinity of @xmath20 and @xmath21 the sum of @xmath63 up to @xmath65 equals 1 approximately ( see the seventh column in table [ table1 ] ) . nevertheless , to calculate @xmath66 , eq . ( [ finalchi ] ) , @xmath67 were normalized appropriately and these results are listed in the fifth column of tables [ table1]-[table3 ] . another way to treat this problem is to create arbitrary the @xmath68st bin for all possible @xmath69 and with @xmath70 . bins with @xmath71 ( @xmath72 equivalently ) do not contribute in eq . ( [ finalchi ] ) ( see ref . ) . in practice , it means that @xmath66 would be calculated also from eq . ( [ finalchi ] ) but without the normalization . it has turned out that that way leads to much greater values of @xmath66 . the method described in sections [ likmeth ] and [ liktest ] requires that all bins in a given data set have the width equal to 1 , so as the experimental probability @xmath73 to measure a signal in the @xmath31th bin was equivalent to the probability of the measurement of @xmath74 charged particles ( the first bin is the bin of 0 charged particles detected ) . this is fulfilled for all bins of the considered data sets except the ends of their tails . in these tails the measured values of @xmath73 have been uniformly distributed over the bin range so as the method could be applied directly . if the bin width is not significantly greater than 1 then this approximation should not change substantially the value of @xmath66 given by eq . ( [ finalchi ] ) because in the most cases @xmath73 at tails are two orders smaller than in the main part of distributions . also errors in tails are bigger , in the range @xmath75 , increasing with @xmath31 . since the test statistic @xmath76 has a @xmath77 distribution approximately in the large sample limit , it can be used as a test of the goodness - of - fit . the result of the test is given by the so - called @xmath12-value which is the probability of obtaining the value of the statistic , eq . ( [ poissonchi ] ) , equal to or greater then the value just obtained by the ml method for the present data set , when repeating the whole experiment many times : @xmath78 where @xmath79 is the @xmath77 p.d.f . and @xmath80 the number of degrees of freedom , @xmath81 here . @cccccccccc@ & & & & @xmath11/@xmath80 & & & + experiment & n & @xmath21 & @xmath20 & @xmath11 & p - value & @xmath82 & quadrature & sum & statistical + @xmath0 & & & & ( @xmath80 ) & [ % ] & & sum & & only + ua5 & 8550.0 & 1.5574 & 0.3012 & 0.339 & 99.97 & 0.99996 & 0.375 & na & na + @xmath83 tev & ( 80 % _ eff_. ) & @xmath84 & @xmath85 & 10.16 & & & & & + @xmath86 0.5 & & & & ( 30 ) & & & & & + & & & & & + ua5 & 10000.0 & 1.5574 & 0.3012 & 0.396 & 99.87 & 0.99996 & 0.375 & na & na + @xmath83 tev & ( 70 % _ eff_. ) & @xmath87 & @xmath88 & 11.88 & & & & & + @xmath86 0.5 & & & & ( 30 ) & & & & & + & & & & & + alice & 149663.16 & 1.3764 & 0.2767 & 14.155 & 0 & 0.99960 & 1.116 & 0.576 & 3.089 + @xmath83 tev & & @xmath89 & @xmath90 & 353.88 & & & & & + @xmath86 0.5 & & & & ( 25 ) & & & & & + & & & & & + alice & 128476.45 & 1.4316 & 0.1625 & 37.761 & 0 & 0.99865 & 1.886 & 1.034 & 11.51 + @xmath83 tev & & @xmath91 & @xmath92 & 1548.21 & & & & & + @xmath86 1.0 & & & & ( 41 ) & & & & & + & & & & & + alice & 60142.77 & 1.4955 & 0.1332 & 22.051 & 0 & 0.99876 & 2.993 & 1.671 & 15.31 + @xmath83 tev & & @xmath93 & @xmath94 & 1168.69 & & & & & + @xmath86 1.3 & & & & ( 53 ) & & & & & + & & & & & + ua5 & 8550.0 & 1.7987 & 0.1385 & 0.812 & 87.81 & 0.99991 & 0.487 & na & na + @xmath83 tev & ( 80 % _ eff_. ) & @xmath95 & @xmath96 & 60.12 & & & & & + @xmath86 1.5 & & & & ( 74 ) & & & & & + & & & & & + ua5 & 10000.0 & 1.7987 & 0.1385 & 0.950 & 59.99 & 0.99991 & 0.487 & na & na + @xmath83 tev & ( 70 % _ eff_. ) & @xmath97 & @xmath98 & 70.31 & & & & & + @xmath86 1.5 & & & & ( 74 ) & & & & & + & & & & & + alice & 38970.79 & 1.1778 & 0.2084 & 6.266 & 0 & 0.99930 & 0.888 & 0.501 & 3.592 + @xmath99 tev & & @xmath100 & @xmath101 & 194.26 & & & & & + @xmath86 0.5 & & & & ( 31 ) & & & & & + & & & & & + alice & 37883.99 & 1.2139 & 0.1180 & 17.416 & 0 & 0.99726 & 2.209 & 1.312 & 17.73 + @xmath99 tev & & @xmath102 & @xmath103 & 853.37 & & & & & + @xmath86 1.0 & & & & ( 49 ) & & & & & + & & & & & + alice & 22189.40 & 1.2123 & 0.0927 & 15.561 & 0 & 0.99644 & 4.0557 & 2.4537 & 34.40 + @xmath99 tev & & @xmath104 & @xmath103 & 949.22 & & & & & + @xmath86 1.3 & & & & ( 61 ) & & & & & + & & & & & + [ table1 ] @cccccccccc@ & & & & @xmath11/@xmath80 & & + experiment & n & @xmath21 & @xmath20 & @xmath11 & p - value & quadrature & sum & statistical & @xmath105 + @xmath0 & & & & ( @xmath80 ) & [ % ] & sum & & only & + ua5 & 8550.0 & 1.5574 & 0.3012 & 0.211 & 99.998 & 0.072 & na & na & 0.203 + @xmath83 tev & ( 80 % _ eff_. ) & @xmath84 & @xmath85 & 4.859 & & & & & + @xmath86 0.5 & & & & ( 23 ) & & & & & + & & & & & + ua5 & 10000.0 & 1.5574 & 0.3012 & 0.247 & 99.991 & 0.072 & na & na & 0.237 + @xmath83 tev & ( 70 % _ eff_. ) & @xmath87 & @xmath88 & 5.683 & & & & & + @xmath86 0.5 & & & & ( 23 ) & & & & & + & & & & & + alice & 149663.16 & 1.3764 & 0.2767 & 14.498 & 0 & 0.728 & 0.381 & 2.458 & 15.107 + @xmath83 tev & & @xmath89 & @xmath90 & 347.95 & & & & & + @xmath86 0.5 & & & & ( 24 ) & & & & & + & & & & & + alice & 128476.45 & 1.4316 & 0.1625 & 36.855 & 0 & 1.718 & 0.948 & 11.010 & 38.017 + @xmath83 tev & & @xmath91 & @xmath92 & 1547.91 & & & & & + @xmath86 1.0 & & & & ( 42 ) & & & & & + & & & & & + alice & 60142.77 & 1.4955 & 0.1332 & 24.323 & 0 & 2.213 & 1.276 & 15.201 & 25.771 + @xmath83 tev & & @xmath93 & @xmath94 & 1167.51 & & & & & + @xmath86 1.3 & & & & ( 48 ) & & & & & + & & & & & + ua5 & 8550.0 & 1.7987 & 0.1385 & 1.099 & 28.94 & 0.362 & na & na & 1.14 + @xmath83 tev & ( 80 % _ eff_. ) & @xmath95 & @xmath96 & 57.16 & & & & & + @xmath86 1.5 & & & & ( 52 ) & & & & & + & & & & & + ua5 & 10000.0 & 1.7987 & 0.1385 & 1.286 & 8.06 & 0.362 & na & na & 1.33 + @xmath83 tev & ( 70 % _ eff_. ) & @xmath97 & @xmath98 & 66.85 & & & & & + @xmath86 1.5 & & & & ( 52 ) & & & & & + & & & & & + alice & 38970.79 & 1.1778 & 0.2084 & 7.030 & 0 & 0.761 & 0.428 & 3.805 & 7.465 + @xmath99 tev & & @xmath100 & @xmath101 & 189.82 & & & & & + @xmath86 0.5 & & & & ( 27 ) & & & & & + & & & & & + alice & 37883.99 & 1.2139 & 0.1180 & 18.535 & 0 & 2.288 & 1.362 & 18.802 & 20.282 + @xmath99 tev & & @xmath102 & @xmath103 & 852.59 & & & & & + @xmath86 1.0 & & & & ( 46 ) & & & & & + & & & & & + alice & 22189.40 & 1.2123 & 0.0927 & 18.233 & 0 & 4.245 & 2.599 & 39.647 & 19.980 + @xmath99 tev & & @xmath104 & @xmath103 & 948.11 & & & & & + @xmath86 1.3 & & & & ( 52 ) & & & & & + & & & & & + [ table2 ] the results of the analysis are presented in table [ table1 ] . note that for ua5 cases two possibilities of the corrected number of events are listed . this is because only the measured number of events , @xmath106 , is given in ref . . however , the fits have been done to the corrected distributions , so also the corrected number of events should be put into eq . ( [ finalchi ] ) . the number have been estimated in the following way : in fig.4 of ref . the mean of the observed distribution versus the corrected ( true ) number of particles is plotted , the curve is a straight line roughly with the tangent equal to @xmath107 , so one can guess that the efficiency is also about @xmath108 . just to check how results are stable with respect to a change in the number of events , the case with @xmath109 efficiency has been also calculated . as one can see , for all alice cases the hypothesis in question should be rejected , whereas for the listed ua5 cases should be accepted . but it was claimed that charged - particle multiplicities measured in the limited pseudo - rapidity windows by the alice collaboration are distributed according to the nbd @xcite . however that conclusion was the result of the @xmath11 minimization ( the ls method ) . therefore it seems to be reasonable to check what are the values of the ls @xmath11 function at the ml estimators listed in the third and fourth columns of table [ table1 ] . for the sample described in sect . [ liktest ] one can define the ls @xmath77 function as : @xmath110 where @xmath111 is the uncertainty of the @xmath31th measurement . here this function * is not minimized * with respect to @xmath12 and @xmath4 as in the ls method but is calculated at ml estimates of @xmath12 and @xmath4 , _ i.e. _ at @xmath20 and @xmath21 . one can see from the eight and ninth columns of table [ table1 ] that @xmath112/@xmath80 values are significant for the alice narrowest pseudo - rapidity windows , what agrees with the results of ref . . since the determination of @xmath21 and @xmath20 has been done for the distributions modified in their tails , as it has been just explained , one should check what values of @xmath11 and @xmath112 are at @xmath21 and @xmath20 for the original data sets . it means that if the @xmath31th bin width is greater than 1 , instead of @xmath67 in eq . ( [ finalchi ] ) the appropriate sum @xmath113 over @xmath114 is taken . the results of the check are presented in table [ table2 ] . qualitatively the results are the same as in table [ table1 ] , only slight differences in numbers can be noticed except the ua5 cases ( for @xmath86 0.5 @xmath11 has decreased more than 2 times , but the change is in the good direction ) . this is because the maximal width of a tail bin is 2 for all alice cases , but is 8 and 17 for ua5 windows @xmath86 0.5 and @xmath86 1.5 , respectively . of course , the assumption of the uniform distribution inside a wider bin causes greater discrepancies . nevertheless , the results of the test for both ua5 cases are positive even if ( @xmath21 , @xmath20 ) is not the maximum of the exact likelihood function ( in fact , values of @xmath21 are the same as those obtained by ua5 collaboration in ref . ) . this is guaranteed by the wilks s theorem ( see appendix b ) , which allows for the test of a single point in the parameter space . then the tested point might not be the best estimate of the true value but the hypothesis in question becomes the hypothesis only about a particular distribution ( a _ simple _ hypothesis ) . this is also the reason why @xmath115 in table [ table2 ] . in terms of rigorous statistics single points are tested in there . in all alice cases @xmath11 values listed in the fifth column of table [ table2 ] are only slightly smaller than corresponding ones from table [ table1 ] . for @xmath86 0.5 the decrease is about @xmath116 , for other cases is less than @xmath117 . also @xmath11/@xmath80 values are much greater than 1 . therefore it is reasonable to recognize @xmath21 and @xmath20 determined for modified data sets as a good approximations of the ml estimators . thus the hypothesis about the nbd should be rejected on the basis of obtained values of @xmath11/@xmath80 and @xmath12-values . one can also compare @xmath11/@xmath80 with @xmath112/@xmath80 calculated for the original data sets and the same @xmath21 and @xmath20 . the results are listed in four last columns of table [ table2 ] for various treatment of errors . note that for ua5 conclusions from both statistics are exactly the same . in the alice both cases of the window @xmath86 0.5 , @xmath112/@xmath118 is acceptable for errors expressed as the quadrature sum of statistical and systematical components and is smaller than the corresponding values in table [ table1 ] . in other alice cases @xmath112/@xmath80 is substantially greater than 1 for the same treatment of errors . this is in the full agreement with the results of ref . . one can also check what @xmath112/@xmath80 is if only statistical errors are taken into account . the results are listed in the next to last column of table [ table2 ] . for all alice cases the values are much greater than 1 . this means that acceptable @xmath112/@xmath80 was obtained only because of significant systematic errors of alice measurements . significant is subjective , here means significant with respect to the sample size , not to the value of @xmath73 . the crucial question is now why the conclusions from @xmath11 and @xmath112 test statistics are the same for ua5 data but entirely opposite for alice measurements ? the main difference between both statistics is that @xmath11 depends explicitly on the number of events but @xmath112 does not . on opposite , @xmath11 does not depend on the actual errors but @xmath112 does . in fact , @xmath11 statistic implicitly assumes errors of the type @xmath119 , what is the straightforward result of the form of the likelihood function , eqs . ( [ jointpdf ] ) and ( [ ljointpdf ] ) , namely the product of poisson distributions . this is revealed when one compare @xmath11/@xmath80 and @xmath112/@xmath80 with errors @xmath120 ( the fifth and last column in table [ table2 ] ) . the values are practically the same . @cccccccccc@ & & & & @xmath11/@xmath80 & & + experiment & n & @xmath21 & @xmath20 & @xmath11 & p - value & quadrature & sum & statistical & @xmath105 + @xmath0 & & & & ( @xmath80 ) & [ % ] & sum & & only & + ua5 & 8550.0 & 1.5574 & 0.3012 & 0.211 & 99.998 & 0.072 & na & na & 0.203 + @xmath83 tev & ( 80 % _ eff_. ) & @xmath84 & @xmath85 & 4.859 & & & & & + @xmath86 0.5 & & & & ( 23 ) & & & & & + & & & & & + alice & 8550.0 & 1.3764 & 0.2767 & 0.828 & 70.37 & 0.728 & 0.381 & 2.458 & 0.863 + @xmath83 tev & & @xmath121 & @xmath122 & 19.88 & & & & & + @xmath86 0.5 & & & & ( 24 ) & & & & & + & & & & & + alice & 8550.0 & 1.4316 & 0.1625 & 2.453 & 5 @xmath123 & 1.718 & 0.948 & 11.010 & 2.530 + @xmath83 tev & & @xmath124 & @xmath125 & 103.01 & & & & & + @xmath86 1.0 & & & & ( 42 ) & & & & & + & & & & & + alice & 8550.0 & 1.4955 & 0.1332 & 3.458 & 7 @xmath126 & 2.213 & 1.276 & 15.201 & 3.664 + @xmath83 tev & & @xmath127 & @xmath128 & 165.97 & & & & & + @xmath86 1.3 & & & & ( 48 ) & & & & & + & & & & & + ua5 & 8550.0 & 1.7987 & 0.1385 & 1.099 & 28.94 & 0.362 & na & na & 1.14 + @xmath83 tev & ( 80 % _ eff_. ) & @xmath95 & @xmath96 & 57.16 & & & & & + @xmath86 1.5 & & & & ( 52 ) & & & & & + & & & & & + [ table3 ] to find out what is the reason for the above - mentioned disagreement the calculations of table [ table2 ] have been repeated for alice measurements at @xmath83 tev but with the arbitrary assumption that all cases have the same number of events as ua5 ones . the results are listed in table [ table3 ] . one can see that now there is full agreement between @xmath11 and @xmath112 test statistic results for all alice cases . this means that the accuracy with which experimental distributions approximate the nbd has not increased in alice data even though the sample sizes are one order greater . but the accuracy should increase with the sample size because if the hypothesis is true the postulated form of distribution is exact for the whole population . so with the growing number of events , the experimental distribution should be closer to the postulated one . this is also seen in the form of @xmath66 , eq . ( [ finalchi ] ) , where the linear dependence on @xmath14 is explicit . to keep @xmath66 at least constant when @xmath14 ( the sample size ) is growing the relative differences between @xmath129 and @xmath73 have to decrease . the main conclusion is that the hypothesis of the nbd of charged - particle multiplicities measured by the alice collaboration in proton - proton collisions at @xmath0 = 0.9 and 2.36 tev should be rejected for all pseudo - rapidity window classes . this is the result of likelihood ratio tests performed for the corresponding data samples . the significant systematic errors are the reasons for acceptable values of the least squares test statistic for the narrowest pseudo - rapidity window measurements . the second conclusion is that the size of proper errors ( _ i.e. _ not too big and not too small , both extremes cause the false inference from @xmath112/@xmath80 values ) is somehow related to the sample size . here , for instance , errors of the type @xmath119 could be a frame of reference as it has been revealed from the results gathered in tables [ table2 ] and [ table3 ] . this is connected with the meaning of the formulation of a hypothesis . if the hypothesis is true , it means that the form of a distribution postulated by this hypothesis is exact for the whole population . thus for the very large samples ( as in all alice cases ) the measured distribution should be very close to that postulated . the performed analysis has shown that the alice experimental errors are much bigger than the acceptable discrepancies ( acceptable for these sample sizes ) . therefore @xmath11 and @xmath112 test statistics give the opposite answers in the narrowest pseudo - rapidity windows of the alice measurements . for the ua5 sample sizes , which are much smaller than the alice ones , the experimental errors have turned out to be of the order of acceptable discrepancies , so both test statistics give the same answer . the author thanks jan fiete grosse - oetringhaus for providing him with the numbers of entries in the alice event classes . this work was supported in part by the polish ministry of science and higher education under contract no . n n202 231837 . [ [ section ] ] the sample defined in sect . [ likmeth ] can be divided into @xmath27 bins with the different value of measured @xmath13 in each bin . let @xmath30 be the number of events in the @xmath31th bin , _ i.e. _ events with the same measured value of @xmath13 , say @xmath28 . then the number of events in the sample equals @xmath130 dividing by @xmath14 one can obtain the condition for experimental probabilities ( frequencies ) @xmath73 : @xmath131 now the likelihood function , eq . ( [ likelfun ] ) , can be rewritten as @xmath132^{n } \cr \cr & & = \bigg [ \ ; \prod_{i=1}^{m}\ ; p(y_i ; p , k)^{p_i^{ex } } \bigg ] ^{n}\ ; , \label{likfunbin}\end{aligned}\ ] ] and the corresponding log - likelihood function reads @xmath133 since the logarithm of the nbd is given by @xmath134 the necessary conditions for the existence of the maximum , eqs . ( [ loglikeqp ] ) , have the following form : @xmath135 \cr \cr & & = n \bigg [ -\frac{1}{1-p } \sum_{i=1}^{m}\;p_i^{ex } y_i + \frac{k}{p } \sum_{i=1}^{m}\;p_i^{ex } \bigg ] \cr \cr & & = n \bigg [ -\frac{1}{1-p } \langle n_{ch } \rangle + \frac{k}{p } \bigg ] = 0 \ ; , \label{dloglikeqp}\end{aligned}\ ] ] @xmath136 \cr \cr & & = n \bigg [ \sum_{i=1}^{m}\;p_i^{ex}\ ; \sum_{j=1}^{y_i}\ ; \frac{1}{k+j-1 } + \ln{p } \bigg ] = 0 \ ; , \label{dloglikeqk}\end{aligned}\ ] ] where the sum over @xmath137 is 0 if @xmath138 . from eqs . ( [ dloglikeqp ] ) and ( [ parametpk ] ) one can obtain : @xmath139 expressing @xmath12 as a function of @xmath4 and @xmath140 @xmath141 and substituting it to eq . ( [ dloglikeqk ] ) the equation which determines @xmath21 is obtained : @xmath142 = 0 \;. \cr & & \label{dloglikfk}\end{aligned}\ ] ] the above equation can be solved numerically . having obtained @xmath21 and substituting it into eq . ( [ oneoverp ] ) @xmath20 is derived . let @xmath143 be a random variable with p.d.f @xmath144 , which depends on parameters @xmath145 , where a parameter space @xmath146 is an open set in @xmath147 . for the set of @xmath14 independent observations of @xmath143 , @xmath148 , one can defined the likelihood function this is a statistic because it does not depend on parameters @xmath153 no more , in the numerator and the denominator there are likelihood function values at the ml estimators of parameters @xmath153 with respect to sets @xmath150 and @xmath146 , respectively . the wilks s theorem says that under certain regularity conditions if the hypothesis @xmath150 is true ( _ i.e. _ it is true that @xmath154 ) , then the distribution of the statistic @xmath155 converges to a @xmath11 distribution with @xmath156 degrees of freedom as @xmath157 @xcite . the proof can be found in ref . . note that @xmath158 is possible , so one point in the parameter space ( one value of the parameter ) can be tested as well . 00 ua5 collab . ( g. j. alner _ _ ) , _ phys . lett . b _ * 160 * , 193 ( 1985 ) . ua5 collab . ( r. e. ansorge _ et al_. ) , _ z. phys . c _ * 43 * , 357 ( 1989 ) . alice collab . ( k. aamodt _ et al_. ) , _ eur . c _ * 68 * , 89 ( 2010 ) . alice collab . ( k. aamodt _ et al_. ) , _ eur . c _ * 68 * , 345 ( 2010 ) . g. cowan , _ statistical data analysis _ , ( oxford university press , oxford , 1998 ) f. james , _ statistical methods in experimental physics _ , ( world scientific , singapore , 2006 ) s. baker and r. d. cousins , _ nucl . meth . _ * 221 * , 437 ( 1984 ) . e-802 collab . ( t. abbott _ et al_. ) , _ phys . c _ * 52 * , 2663 ( 1995 ) . j. f. grosse - oetringhaus , private communication . r. m. dudley , _ 18.466 mathematical statistics , spring 2003 _ , ( massachusetts institute of technology : mit opencourseware ) , http://ocw.mit.edu/courses/ mathematics/18 - 466-mathematical - statistics - spring-2003/lecture - notes/

likelihood ratio tests are performed for the hypothesis that charged - particle multiplicities measured in proton-(anti)proton collisions at @xmath0 = 0.9 and 2.36 tev are distributed according to the negative binomial form . results indicate that the hypothesis should be rejected in the all cases of alice - lhc measurements in the limited pseudo - rapidity windows , whereas should be accepted in the corresponding cases of ua5 data . possible explanations of that and of the disagreement with the least - squares fitting method are given .

first , the problem of the influence of the weak microlensing effect on the pulsar timing observations was discussed in @xcite . it was considered as interstellar shapiro effect . the massive body that flies not far from the line pulsar - observer produces changes in the observing frequency of the pulsar similar to glitches . estimations were made for crab and vela pulsars , glitches in these pulsars can be partially explained by the influence of the effect . substantial contribution to the problem was made by @xcite ; they mainly investigated the case of microlensing ( i.e. the gravitational deflector flied very close to the line observer - pulsar ) . it was shown that the microlensing effect would cause short - term growth of the residuals and follow - up relaxation . whole interaction would take less then several years and the maximum amplitude of the residuals would be 20 - 30 ms . such remarkable events are very rare , but all the pulsars are affected by the weak microlensing effect to a greater or lesser extent . this effect was considered in @xcite , where timing of millisecond pulsars was proposed as detection method for machos . growth of number of observed pulsars and time span of observation would make such detection easier . numerical estimates were made in @xcite . they stated that even when the measurement accuracy reaches to 10 ns , probability of the remarkable influence would be in the order of @xmath0 for the pulsar of a few kpc distance from us observed over ten years . on the other hand there s well developed formalism for the effect that came from the optics . the weak microlensing effect causes distant sources like quasars from icrf to `` tremble '' on the level of tens of mas . it was shown in @xcite that these angular fluctuations range from a few up to hundreds of microarcseconds and this leads to a small rotation of the celestial reference frame . in @xcite influence of the effect on parallax measurements was considered - apparent parallax can be even negative due to the influence of the effect . also , the weak microlensing effect can affect vlbi observations @xcite and it should be taken into account with new generation of space - based vlbi . in @xcite some statistical studies with toy - models were made , that was applied later to real model of the galaxy . in fact , both weak microlensing effect and fly - by effect on timing are very similar and can be considered as manifestation of 4d ( four - dimensional ) astrometry @xcite in this work we tried to apply eikonal formalism that was developed earlier for investigation of weak microlensing effect for use in pulsar timing studies . the paper is organized as follows . in section 2 we give a short review of influence of a passing body on pulsar timing in eikonal approximation . in section 3 we apply a model of distribution of stars in our galaxy to numerical estimations of their influence on pulsar timing and conclude our consideration in section 4 . change of phase during the propagation of electromagnetic wave can be obtained as a solution of hamilton - jacobi equation for a massless particle : @xmath1 though @xmath2 formally is a function of action , we hereafter identify it as eikonal or wave phase along the trajectory of the ray of light . in weak field approximation the metric tensor of gravitational field can be written down in a following form @xmath3 here @xmath4 is flat minkowskian metric , @xmath5 small additions to the flat metric that describes gravitational field of spherically symmetric body ( star ) equation ( [ eik1 ] ) can be solved in the following form : we take an exact solution @xcite and then take its asymptotic when the impact parameter of the propagating ray is much larger then the schwarzschild radius @xmath6 ( @xmath7-mass of the deflector ) @xmath8 here @xmath9 is full change of the phase along the trajectory , @xmath10- change of the phase along the trajectory that corresponds to the propagation in the flat space and time , @xmath11 - schwarzschild radius of the deflector , @xmath12-frequency of the electromagnetic wave , r- some point on the trajectory , @xmath13 - impact parameter ( i.e. minimal distance between deflector @xmath14 and curve of photon propagation ) . -deflecting object . ] only the second term in ( [ eik3 ] ) is a matter of interest to us , though it s only a small addition to the usual change of phase during the propagation.the complete phase shift can be obtained as a sum of two solutions . the first is a phase shift during propagation from the source of the electromagnetic waves ( which is located in @xmath15 ) to the closest approach to the deflector ( we set the point of origin to the center of the deflector ) : @xmath16 the second - is a phase shift during propagation from the closest approach to the deflector to the observers ( at @xmath17 ): @xmath18 and the total phase shift is : @xmath19 we treated the deflector as a motionless body in this solution . in fact , all stars , including machos of our galaxy are moving . approximate solution of space - time metric in the case of moving deflector and trajectory of photon in such a variable gravitational field was calculated in @xcite . the metric which originates from a moving body and small perturbations of photon trajectory in gravitational field of this body , differ in @xmath20 terms from our solutions and we will omit this difference . to describe the motion we take @xmath13 ( impact parameter ) as function of @xmath21 only : @xmath22 \end{array}\ ] ] indices denotes values at different epochs @xmath23 and @xmath24 . the first two terms are negligibly small , so we can rewrite expression ( [ eik7 ] ) : @xmath25 also we can write out time dependence of @xmath26 : + @xmath27 , here @xmath28-minimal impact parameter , @xmath29- velocity of relative motion of pulsar and deflector , @xmath30 - epoch of the closest approach . we can rewrite the equation for the phase shift and obtain equation for time delays or residuals of time of arrival ( toa ) . it s worth noting , that these delays do nt depend on frequency of electromagnetic wave : @xmath31 and impact parameter @xmath32 is passing by near this position ] we can set the first epoch @xmath23 equal to zero and discard the second index , @xmath33 : @xmath34 here , @xmath21 is time span of observations ( we set the epoch of initial observations equal to 0 ) , @xmath30 - is the epoch of the closest approach of the deflector to the line of propagation . it s convenient to consider this problem on the `` plane of deflector '' . thus we convert all linear measures into angular ones : @xmath35 , @xmath36 , @xmath32 -angular distance of the closest approach of deflector to pulsar , @xmath37 - angular velocity of the relative motion ( mainly due to the proper motion of pulsar ) , @xmath38- distance between the deflector and the observer . hereafter phrases like `` deflector s close to pulsar '' mean we observe close angular coincidence of the bodies , not in 3d space . @xmath39 value @xmath32 depends on location of pulsar in galaxy and its proper motion . the higher is density of deflectors in the neighborhood of pulsar on the celestial sphere , the smaller that value would be . we take into consideration only deflectors between the pulsar and the observer , because they make the largest contribution on the effect . we chose two pulsars j1643 - 1224 and b1937 + 21 for further estimates , because they re quite distant and located in populated regions of our galaxy ( b1937 + 21 : @xmath40 ; j1643 - 1224 : @xmath41 ) @xcite , so probability that effect would have place is much higher than for other millisecond pulsars . it s essential to define values @xmath32 and @xmath42 - average duration of influence . they can be approximately found in such way @xcite : stars are nearly uniformly distributed in the neighborhood of the pulsar on the celestial sphere ; the angular distance to the nearest star , which would affect the pulsar timing depends on the location of pulsar . we calculated the density of the stars in the neighborhood , using accepted model of the disk of our galaxy @xcite . @xmath43 @xmath44 sought density in the direction of the pulsar , which is assigned by the angles @xmath45,@xmath46 . @xmath45- angle between the line of sight and the galactic plane , @xmath46- angle between the projection of the line observer - pulsar to the galactic plane and the line solar system - galactic center;@xmath47- distance from the observer . @xmath48 @xmath49 -density of the stars in sun s neighborhood , @xmath50 -distance from the axis of the galaxy , z - distance from the galactic plane , @xmath51 -distance between the solar system and the galactic center , @xmath52 and @xmath53 - radial and vertical scales of the model , accordingly . @xmath54 @xmath55 @xmath56 shows that the influence of higher - power order items should be taken into consideration . ] shows that the influence of higher - power order items should be taken into consideration . ] average angular distance @xmath57 between the pulsar and the closest deflector ( star ) can be found with taking into account @xmath44 . values @xmath32 and @xmath42 were calculated using monte - carlo simulation : a circle of @xmath57 were circumscribed around the pulsar on the celestial sphere , then a large amount ( 1000 ) of test deflectors with proper motion @xmath37 were started from this circle under random angles @xmath58 . as a result we found distributions for values @xmath32 and @xmath42 , and their averages , that were used in following estimates . only known distribution of stars in our galaxy was used in our estimates and if we take into account possible influence of dark matter , then sought values can be lower in 2 - 3 times , because mass of dm does nt exceed mass of ordinary matter more than 4 - 5 times . also , we set mass of deflectors equal to @xmath59 . values that are essential for further estimations ( j1643 - 1224 , b1937 + 21 ) are given in the table below . @xmath32 & @xmath42 + j1643 - 1224 & 7.3 & @xmath60 & 4.7 `` & 470 yr + b1937 + 21 & 2.5 '' & @xmath60 & 1.5 " & 150 yr + we can see the influence of the effect on the residuals , but only trends of cubic order and higher will survive during usual fitting procedure @xcite . linear and quadratic terms will redefine apparent period of pulsar @xmath61 and its first derivative @xmath62 and ca nt be found . residuals of toa due to the weak microlensing effect can be written as follows : @xmath63 @xmath64 are coefficients in taylor s series of function ( 5 ) where @xmath65 . plotted coefficient @xmath66 depending on @xmath30 is represented in fig . [ fig : c ] ( plotted for b1937 + 21 ) . one can see from the plot that the fastest increase of residuals takes place when the epoch of the initial observation are 50 - 150 years away from the epoch @xmath30 , because the third derivative have maximum in that interval maximal . if the initial observation coincides with the closest approach , then only fourth and higher orders term will affect timing and the residuals will increase much slowly . magnitude of the residuals after subtraction of linear and quadratic terms can be expressed as follows : @xmath67 , where @xmath68 , @xmath69 -linear and quadratic coefficients at @xmath65 . @xmath70 @xmath71 the plot in fig . [ fig : timing ] shows magnitude of the residuals at different @xmath30 ( 0 , 50 , 100 years ; blue , green and red graphs accordingly ) . module of that magnitude depends only on module @xmath30 . residuals of 10 ns magnitude due to the effect of weak microlensing will appear with probability of @xmath72 if time span of observations exceeds 20 years . we can also calculate allan variance ( avar ) for pulsar time scale with time residuals caused by the effect . toa residuals due to the effect can be significant , if @xmath57 ( angular distance between the pulsar and the nearest affecting body ) is much smaller than average . the plot fig.7 represents situation when @xmath73 . this situation has @xmath74 chance of probability in case of b1937 + 21 ; probability reduces like @xmath32 inverse squared . the magnitude of the residuals can be as a great as 800 - 1000 ns in the same 20 years span . however , if we used in fitting procedure terms of cubic and higher orders , then the magnitude of the effect can be effectively set to 0 . the magnitude can be much greater for pulsars in gc ( or pulsars behind gc ) ( @xmath75 , @xmath76 ( length of path of ray in gc)@xmath77 , @xmath38(distance to gc ) = @xmath78 ) ) . @xmath42 and @xmath32 can be much smaller because the density of stars in gc is large , the magnitude of the effect will be much greater ( the same 1 ms in 20 years span ) . time of one significant interaction will be quite small ( 20 - 30 years ) . complete investigation of the question can be found in @xcite . so , we can make several conclusions : average toa residuals due to a weak microlensing effect is about 10 ns ( b1937 + 21 ) in 20 years span . toa residuals can be effectively set to zero by using higher order terms in fitting procedure ( not for pulsars in globular clusters . residuals can be much greater if pulsar is located in a globular cluster , so the pulsars in globular clusters ca nt be recommended for using in pt scale . j. n. bahcall , annual review of astronomy and astrophysics , v. 24 , p. 577 , 1986 baker&hellings , annual review of astronomy and astrophysics , v. 24 , p. 537 hosokava , ohnishi , fukushima , astronomy and astrophysics , v. 351 , p. 393,1999 ilyasov et al . , iau symp . 141 , p. 213 , 1989 kalinina , pshirkov , astronomy reports , vol . 50 , issue 6,p . 427 - 431 , 2006 6 . kopeikin s.m.,schaffer g. , phys . d 60 , no.12 , p. 4002 , 1999 or arxiv : gr - qc 9902030 larchenkova , doroshenko , astronomy and astrophysics , v. 297 , p. 607 , 1995 larchenkova , kopeikin , astronomy letters , v. 32 , issue 1 , p. 18 , 2006 manchester , r. n. , hobbs , g. b. , teoh , a. & hobbs , m. , aj , 129 , 1993 - 2006 ( 2005 ) ohnishi et al . in asp conference series v. 105 , p. 1250 , 1996 sazhin m.v . , in proc . conference on general relativity and gravity , stockholm , p. 519 sazhin m.v . , astronomy letters , vol . 22 , issue 5 , p. 573 , 1996 sazhin m.v . , zharov a.f . , kalinina t.a . , mnras , v . 300 , p. 287 , 1998 sazhin m.v . , zharov a.f . , kalinina t.a . , mnras , v . 323 , p. 952 , 2001 sazhin , saphonova , astrophysics and space science , v. 208 , p. 93 , 1993 sazhin , pshirkov , http://zhurnal.ape.relarn.ru/articles/2005/119.pdf , ( in russian ) , 2005 gravitation and cosmology : principles and applications of the general theory of relativity , 1972

an influence of the weak microlensing effect on the pulsar timing is investigated for pulsar b1937 + 21 . average residuals of time of arrival ( toa ) due to the effect would be as large as 10 ns in 20 years observation span . these residuals can be much greater ( up to 1 ms in 20 years span ) if pulsar is located in globular cluster ( or behind it ) .

studies of galaxy evolution have revealed surprisingly recent changes in galaxy populations . comparisons of present day galaxies with those at moderate ( @xmath13 ) and high ( @xmath14 ) redshift have uncovered trends which are often dramatic , and may trace galaxies to the time at which they were first assembled into recognizable entities . these discoveries have shed new light on the formation of galaxies , and have provided clues as to the nature of their evolution . at @xmath15 , the picture that is emerging is one in which early type galaxies evolve slowly and passively , while late type galaxies become more numerous with increasing redshift ( e.g. , @xcite ) . at higher redshifts , deep surveys such as the hubble deep field ( @xcite ) indicate an increase in the cosmic star formation rate out to @xmath16 ( e.g. , madau , pozzetti , and dickinson 1998 ) . while considerable progress has been made in the observational description of galaxy evolution , important questions remain regarding the physical processes driving this evolution . mechanisms that have been postulated include galaxy - galaxy mergers , luminosity - dependent luminosity evolution , and the existence of a new population of galaxies that has faded by the present epoch ( see reviews by @xcite and @xcite ) . in this study , we will investigate the relative importance of mergers in the evolution of field galaxies . mergers transform the mass function of galaxies , marking a progression from small galaxies to larger ones . in addition , mergers can completely disrupt their constituent galaxies , changing gas - rich spiral galaxies into quiescent ellipticals ( e.g. , toomre and toomre 1972 ) . during a collision , a merging system may also go through a dramatic transition , with the possible onset of triggered star formation and/or accretion onto a central black hole ( see review by barnes & hernquist 1992 ) . it is clear that mergers do occur , even during the relatively quiet present epoch . however , the frequency of these events , and the distribution of masses involved , has yet to be accurately established . this is true at both low and high redshift . furthermore , while a number of attempts have been made , a secure measurement of evolution in the galaxy merger rate remains elusive , and a comparable measure of the accretion rate has yet to be attempted . in this study , we introduce a new approach for relating dynamically close galaxy pairs to merger and accretion rates . these new techniques yield robust measurements for disparate samples , thereby allowing meaningful comparisons of mergers at low and high redshift . in addition , these pair statistics can be adapted to a variety of redshift samples , and to studies of both major and minor mergers . we apply these techniques to a large sample of galaxies at low redshift ( ssrs2 ) , providing a much needed local benchmark for comparison with samples at higher redshift . in a forthcoming paper ( patton et al . 2000 ) , we will apply these techniques to a large sample of galaxies at moderate redshift ( cnoc2 ; @xmath17 ) , yielding a secure estimate for the rate of evolution in the galaxy merger and accretion rates . an overview of earlier pair studies , and a discussion of their limitations and shortcomings , are given in the next section . the ssrs2 data are described in [ ssrs2mr : data ] . section [ ssrs2mr : mrate ] discusses the connection between close pairs and the merger and accretion rates , while [ ssrs2mr : nclc ] introduces new statistics for relating these quantities . section [ ssrs2mr : flux ] describes how these statistics can be applied to flux - limited surveys in a robust manner . a pair classification experiment is presented in [ ssrs2mr : class ] , giving empirical justification for our close pair criteria . pair statistics are then computed for the ssrs2 survey in [ ssrs2mr : sample ] , and the implications are discussed in [ ssrs2mr : discuss ] . conclusions are given in the final section . throughout this paper , we use a hubble constant of @xmath18 km s@xmath19 mpc@xmath19 . we assume @xmath20=1 and @xmath21=0.1 , unless stated otherwise . every estimate of evolution in the merger and/or accretion rate begins with the definition of a merger statistic . ideally , this statistic should be independent of selection effects such as optical contamination due to unrelated foreground / background galaxies , redshift incompleteness , redshift - dependent changes in minimum luminosity resulting from flux limits , contamination due to non - merging systems , @xmath22-corrections , and luminosity evolution . in addition , it should be straightforward to relate the statistic to the global galaxy population , and to measurements on larger scales . the statistic should then be applied to large , well - defined samples from low to high redshift , yielding secure estimates of how the merger and/or accretion rates vary with redshift . within the past decade , there have been a number of attempts to estimate evolution in the galaxy merger rate using close pairs of galaxies ( e.g. , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ) . the statistic that has been most commonly employed is the traditional pair fraction , which gives the fraction of galaxies with suitably close physical companions . this statistic is assumed to be proportional to the galaxy merger rate . the local ( low - redshift ) pair fraction was estimated by patton et al . ( 1997 ) , using a flux - limited ( @xmath23 ) sample of galaxies from the ugc catalog ( @xcite ) . using pairs with projected physical separations of less than 20 @xmath5 kpc , they estimated the local pair fraction to be @xmath24 . this result was shown to be consistent with the local pair fraction estimates of carlberg et al . ( 1994 ) and yee & ellingson ( 1995 ) , both of whom also used the ugc catalog . the pair fraction has been measured for samples of galaxies at moderate redshift ( @xmath13 ) , yielding published estimates ranging from approximately 0% ( @xcite ) to 34% @xmath25 9% ( @xcite ) . evolution in the galaxy merger rate is often parameterized as @xmath26 . close pair studies have yielded a wide variety of results , spanning the range @xmath27 . there are several reasons for the large spread in results . first , different methods have been used to relate the pair fraction to the merger rate . in addition , some estimates have been found to suffer from biases due to optical contamination or redshift completeness . after taking all of these effects into account , patton et al . ( 1997 ) demonstrated that most results are broadly consistent with their estimate of @xmath28 , made using the largest redshift sample ( 545 galaxies ) to date . while this convergence seems promising , all of these results have suffered from a number of very significant difficulties . the central ( and most serious ) problem has been the comparison between low and moderate redshift samples . low-@xmath4 samples have been poorly defined , due to a lack of suitable redshift surveys . in addition , the pair fraction depends on both the clustering and mean density of galaxies . the latter is very sensitive to the limiting absolute magnitude of galaxies , leading to severe redshift - dependent biases when using flux - limited galaxy samples . these biases have not been taken into account in the computation of pair fractions , or in the comparison between samples at different redshifts . while these problems are the most serious , there are several other areas of concern . a lack of redshift information has meant dealing with optical contamination due to unrelated foreground and background galaxies . moreover , while one can statistically correct for this contamination , it is still not possible to discern low velocity companions from those that are physically associated but unbound , unless additional redshift information is available . finally , there is no direct connection between the pair fraction and the galaxy correlation function ( cf ) and luminosity function ( lf ) , making the results more difficult to interpret . to address these issues , we have developed a novel approach to measuring pair statistics . we will introduce new statistics that overcome many of the afflictions of the traditional pair fraction . we will then apply these statistics to a large , well - defined sample of galaxies at low redshift . the second southern sky redshift survey ( @xcite ; hereafter ssrs2 ) consists of 5426 galaxies with @xmath29 , in two regions spanning a total of 1.69 steradians in the southern celestial hemisphere . the first region , denoted ssrs2 south , has boundaries @xmath30 and @xmath31 . the second region , ssrs2 north , is a more recent addition , and is bounded by @xmath32 and @xmath33 . galaxies were selected primarily from the list of non - stellar objects in the _ hubble space telescope _ guide star catalog , with positions accurate to @xmath10 1 and photometry with an rms scatter of @xmath10 0.3 magnitudes ( alonso et al . 1993 , alonso et al . 1994 ) . steps were taken to ensure that single galaxies were not mistakenly identified as close pairs , due to the presence of dust lanes , etc . ( @xcite ) . in addition , careful attention was paid to cases where a very close pair might be mistaken for a single galaxy . this was found to make a negligible contribution to the catalog as a whole ( @xmath34 of galaxies are affected ) . the effect on the pairs analysis in this paper is further reduced by imposing a minimum pair separation of 5 @xmath5 kpc ( see [ ssrs2mr : class ] ) . the sample now includes redshifts for all galaxies brighter than @xmath29 . we correct all velocities to the local group barycenter using equation 6 from courteau and van den bergh ( 1999 ) . we restrict our analysis to the redshift range @xmath35 . this eliminates nearby galaxies , for which recession velocities are dominated by peculiar velocities , giving poor distance estimates . we also avoid the sparsely sampled high redshift regime . this leaves us with a well - defined sample of 4852 galaxies . the primary goal of earlier close pair studies has been to determine how the galaxy merger rate evolves with redshift . the merger rate affects the mass function of galaxies , and may also be connected to the cosmic star formation rate . before attempting to measure the merger rate , it is important to begin with a clear definition of a merger and a merger rate . here , we refer to mergers between two galaxies which are both above some minimum mass or luminosity . if this minimum corresponds roughly to a typical bright galaxy ( @xmath36 ) , this criterion can be thought of as selecting so - called major mergers . we consider two merger rate definitions . first , it is of interest to determine the number of mergers that a typical galaxy will undergo per unit time . in this case , the relevant rate may be termed the galaxy merger rate ( hereafter @xmath37 ) . a related quantity is the total number of mergers taking place per unit time per unit co - moving volume . we will refer to this as the volume merger rate ( hereafter @xmath38 ) . clearly , @xmath39 , where @xmath40 is the co - moving number density of galaxies . while both of these merger rates provide useful measures of galaxy interactions , they have their limitations . as one probes to faint luminosities , one will find an increasing number of faint companions ; hence , the number of inferred mergers will increase in turn . for all realistic lfs , this statistic will become dominated by dwarf galaxies . in addition , it is of interest to determine how the mass of galaxies will change due to mergers . to address these issues , we will also investigate the rate at which mass is being accreted onto a typical galaxy . this quantity , the total mass accreted per galaxy per unit time , will be referred to as the galaxy accretion rate ( hereafter @xmath41 ) . this is related to the rate of mass accretion per unit co - moving volume ( @xmath42 ) by @xmath43 . the mass ( or luminosity ) dependence of the accretion rate means that it will be dominated by relatively massive ( or luminous ) galaxies , with dwarfs playing a very minor role unless the mass function is very steep . in order to determine @xmath37 observationally , one may begin by identifying systems which are destined to merge . by combining information about the number of these systems and the timescale on which they will undergo mergers , one can estimate an overall merger rate . specifically , if one identifies @xmath44 ongoing mergers per galaxy , and if the average merging timescale for these systems is @xmath45 , then @xmath46 . if the mass involved in these mergers ( per galaxy ) is @xmath47 , then @xmath48 . in practice , direct measurement of these quantities is a daunting task . it is difficult to determine if a given system will merge ; furthermore , estimating the merger timescale for individual systems is challenging with the limited information generally available . however , if one simply wishes to determine how the merger rate is _ changing _ with redshift , then the task is more manageable . if one has the same definition of a merger in all samples under consideration , then it is reasonable to assume that the merger timescale is the same for these samples . in this case , we are left with the task of measuring quantities which are directly proportional to the number or mass of mergers per galaxy or per unit co - moving volume . if one wishes to consider luminosity instead of mass , the relation between mass and luminosity must either be the same at all epochs , or understood well enough to correct for the differences . we have considered several quantities that fit this description . all involve the identification of close physical associations of galaxies . a `` close companion '' is defined as a neighbour which will merge within a relatively short period of time ( @xmath49 ) , which allows an estimate of the instantaneous merger / accretion rate . if a galaxy is destined to undergo a merger in the very near future , it must have a companion close at hand . one might attempt to estimate the number of mergers taking place within a sample of galaxies . for example , a close pair of galaxies would be considered one merger , while a close triple would lead to two mergers , etc . owing to the difficulty of determining with certainty which systems are undergoing mergers , we will not use this approach . one alternative is to estimate the number of galaxies with one or more close companions , otherwise known as the pair fraction . one drawback of this approach is that close triples or higher order n - tuples complicate the analysis , since they are related to higher orders of the correlation function . this also makes it difficult to correct for the flux - limited nature of most redshift surveys . as a result , we choose to steer clear of this method also . in this study , we choose instead to use the _ number and luminosity of close companions _ per galaxy . the number of close companions per galaxy , hereafter @xmath0 , is similar in nature to the pair fraction . in fact , they are identical in a volume - limited sample with no triples or higher order n - tuples . however , @xmath0 will prove to be much more robust and versatile . we assume that @xmath0 is directly proportional to the number of mergers per galaxy , such that @xmath50 ( @xmath22 is a constant ) . this pairwise statistic is preferable to the number of mergers per galaxy or the fraction of galaxies in merging systems , in that it is related , in a direct and straightforward manner , to the galaxy two - point cf and the lf ( see section [ ssrs2mr : nclc ] ) . we note that it is not necessary that there be a one - to - one correspondence between companions and mergers , as long as the correspondence is the same , on average , in all samples under consideration . using this approach to estimate the number of mergers per galaxy , the merger rate is then given by @xmath51 . the actual value of @xmath22 depends on the merging systems under consideration . if one identifies a pure set of galaxy pairs , each definitely undergoing a merger , then each pair , consisting of 2 companions , would lead to one merger , giving @xmath22=0.5 . for a pair sample which includes some triples and perhaps higher order n - tuples , if the merging sample under investigation contains some systems which are not truly merging ( for instance , close pairs with hyperbolic orbits ) , then @xmath22 will also be reduced . while @xmath22 clearly varies with the type of merging system used , the key is for @xmath22 to be the same for all samples under consideration . we take a similar approach with the accretion rate . we again use close companions , and in this case we simply add up the luminosity in companions , per galaxy ( @xmath1 ) . defining the mean companion mass - to - light ratio as @xmath53 , it follows that @xmath54 and @xmath55 . when comparing different samples , any significant differences in @xmath53 must be accounted for . in order to motivate further the need for merger rate measurements , and to set the stage for future work relating pair statistics to the mass and luminosity function , we develop a simple model which relates these important quantities . suppose the galaxy mass function is given by @xmath56 . this function gives the number density of galaxies of mass @xmath57 at time @xmath58 , per unit mass . the model that follows can also be expressed in terms of luminosity or absolute magnitude , rather than mass . we begin by assuming that all changes in the mass function are due to mergers . while this is clearly simplistic , this model will serve to demonstrate the effects that various merger rates can have on the mass function . in order to relate the mass function to the observable luminosity function , we further assume that mergers do not induce star formation . again , this is clearly an oversimplification ; however , this simple case will still provide a useful lower limit on the relative contribution of mergers to lf evolution . finally , we assume that merging is a binary process . following @xcite , we consider how @xmath56 evolves as the universe ages from time @xmath58 to time @xmath59 . each merger will remove two galaxies from the mass function , and produce one new galaxy . let @xmath60 represent the decrease in the mass function due to galaxies removed by mergers , while @xmath61 gives the increase due to the remnants produced by these mergers . evolution in the mass function can then be given by @xmath62 we model this function by considering all galaxy pairs , along with an expression for the merging likelihood of each . let @xmath63 denote the probability that a galaxy of mass @xmath57 will merge with a galaxy of mass @xmath64 in time interval @xmath65 . in order to estimate @xmath60 , we need to take all galaxies of mass @xmath57 , and integrate over all companions , yielding @xmath66 ^ 3dm^{\prime}.\end{aligned}\ ] ] we devise a comparable expression for @xmath61 by integrating over all pairs with end@xmath67products of mass @xmath57 . this is achieved by considering all pairs with component of mass @xmath68 and @xmath64 , such that @xmath69 ^ 3dm^{\prime}.\end{aligned}\ ] ] we can also express equation [ ssrs2mr : eqinout ] in terms of the pair statistics outlined in section [ ssrs2mr : observables ] . if one considers close companions of mass @xmath70 next to primary galaxies of mass @xmath57 , the volume merger rate can be expressed as @xmath71 , yielding @xmath72 similarly , if one defines a merger remnant statistic , @xmath73 , to be the co - moving number density of merger remnants per unit time corresponding to these same mergers , then @xmath74 therefore , it is possible , in principle , to use pair statistics to measure the evolution in the mass or luminosity function due to mergers . however , current pair samples are too small to permit useful pair statistics for different mass combinations . in addition , present day observations of close pairs are not of sufficient detail to determine the proportion of pairs that will result in mergers ( factor @xmath22 in previous section ) . moreover , timescale estimates for these mergers are not known with any degree of certainty . hence , useful observations of mass function evolution due to mergers will have to wait for improved pair samples and detailed estimates of merger timescales . in this section , we outline the procedure for measuring the mean number ( @xmath0 ) and luminosity ( @xmath1 ) of close companions for a sample of galaxies with measured redshifts . we begin by defining these statistics in real space , demonstrating how they are related to the galaxy lf and cf . we then show how these statistics can be applied in redshift space . in this study , we will measure pair statistics for a complete low - redshift sample of galaxies ( ssrs2 ) . however , we wish to make these statistics applicable to a wide variety of redshift samples . we would also like this method to be useful for studies of minor mergers , where one is interested in faint companions around bright galaxies . moreover , these techniques should be adaptable to redshift samples with varying degrees of completeness ( that is , with redshifts not necessarily available for every galaxy ) . therefore , in the following analysis , we treat host galaxies and companions differently . consider a primary sample of @xmath75 host galaxies with absolute magnitudes @xmath76 , lying in some volume @xmath77 . suppose this volume also contains a secondary sample of @xmath78 galaxies with @xmath79 . in the general case , the primary and secondary samples may have galaxies in common . this includes the special case in which the two samples are identical . if @xmath80 , this will tend to probe major mergers . if @xmath81 is chosen to be significantly fainter than @xmath82 , this will allow for the study of minor mergers . we assume here that both samples are complete to the given absolute magnitude limits ; in section [ ssrs2mr : flux ] , we extend the analysis from volume - limited samples to those that are flux - limited . we wish to determine the mean number and luminosity of companions ( in the secondary sample ) for galaxies in the primary sample . in real space , we define a close companion to be one that lies at a true physical separation of @xmath83 , where @xmath84 is some appropriate maximum physical separation . to compute the observed mean number ( @xmath0 ) and luminosity ( @xmath1 ) of companions , we simply add up the number ( @xmath85 ) and luminosity ( @xmath86 ) of companions for each of the @xmath75 galaxies in the primary sample , and then compute the mean . therefore , @xmath87 and @xmath88 we can also estimate what these statistics should be , given detailed knowledge of the galaxy two - point cf @xmath2 and the lf @xmath3 . this is necessary if one wishes to relate these pair statistics to measurements on larger scales . consider a galaxy in the primary sample with absolute magnitude @xmath89 at redshift @xmath90 . we would first like to estimate the number and luminosity of companions lying in a shell at physical distance ( _ proper _ co - ordinates ) @xmath91 from this primary galaxy . to make this estimate , we need to know the mean density of galaxies ( related to the lf ) , and the expected overdensity in the volume of interest ( given by the cf ) . the mean physical number density of galaxies at redshift @xmath90 in the secondary sample , with absolute magnitudes @xmath92 , is given by @xmath93 where @xmath94 is the differential galaxy lf , which specifies the _ co - moving _ number density of galaxies at redshift @xmath90 , in units of @xmath95 . the actual density of objects in the region of interest is determined by multiplying the mean density by ( 1+@xmath2 ) , where @xmath2 is the overdensity given by the two - point cf ( @xcite ) . in general , @xmath2 depends on the pair separation @xmath96 , the mean redshift @xmath90 , the absolute magnitude of each galaxy ( @xmath89 , @xmath97 ) , and the orbits involved , specified by components parallel ( @xmath98 ) and perpendicular ( @xmath99 ) to the line of sight . it follows that the mean number of companions with @xmath92 and @xmath91 is given by @xmath1004\pi r^2dr.\end{aligned}\ ] ] we must now integrate this expression for all companions with @xmath79 and @xmath83 . integration over the lf yields @xmath101 integration over the cf is non - trivial , because of the complex nature of @xmath102 . with redshift samples that are currently available , it is not possible to measure this dependence accurately for the systems of interest . hence , we must make three important assumptions at this stage . first , we assume that @xmath2 is independent of luminosity . later in the paper , we demonstrate empirically that this is a reasonable assumption , provided one selects a sample with appropriate ranges in absolute magnitude ( see section [ ssrs2mr : clust ] ) . secondly , we assume that the distribution of velocities is isotropic . if one averages over a reasonable number of pairs , this is bound to be true , and therefore @xmath2 is independent of @xmath99 and @xmath98 . finally , we assume that the form of the cf , as measured on large scales , can be extrapolated to the small scales of interest here . this assumption applies only to the method of relating pairs to large scale measures , and not to the actual measurement of pair statistics . it is now straightforward to integrate equation [ ssrs2mr : ncdiff ] . the mean number of companions with @xmath79 and @xmath83 for a primary galaxy at redshift @xmath90 is given by @xmath1034\pi r^2dr.\end{aligned}\ ] ] we derive an analogous expression for the mean luminosity in companions . the integrated luminosity density is given by @xmath104 where @xmath105 therefore , @xmath1064\pi r^2dr.\end{aligned}\ ] ] given measurements of the cf on large scales , it is then straightforward to integrate these equations to arrive at predicted values of @xmath85 and @xmath86 . it is important to note that these statistics are directly dependent on @xmath81 , which affects the mean density of galaxies in the secondary sample . this is different from statistics such as the correlation function , which are independent of density . hence , this serves as a reminder that we must exercise caution when choosing our samples , to ensure that differences in the pair statistics ( and hence in the merger and accretion rates ) are not simply due to apparent density differences resulting from selection effects . in addition , note that the choice of @xmath82 has no density - related effects on @xmath0 and @xmath1 . while it is preferable to identify companions based on their true physical pair separation , this is clearly not feasible when dealing with data from redshift surveys . in the absence of independent distance measurements for each galaxy , one must resort to identifying companions in redshift space . in this section , we outline a straightforward approach for measuring our new pair statistics in redshift space . we then attempt to relate these statistics to their counterparts in real space . for any given pair of galaxies in redshift space , one can compute two basic properties which describe the intrinsic pair separation : the projected physical separation ( hereafter @xmath107 ) and the rest - frame relative velocity along the line of sight ( hereafter @xmath108 ) . for a pair of galaxies with redshifts @xmath90 ( primary galaxy ) and @xmath109 ( secondary ) , with angular separation @xmath110 , these quantities are given by @xmath111 and @xmath112 , where @xmath113 is the angular diameter distance at redshift @xmath90 . note that @xmath107 gives the projected separation at the redshift of the primary galaxy . we define a close companion as one in which the separation ( both projected and line - of - sight ) is less than some appropriate separation , such that @xmath114 and @xmath115 . the line - of - sight criterion depends on both the physical line - of - sight separation and the line - of - sight peculiar velocity of the companion . it is of course not possible to determine the relative contributions of these components without distance information . however , for the small companion separations we will be concerned with , the peculiar velocity component is likely to be dominant in most cases , as we will be dealing with a field sample of galaxies ( the same would not be true in the high velocity environment of rich clusters ) . hence , this criterion serves primarily to identify companions with low peculiar velocities . while this is fundamentally different from the pure separation criterion used in real space , it too will serve to identify companions with the highest likelihood of undergoing imminent mergers . using this definition of a close companion , it is straightforward to compute @xmath0 and @xmath1 , using equations [ ssrs2mr : eqncbasic ] and [ ssrs2mr : eqlcbasic ] . thus , the complexities of redshift space do not greatly complicate the computation of these pair statistics . as in real space , we wish to relate these statistics to measurements on larger scales , given reasonable assumptions about the lf and cf . the situation is more complicated in redshift space , and therefore involves additional assumptions . we stress , however , that these assumptions apply only to the method of relating pair statistics to large scale measures , and not to the measured pair statistics themselves . to outline an algorithm for generating these predictions , we follow the approach of the previous section . we begin by modifying equations [ ssrs2mr : ncint ] and [ ssrs2mr : lcint ] , integrating over the new pair volume defined in redshift space . in order to do this , we use the two dimensional correlation function in redshift space , @xmath116 , giving @xmath1172\pi r_pdr_pdr_v\end{aligned}\ ] ] and @xmath1182\pi r_pdr_pdr_v.\end{aligned}\ ] ] the two dimensional correlation function is the convolution of the velocity distribution in the redshift direction , @xmath119 , with the spatial correlation function @xmath120 , given by @xmath121)dy.\ ] ] here , @xmath122 is the hubble constant at redshift @xmath4 , given by @xmath123 . we have ignored the effect of infall velocities , which must be taken into account at larger radii but is an acceptable approximation for small separations . if the form of the cf and lf are known , it is straightforward to integrate equations [ ssrs2mr : nc2int ] and [ ssrs2mr : lc2int ] , yielding predictions of @xmath0 and @xmath1 . it is not always possible to have precise redshifts for all galaxies of interest in a sample . a common scenario with redshift surveys is to have redshifts available for a subset of galaxies identified in a flux - limited photometric sample . the photometric sample used to select galaxies for follow - up spectroscopy probes to fainter apparent magnitudes than the spectroscopic sample . in addition , the spectroscopic sample may be incomplete , even at the bright end of the sample . in this section , we will describe the procedure for applying pair statistics to this class of samples . suppose the primary sample is defined as all galaxies in the spectroscopic sample with absolute magnitudes @xmath76 . the secondary sample consists of all galaxies lying in the photometric sample , regardless of whether or not they have measured redshifts . once again , there may be some overlap between the primary and secondary samples . we must now identify close pairs . for each primary - secondary pair , we can compute @xmath107 in precisely the same manner as before ( see previous section ) , since we need only the redshift of the primary galaxy and the angular separation of the pair . however , we are no longer able to compute the relative velocity along the line of sight , since this requires redshifts for both members of the pair . thus , we do not have enough information to identify close dynamical pairs . however , it is still possible to determine , in a statistical manner , how many physically associated companions are present . this is done by comparing the number ( or luminosity ) of observed companions with the number ( or luminosity ) expected in a random distribution . as stressed in section [ ssrs2mr : realspace ] , pair statistics depend on the minimum luminosity @xmath81 imposed on the secondary sample . while we are now unable to compute the actual luminosity for galaxies in the secondary sample , we must still impose @xmath81 if the ensuing pair statistics are to be meaningful . to do this , we make use of the fact that all physical companions must lie at approximately the same redshift as the primary galaxy under consideration . therefore , @xmath81 corresponds to a limiting _ apparent _ magnitude @xmath124 at redshift @xmath90 , such that @xmath125 where @xmath126 is the luminosity distance at redshift @xmath90 , and @xmath127 is the @xmath22-correction . to begin , one finds all observed close companions with @xmath128 , using only the @xmath107 criterion . this results in the quantities @xmath129 and @xmath130 , where the `` d '' superscript denote companions found in the data sample . one must then estimate the number ( @xmath131 ) and luminosity ( @xmath132 ) of companions expected at random . the final pair statistics for close physical companions are then given by @xmath133 and @xmath134 . we will now describe how to predict these statistics using the known lf and cf . this is relatively straightforward , since the excess @xmath2 given by the cf is determined by the relative proportions of real and random companions . the pair statistics are once again integrals over the two dimensional cf in redshift space , as specified by equations [ ssrs2mr : nc2int ] and [ ssrs2mr : lc2int ] . in the `` @xmath135 '' term , the first part gives the random contribution , while the second gives the excess over random . thus , these pair statistics give the true density of companions , rather than the `` excess '' density . this is intentional , since mergers will occur even in an uncorrelated , randomly distributed sample of galaxies . at the small separations of interest , usually less than 1% of the correlation length , the difference between the mean density and the mean overdensity is less than about 0.01% in real space . for practical measurements in redshift space , where @xmath136 is of order the correlation length , the background contribution is substantially larger than real space , but still amounts to less than 1% . thus , for the close pairs considered in this study , it is reasonable to ignore the contribution that random companions make to the sample of physical companions . that is , we take @xmath137 . this allows us to relate the predictions to the measured pair statistics set out above . in principle , equations [ ssrs2mr : nc2int ] and [ ssrs2mr : lc2int ] can be integrated over the range @xmath138 to obtain predictions of @xmath0 and @xmath1 . to illustrate the concepts introduced so far , and to emphasize how these statistics depend on @xmath81 , we apply these techniques to volume - limited monte carlo simulations , which mimic the global distribution of galaxies in the ssrs2 north and south catalogs . using @xmath21=0.5 , galaxies are distributed randomly within the co - moving volume enclosed by @xmath139 and the ssrs2 boundaries on the sky ( see section [ ssrs2mr : data ] ) . all peculiar velocities are set to zero . to create a volume - limited sample , we impose a minimum luminosity of @xmath140=@xmath141 , and assign luminosities using the ssrs2 lf ( @xcite ) , which has schechter function parameters @xmath142=@xmath143 , @xmath144=@xmath145 , and @xmath146 = @xmath147 . an arbitrarily large number of galaxies can be generated , which is of great assistance when looking for small systematic effects . we produce 16000 galaxies in the south , and 8070 in the north ; this gives the same density of galaxies in both regions . using these simulations , we compute @xmath0 and @xmath1 . as these galaxies are distributed randomly ( as opposed to real galaxies which are clustered ) , close pairs are relatively rare . to ensure a reasonable yield of pairs , we use a pair definition of @xmath148=1 @xmath5 mpc and @xmath149=1000 km / s . we note that there are no peculiar velocities in these simulations ; hence , the @xmath149 criterion provides upper and lower limits on the line - of - sight distance to companions . also , recall from the preceding section that the choice of @xmath82 has no effect on the pair statistics if clustering is independent of luminosity . hence , we choose @xmath82=@xmath81 , which maximizes the size of the primary sample , and therefore minimizes the measurement errors in @xmath0 and @xmath1 . with these assumptions , we compute pair statistics for a range of choices of @xmath81 . errors are computed using the jackknife technique . for this resampling method , partial standard deviations , @xmath150 , are computed for each object by taking the difference between the quantity being measuring , @xmath151 , and the same quantity with the @xmath152 galaxy removed from the sample , @xmath153 , such that @xmath154 . for a sample of @xmath155 galaxies , the variance is given by @xmath156^{1/2}$ ] ( efron 1981 ; efron & tibshirani 1986 ) . results are given in figure [ ssrs2mr : figm2vl ] . both statistics continue to increase as @xmath81 becomes fainter . @xmath0 diverges at faint magnitudes , while @xmath1 is seen to converge . this behaviour is a direct consequence of the shape of the lf ; @xmath0 converges for @xmath157 , while @xmath1 converges for @xmath158 . the existence and magnitude of these trends clearly demonstrate the need to specify @xmath81 when measuring pair statistics . the preceding section gives a straightforward prescription for computing pair statistics in volume - limited samples . however , redshift surveys are generally flux - limited . by defining a volume - limited sample within such a survey , one must discard a large proportion of the data . in this section , we will outline how these pair statistics can be applied to flux - limited surveys . pair statistics necessarily depend on both clustering and mean density , as shown by equations [ ssrs2mr : nc2int ] and [ ssrs2mr : lc2int ] . in a flux - limited sample , both clustering and mean density will vary throughout the sample . we will use these equations to account for redshift - dependent changes in mean density , and we will demonstrate how to minimize the effects of clustering differences . these techniques will then be tested with monte carlo simulations . by removing the fixed luminosity limit , the overall distribution of galaxy luminosities will vary with redshift within the sample , and the mean luminosity of the sample will differ from the volume - limited sample . however , galaxy clustering is known to be luminosity dependent . measures of the galaxy correlation function ( e.g. , loveday et al . 1995 , willmer et al . 1998 ) , power spectrum ( e.g. , vogeley 1993 ) , and counts in cells ( benoist et al . 1996 ) all find that luminous galaxies ( @xmath159 ) are more clustered than sub-@xmath36 galaxies , typically by a factor of @xmath160 . this increase in clustering may be particularly strong ( factor @xmath161 ) for very luminous galaxies ( @xmath162 ) . clearly , this effect should not be ignored when computing pair statistics . in principle , this could be incorporated into the measurement of these pair statistics . however , available pair samples are too small to measure this dependence . we choose instead to minimize these effects by restricting the analysis to a fixed range in absolute magnitude , within which luminosity - dependent clustering is small or negligible . this is done by imposing additional bright ( @xmath163 ) and faint ( @xmath164 ) absolute magnitude limits on the sample . having thereby reduced the effects of luminosity segregation , we then assume that the remaining differences will not have a significant effect on the measured pair statistics . in section [ ssrs2mr : sense ] , we use the ssrs2 sample to demonstrate empirically that this is in fact a reasonable assumption . in section [ ssrs2mr : nclc ] , we demonstrated that these pair statistics are meaningful only if one specifies the minimum luminosity of the primary and secondary samples . for a flux - limited sample , however , the minimum luminosity of the sample increases with redshift . one must therefore decide on a representative minimum luminosity , and account for differences between the desired minimum luminosity and the redshift - dependent minimum imposed by the apparent magnitude limit of the sample . if the lf is known , this can be achieved by weighting each galaxy appropriately . in this section , we outline a weighting scheme which makes this correction . consider a flux - limited sample in which host galaxies are located at a variety of redshifts . those at low redshift will have the greatest probability of having close companions that lie above the flux limit , since the flux limit corresponds to an intrinsic luminosity that is fainter than that for galaxies at higher redshift . if we wish to avoid an inherent bias in the pair statistics , we must correct for this effect . furthermore , we must account for any limits in absolute magnitude imposed on the sample to reduce the effects of luminosity - dependent clustering ( [ ssrs2mr : clust ] ) . finally , we have demonstrated the importance of specifying a limiting absolute magnitude for companions ( @xmath81 ) when computing pair statistics . therefore , we must attempt to correct the pair statistics to the values that would have been achieved for a volume - limited secondary sample with @xmath165 . qualitatively , this correction should assign greater importance ( or weight ) to the more rare companions found at the high redshift end of the flux - limited sample . to make this correction as rigorous as possible , we will use the galaxy lf . by integrating the lf over a given range in absolute magnitude , one can obtain an estimate of the mean number or luminosity density of galaxies in the sample . by performing this integration at any given redshift , accounting for the allowed ranges in absolute magnitude and the flux limit , it is possible to quantify how the mean density varies with redshift within the defined sample . this information can be used to remove this unwanted bias from the pair statistics . we assign a weight to each galaxy in the secondary sample , which renormalizes the sample to the density corresponding to @xmath165 . we must first determine @xmath166 , which gives the limiting absolute magnitude allowed at redshift @xmath90 . at most redshifts , this is imposed by the limiting apparent magnitude @xmath167 , such that @xmath168 . at the low redshift end of the sample , however , @xmath164 ( defined in [ ssrs2mr : clust ] ) will take over . that is , the limiting absolute magnitude used for identifying galaxies in the secondary sample is given by @xmath169.\ ] ] the selection function , denoted @xmath170 , is defined as the ratio of densities in flux - limited versus volume - limited samples . this function , given in terms of number density ( @xmath171 ) and luminosity density ( @xmath172 ) , is as follows : @xmath173 @xmath174 where @xmath175 is defined in equation [ ssrs2mr : eql ] . in order to recover the correct pair statistics , each companion must be assigned weights @xmath176 and @xmath177 . the total number and luminosity of close companions for the @xmath178 primary galaxy , computed by summing over the @xmath179 galaxies satisfying the `` close companion '' criteria , is given by @xmath180 and @xmath181 respectively . by applying this weighting scheme to all galaxies in the secondary sample , we will retrieve pair statistics that correspond to a volume - limited sample with @xmath165 . the above weighting scheme ensures that the number and luminosity of companions found around each primary galaxy is normalized to @xmath165 . however , these estimates are obviously better for galaxies at the low redshift end of the primary sample , since they will have the largest number of _ observed _ companions . recall that @xmath0 and @xmath1 are quantities that are averaged over a sample of primary galaxies . in order to minimize the errors in these statistics , we assign weights to the primary galaxies ( denoted @xmath182 and @xmath183 ) which are inversely proportional to the square of their uncertainty . if the observed number and luminosity of companions around the @xmath178 primary galaxy are given by @xmath184 and @xmath185 respectively , and if we assume that the uncertainties are determined by poisson counting statistics , then @xmath186 and @xmath187 . on average , these quantities will be related to expectation values @xmath188 and @xmath189 by @xmath190 and @xmath191 . combining these relations yields @xmath192 @xmath193 that is , the optimal weighting is the reciprocal of the weighting scheme used for companions . therefore , weights @xmath194 and @xmath195 should be assigned to primary galaxies . the pair statistics are then computed as follows : @xmath196 @xmath197 it is worth noting that , for a close pair , both galaxies will lie at roughly the same redshift , meaning that @xmath198 . we choose not to make this approximation , in order to keep these relations valid for pairs that are not close , and to allow for future application to pairs with additional selection weights . however , we stress that , with or without this approximation , the primary weights in the denominator provide an overall correction for the flux limit , unlike the traditional pair fraction . note also that , for a volume - limited sample , weights for all galaxies in the primary and secondary samples are equal , reducing these equations to @xmath199 and @xmath200 , as defined in section [ ssrs2mr : realspace ] . a small correction must be made to these weights if a primary galaxy lies close to a region of space that is not covered by the survey . this will happen if a galaxy lies close to the boundaries on the sky , or close to the minimum or maximum redshift allowed . if this is the case , it is possible that close companions will be missed , leading to an underestimate of the pair statistics . therefore , we must account for these effects . first , we consider galaxies lying close to the survey boundaries on the sky , as defined in section [ ssrs2mr : data ] . for each galaxy in the primary sample , we compute the fraction of sky with @xmath201 that lies within the survey boundaries . this fraction will be denoted @xmath202 . for ssrs2 , our usual choices of @xmath203 and @xmath148 ( see [ ssrs2mr : class ] ) make this a very small effect , with @xmath204=1 for 99.75% of galaxies in the primary sample . having measured @xmath202 for each galaxy in the primary sample , we must incorporate this into the measurement of the pair statistics . the first task is to ensure that we correct the number of companions to match what would be expected if coverage was complete . we do this by assigning each companion a boundary weight @xmath205 = @xmath206 , where @xmath202 is associated with its host galaxy from the primary sample . by multiplying each companion by its boundary weight , we will recover the correct number of companions . we must also adjust weights for the primary galaxies . following the method described in the previous section , we wish to give less weight to galaxies that are likely to have fewer observed companions . therefore , each primary galaxy is assigned a boundary weight @xmath207 = @xmath202 . we now consider galaxies which lie near the survey boundaries along the line of sight . if a primary galaxy lies close to the minimum or maximum redshift allowed , it is possible that we will miss companions because they lie just across this redshift boundary . in order to account correctly for this effect , one would need to model the velocity distribution of companions . as this requires several assumptions , we choose instead to exclude all companions that lie between a primary galaxy and its nearest redshift boundary , provided the boundary lies within @xmath149 of the primary galaxy . to account for this exclusion , we assume that the velocity distribution is symmetric along the line of sight . thus , as we will miss half of the companions for these galaxies , we assign a weight of @xmath208=2 to any companions found in the direction opposite to the boundary . we must also consider how to weight the primary galaxies themselves . clearly , primary galaxies close to the redshift boundaries will be expected to have half as many _ observed _ companions as other primary galaxies . to minimize the errors in computing the pair statistics , we assign these primary galaxies weights @xmath209=0.5 . to summarize , weights for companions in the secondary sample are given by @xmath210 @xmath211 while primary galaxies are assigned weights @xmath212 @xmath213 we will now perform a test to see if this weighting scheme achieves the desired effects . to do this , we will use flux - limited monte carlo simulations , for which the intrinsic density and clustering are fixed . therefore , the _ intrinsic _ pair statistics do not depend on redshift or luminosity . if the secondary sample weights are correct , the measured pair statistics will be the same everywhere ( within the measurement errors ) , regardless of redshift or luminosity . we will also check to see if the weights for the primary sample are correct . if they are , the errors on the pair statistics will be minimized , as desired . the flux - limited monte carlo simulations were generated in a similar manner to the simulations described in section [ ssrs2mr : mc ] ; however , a limiting apparent magnitude of @xmath29 was imposed . sample sizes of 8000 ( south ) and 4035 ( north ) were used , providing a good match to the overall density in ssrs2 . the resulting simulations are similar to ssrs2 in all respects , except for the absence of clustering . we have already established how the pair statistics depend on the choice of @xmath82 and @xmath81 . in the following analysis , we choose @xmath81=@xmath82=@xmath214 . in section [ ssrs2mr : clust ] , we outlined reasons for restricting the sample to a fixed range in absolute magnitude . here , we demonstrate how the chosen range affects @xmath0 and @xmath1 . for comparison , we also compute @xmath0 without normalizing to a specified range in absolute magnitude ( in this case , @xmath215=@xmath215=1 ) . this provides some insight into the behaviour of the traditional ( uncorrected ) pair fraction . these tests are most straightforward if the intrinsic pair statistics are the same everywhere in the enclosed volume . this is not quite true for these simulations , however . galaxies are distributed randomly within the enclosed _ co - moving _ volume . as a result , the physical density varies with redshift as @xmath216 . in addition , the volume element encompassed by the line - of - sight pair criterion @xmath108 varies with redshift as @xmath217 for @xmath21=0.5 . in order to have the simulations mimic a sample with universal pair statistics , we normalize the sample for these effects by weighting each galaxy by @xmath217 . we stress that this is done only for the monte carlo simulations . one should _ not _ apply either of these corrections to real redshift data . in figure [ ssrs2mr : figrmfaint ] , the pair statistics are computed for a range of @xmath164 . in addition , we compute @xmath0 without weighting by the luminosity function , to demonstrate the danger of ignoring this important correction . this statistic is directly analagous to the traditional ( uncorrected ) pair fraction used in the literature . it is clear that both @xmath0 and @xmath1 are independent of the choice of @xmath164 , within the errors . this verifies that we have correctly accounted for the biases introduced by the apparent magnitude limit . in contrast , the unweighted @xmath0 is seen to have a strong dependence on @xmath164 . as expected , it increases as @xmath164 becomes fainter , due to the increase in sample density . we stress that this does not happen with the normalized @xmath0 and @xmath1 statistics , because both are corrected to a fixed range in limiting absolute magnitude . finally , we demonstrate that the weighting scheme used for the primary sample ( [ ssrs2mr : w1 ] ) does in fact minimize errors in @xmath0 and @xmath1 . recall that the weighting used was the reciprocal of the weights for the secondary sample . here we will assume that @xmath218 and @xmath219 . in section [ ssrs2mr : w1 ] , justification was given for setting @xmath220=@xmath221 . here , we will allow @xmath220 to vary , in order to investigate empirically which value minimizes the errors . special cases of interest are @xmath220=0 ( no weighting ) and @xmath220=1 ( same weighting as _ secondary _ sample ) . the results are given in figure [ ssrs2mr : figw1 ] . the relative errors in @xmath0 and @xmath1 reach a minimum at @xmath222 , as expected . errors are @xmath10 40% larger if no weighting is used ( @xmath220=0 ) . for @xmath220=1 , errors are much larger , increasing by nearly a factor of 5 . while errors increase dramatically for @xmath223 , they change slowly around @xmath220=@xmath221 . clearly , @xmath220 = @xmath221 is an excellent choice . the first step in applying these techniques to a real survey of galaxies is to decide on a useful close pair definition . this involves imposing a maximum projected physical separation ( @xmath148 ) and , if possible , a maximum line - of - sight rest - frame velocity difference ( @xmath149 ) . the limits should be chosen so as to extract information on mergers in an optimal manner . this involves a compromise between the number and merging likelihood of pairs . while one should focus on companions which are most likely to be involved in mergers , a very stringent pair definition may yield a small and statistically insignificant sample . in previous close pair studies , the convention has been to set @xmath148 = 20 @xmath5 kpc . pairs with separations of @xmath224 20 @xmath5 kpc are expected to merge within 0.5 gyr ( e.g. , @xcite , @xcite ) . we note , however , that timescale estimates are approximate in nature , and have yet to be verified . in earlier work , it has not been possible to apply a velocity criterion , since redshift samples have been too small to yield useful pair statistics using only galaxies with measured redshifts . instead , all physical companions have been used , with statistical correction for optical contamination ( @xcite ) . with a complete redshift sample , we can improve on this . this can be seen by inspecting a plot of @xmath107 versus @xmath108 for the ssrs2 pairs , given in figure [ ssrs2mr : figrpdelv ] . by imposing a velocity criterion , we can eliminate optical contamination ; furthermore , we are able to concentrate on the physical pairs with the lowest relative velocities , and hence the greatest likelihood of merging . we can now use our large sample of low-@xmath4 pairs to shed new light on these issues . we will use images of these pairs in an attempt to determine how signs of interactions are related to pair separation . we begin by finding all 255 ssrs2 pairs with @xmath225 @xmath5 kpc , computing @xmath107 and @xmath108 for each . images for these pairs were extracted from the digitized sky survey . interactions were immediately apparent in some of these pairs , and the images were deemed to be of sufficient quality that a visual classification scheme would be useful . an interaction classification parameter ( i@xmath226 ) was devised , where i@xmath226=0 indicates that a given pair is `` definitely not interacting '' , and i@xmath226=10 indicates `` definitely interacting '' . in order to avoid a built - in bias , the classifier is not given the computed values of @xmath107 and @xmath108 . the classifier uses all visible information available ( tidal tails and bridges , distortions / asymmetries in member galaxies , apparent proximity , etc . ) . classifications were performed by three of us ( drp , rom , rgc ) , and the median classification was determined for each system . the results are presented in figure [ ssrs2mr : figpc ] . a clickable version of this plot , which allows the user to see the corresponding digitized sky survey image for each pair , is available at http://www.astro.utoronto.ca/@xmath10patton/ssrs2/ic . there are several important features in this plot . first , there is a clear correlation between @xmath227 and @xmath107 , with closer pairs exhibiting stronger signs of interactions . there are several interacting pairs with @xmath228 @xmath5 kpc . while these separations are fairly large , it is not surprising that there would be some early - stage mergers with these separations ( e.g. , barton , bromley , & geller 1998 ) . an excellent example of this phenomenon is the striking tail - bridge system arp 295a / b ( cf . @xcite ) , which has @xmath107 = 95 @xmath5 kpc . however , these systems clearly do not dominate ; almost all pairs with large separations have very low interaction classifications . the majority of pairs showing clear signs of interactions / mergers have @xmath229 @xmath5 kpc . there is also a clear connection with @xmath108 . pairs with @xmath230 km / s do not exhibit signs of interactions , with 61/63 ( 97% ) classified as @xmath231 . this indicates that interactions are most likely to be seen in low velocity pairs , as expected . we note , however , that there are very few optical pairs ( i.e. , small @xmath107 and large @xmath108 ) in this low redshift sample . at higher redshift , increased optical contamination may lead to difficulties in identifying interacting systems when the galaxies are close enough to have overlapping isophotes . clearly , it is necessary to have redshift information for both members of each pair if one is to exclude these close optical pairs . after close inspection of figure [ ssrs2mr : figpc ] , we decided on close pair criteria of @xmath148 = 20 @xmath5 kpc and @xmath149 = 500 km / s . a mosaic of some of these pairs is given in figure [ ssrs2mr : figim ] . in this regime , 31% ( 9/29 ) exhibit convincing evidence for interactions ( i@xmath226 @xmath232 ) , while 69% ( 20/29 ) show some indication of interactions ( i@xmath226 @xmath233 ) . furthermore , the vast majority ( 9/10 ) of pairs with clear signs of interactions ( i@xmath226 @xmath232 ) are found in this regime . these criteria appear to identify a sample of pairs which are likely to be undergoing mergers ; moreover , the resulting sample includes most of the systems classified as interacting . we also impose an inner boundary of @xmath107 = 5 @xmath5 kpc . this limit is chosen so as to avoid the confusion that is often present on the smallest scales . in this regime , it is often difficult to distinguish between small galaxies and sub - galactic units , particularly in merging systems . while we are omitting the most likely merger candidates , those at separations @xmath234 5 @xmath5 kpc are not expected to account for more than @xmath10 5% of the companions within 20 @xmath5 kpc . this expectation , which has yet to be verified , is based both on pair counts in _ hst _ imaging ( @xcite ) and on inward extrapolation of the correlation function ( @xcite ) . while this inner boundary will lead to a slight decrease in @xmath0 and @xmath1 , it should have no significant effect on estimates of merger / accretion rate evolution , provided the same restriction is applied to comparison samples at other redshifts . in the preceding sections , we have outlined techniques for measuring pair statistics in a wide variety of samples . we have demonstrated a robust method of applying this approach to flux - limited samples , accounting for redshift - dependent density changes and minimizing differences in clustering . we have also selected pair definitions that identify the most probable imminent mergers . we will now apply these techniques to the ssrs2 survey . as this is a complete redshift survey , redshifts are available for all close companions ; hence , for the first time , we will measure pair statistics using only close _ dynamical _ pairs . after limiting the analysis to a reasonable range in absolute magnitude , we compute @xmath0 and @xmath1 for the ssrs2 survey . in section [ ssrs2mr : clust ] , we emphasized the importance of restricting the sample in absolute magnitude , to minimize bias due to luminosity - dependent clustering . for ssrs2 , we first impose a bright limit of @xmath163 = @xmath235 . all galaxies brighter than this are hereafter excluded from the analysis . this allows us to avoid the most luminous galaxies , which are probably the most susceptible to luminosity - dependent clustering ; however , this reduces the size of the sample by only 0.5% . we also impose a faint absolute magnitude limit of @xmath164 = @xmath236 , which results in the exclusion of intrinsically faint galaxies at @xmath237 . this guards against the possibility that these intrinsically faint galaxies are clustered differently than the bulk of the galaxies in the sample . this pruning of the sample is illustrated in figure [ ssrs2mr : figmbz ] . these restrictions allow us to minimize concerns about luminosity - dependent clustering while retaining 90% of the sample . the final results are insensitive to these particular choices ( see section [ ssrs2mr : sense ] ) . the above limits in absolute magnitude , along with the flux limit , define the usable sample of galaxies . in order to compute pair statistics , we must also normalize the measurements to a given range in absolute magnitude , for both the primary and secondary samples . the mean limiting absolute magnitude of the primary sample , weighted according to section [ ssrs2mr : w1 ] , is @xmath140 = @xmath238 . for convenience , we set @xmath239 = @xmath240 ( we will compute pair statistics for @xmath241 in the following section ) . for reference , we note that this corresponds to @xmath242=@xmath243 at @xmath4=0.017 . as we are dealing with a complete redshift sample , we set @xmath244=@xmath239 in order to use all of the available information . finally , as we have limited the sample using @xmath163 = @xmath235 , this will be used in conjunction with @xmath239 to derive pair statistics for galaxies with @xmath11 . using these parameters , we identified all close companions in ssrs2 . the north sample yielded 27 companions , and 53 were found in the south , giving a total of 80 . we emphasize that it is _ companions _ that are counted , rather than pairs ; hence , if both members of a pair fall within the primary sample , the pair will usually yield 2 companions . a histogram of companion absolute magnitudes is given in figure [ ssrs2mr : figlh ] . this plot shows that 90% of the companions we observe in our flux - limited sample fall in the range @xmath11 . hence , galaxies with @xmath245 do not dominate the sample . tables [ ssrs2mr : tabcpn ] and [ ssrs2mr : tabcps ] give complete lists of close aggregates ( pairs and triples ) for ssrs2 north and south respectively . these systems contain all companions used in the computation of pair statistics . these tables list system i d , number of members , @xmath107 ( @xmath5 kpc ) , @xmath108 ( km / s ) , ra ( 1950.0 ) , dec ( 1950.0 ) , and recession velocity ( km / s ) . dss images for these systems were given earlier in figure [ ssrs2mr : figim ] . using this sample of companions , the pair statistics were computed . the results are given in table [ ssrs2mr : tabstats ] . errors were computed using the jackknife technique . results from the two subsamples were combined , weighting by jackknife errors , to give @xmath246 and @xmath247 at @xmath4 = 0.015 . results from the two subsamples agree within the quoted 1@xmath248 errors . to facilitate future comparison with other samples , we also generate pair statistics spanning the range @xmath241 ( see table [ ssrs2mr : tabm2 ] ) . we note , however , that while we account for changes in number and luminosity density over this luminosity range ( using lf weights described in section [ ssrs2mr : flux ] ) , there is no correction for changes in clustering . hence , our statistics should be considered most appropriate for @xmath239 = @xmath249 , and more approximate in nature at brighter and fainter levels . the results in table [ ssrs2mr : tabm2 ] indicate that @xmath0 increases by a factor of 5 between @xmath239 = @xmath214 and @xmath239 = @xmath236 , resulting solely from an increase in mean number density . the change in @xmath1 is less pronounced , with an increase by a factor of 2 over the same luminosity range . these substantial changes in both statistics emphasize the need to specify @xmath81 when computing pair statistics and comparing results from different samples . in addition , the smaller change in @xmath1 is indicative of the benefits of using a luminosity statistic such as @xmath1 , which is more likely to converge as one goes to fainter luminosities ( see section [ ssrs2mr : mc ] ) . @xmath1 will always converge faster than @xmath0 , thereby reducing the sensitivity to @xmath81 . furthermore , it is possible to retrieve most of the relevant luminosity information without probing to extremely faint levels . for example , for the ssrs2 lf , 70% of the total integrated luminosity density is sampled by probing down to @xmath239 = @xmath249 . to first order , the same will be true for @xmath1 . going 2 magnitudes fainter would increase the completeness to 95% . while we are currently unable to apply pair statistics down to these faint limits , this will be pursued when deeper surveys become available . in this section , we explore the effects of choosing different survey parameters . earlier in this study , we demonstrated that @xmath0 and @xmath1 are insensitive to the choice of survey limits in absolute magnitude , provided clustering is independent of luminosity and the pair statistics are normalized correctly . here , we test this hypothesis empirically . first , we compute the pair statistics for a range in @xmath164 , normalizing the statistics to @xmath11 in each case . figure [ ssrs2mr : figmfaint ] demonstrates a possible trend of decreasing pair statistics with fainter @xmath164 . this trend , however , is significant only for the brightest galaxies ( @xmath250 ) . this is consistent with the findings of willmer et al . ( 1998 ) , who measure an increase in clustering for bright galaxies in ssrs2 , on scales of @xmath251 @xmath5 mpc . for fainter @xmath164 , there is no significant dependence . the pair statistics vary by @xmath10 5% over the range @xmath252 , which is well within the error bars . therefore , we conclude that our choice of @xmath164 = @xmath236 has a negligible effect on @xmath0 and @xmath1 . this implies that , to first order , clustering is independent of luminosity within this sample . next , we investigate how the pair statistics depend on our particular choices of @xmath148 and @xmath149 , which comprise our definition of a close companion . first , we compute pair statistics for 10 @xmath5 kpc@xmath253@xmath5 kpc , with @xmath254 km / s . results are given in figure [ ssrs2mr : figrp ] . this plot indicates a smooth increase in both statistics with @xmath148 . this trend is expected from measurements of the galaxy cf . the cf is commonly expressed as a power law of the form @xmath255 , with @xmath256=1.8 ( @xcite ) . integration over this function yields pair statistics that vary as @xmath257 , which is in good agreement with the trend found in figure [ ssrs2mr : figrp ] . from this plot , it also appears likely that there are systematic differences between the two subsamples . this is hardly surprising , since there are known differences in density between the subsamples , and it is likely that there are non - negligible differences in clustering as well . this cosmic variance is not currently measurable on the smaller scales ( @xmath258 @xmath5 kpc ) relevant to our main pair statistics . hence , we choose to ignore these differences for now . however , these field - to - field variations are certain to add some systematic error to our quoted pair statistics . we also compute pair statistics for a range in @xmath149 . this is done first for @xmath259 @xmath5 kpc , showing the relative contributions at different velocities to the main pair statistics quoted in this paper . we also compute statistics using @xmath260@xmath5 kpc , in order to improve the statistics . results are given in figure [ ssrs2mr : figrl ] . several important conclusions may be drawn from this plot . first , at small velocities ( @xmath261 km / s ) , both pair statistics increase with @xmath149 , as expected . this simply indicates that one continues to find additional companions as the velocity threshold increases . secondly , it appears that our choice of @xmath149 was a good one . the @xmath258 @xmath5 kpc pair statistics increase very little beyond @xmath254 km / s , while the contamination due to non@xmath67merging pairs would continue to increase ( see figure [ ssrs2mr : figrpdelv ] ) . moreover , as both pair statistics flatten out at around @xmath254 km / s , small differences in the velocity distributions of different samples should not result in large differences in their pair statistics . finally , for @xmath262 @xmath5 kpc , the pair statistics continue to increase out to @xmath263 km / s . this indicates an increase in velocity dispersion at these larger separations . this provides additional confirmation that one is less likely to find low@xmath67velocity pairs at larger separations , thereby implying that mergers should also be less probable . all published estimates of the local pair fraction have been hindered by small sample sizes and a lack of redshifts . in addition , as demonstrated throughout this paper , the traditional pair fraction is not a robust statistic , particularly when applied to flux - limited surveys . the new statistics introduced in this paper , along with careful accounting for selection effects such as the flux limit , yield the first secure measures of pair statistics at low redshift . therefore , strictly speaking , the results in this paper can not be compared directly with earlier pair statistics . however , it is possible to check for general consistency in results , and we will attempt to do so . as discussed in section [ ssrs2mr : background ] , patton et al . ( 1997 ) estimated the local pair fraction to be @xmath264 , using the ugc catalog . the patton et al . ( 1997 ) estimate was based on a flux - limited sample with @xmath23 , and a mean redshift of @xmath4=0.0076 . this corresponds roughly to an average limiting absolute magnitude of @xmath140 = @xmath265 . loosely speaking , this is analogous to @xmath81 . the pair definition used in their estimate was @xmath258 @xmath5 kpc , with no @xmath108 criterion . @xmath0 may be interpreted as an approximation to the traditional pair fraction , provided the relative proportion of triples is small . we recompute the ssrs2 pair statistics , using @xmath266 km / s in an attempt to match the results that would be found using no @xmath108 criterion ( see figure [ ssrs2mr : figrl ] ) . we find @xmath267 . this implies a local pair fraction of @xmath268 . this value is somewhat smaller than the earlier result , with larger errors . we strongly emphasize that , while these results are broadly consistent , we would not expect excellent agreement , due to the improved techniques used in this study . the quantities @xmath0 and @xmath1 are practical measures of the average numbers and luminosities of companions with relatively high merging probabilities . however , the ambiguity of redshift space is such that some of these companions can be entirely safe from ever merging . that is , @xmath270 km / s can correspond either to two galaxies at a small physical separation with a large infall velocity , or , to two galaxies at a separation of 5@xmath5 mpc , with no relative peculiar velocity . in order to transform from @xmath271 ( and @xmath272 ) to an estimate of the incidence of mergers , we must determine what fraction of our close companions have true 3-dimensional physical separations of @xmath273 @xmath5 kpc . this quantity , which we refer to as @xmath274 , has been discussed previously by yee and ellingson ( 1995 ) . this quantity needs to be evaluated based on the small separation clustering and kinematics of the galaxy population , @xmath275 , @xmath256 , and @xmath276 , the parameters @xmath108 and @xmath148 , and the projected separation at which two galaxies `` optically overlap '' in the image . the quantity @xmath274 is then evaluated with a triple integral , first placing the correlation function into redshift space , then integrating over projected and velocity separation . for reasonable choices of pair selection parameters , the outcome is fairly stable at @xmath277 . the most important parameter is @xmath278 , which should lie in the range of 2 to 4 . if this parameter is too small , physical pairs will be missed ; if it is too large , too many distant companions will be incorporated . other reasonable choices include setting the ratio of the overlap separation to maximum pair separation to be at least three , and the ratio of the maximum pair separation to correlation length to be at least a factor of 30 . we will take @xmath274=0.5 to be the best estimate currently available . we can now estimate the merger fraction ( @xmath279 ) at the present epoch . in this study , we have found @xmath271=0.0226@xmath280 . as most companions are found in pairs , rather than triplets or higher order n - tuples , this is comparable to the fraction of galaxies in close pairs . from our estimate of @xmath274 , we infer that half of these galaxies are in merging systems , yielding @xmath281 . this implies that approximately 1.1% of @xmath11 galaxies are undergoing mergers at the present epoch . we stress here that this result applies only to galaxies within the specified absolute magnitude limits . probing to fainter luminosities would cause @xmath279 to increase substantially . in addition , this result applies only to the close companions defined in this analysis . clearly , by modifying this definition ( and therefore changing the typical merger timescale under consideration ) , the merger fraction would also be certain to change . we now have an idea of how prevalent ongoing mergers are at the present epoch . in order to relate this result to the overall importance of mergers , we must estimate the merger timescale ( @xmath45 ) . we will use the properties of our ssrs2 pairs to estimate the mean dynamical friction timescale for pairs in our sample . following binney and tremaine ( 1987 ) , we assume circular orbits and a dark matter density profile given by @xmath282 . the dynamical friction timescale @xmath283 ( in gyr ) is given by @xmath284 where @xmath96 is the initial physical pair separation in kpc , @xmath285 is the circular velocity in km / s , @xmath57 is the mass ( @xmath286 ) , and @xmath287 is the coulomb logarithm . we estimate @xmath96 and @xmath285 using the pairs in tables [ ssrs2mr : tabcpn ] and [ ssrs2mr : tabcps ] . the mean projected separation is @xmath288@xmath5 kpc . as our procedure already includes a correction from projected separation ( @xmath107 ) to 3-dimensional separation ( @xmath96 ) , we take @xmath289 . assuming @xmath20=0.7 , this leads to @xmath290 kpc . the mean line of sight velocity difference is @xmath291 km / s . we assume the velocity distribution is isotropic , which implies that @xmath292 km / s . the mean absolute magnitude of companions is @xmath293 ( see figure [ ssrs2mr : figlh ] ) . we assume a representative estimate of the galaxy mass - to - light ratio of @xmath294 , yielding a mean mass of @xmath295 . finally , dubinski , mihos , and hernquist ( 1999 ) estimate @xmath296 , which fits the orbital decay of equal mass mergers seen in simulations . using equation [ ssrs2mr : eqntfric ] , we find @xmath297 gyr . we caution that this estimate is an approximation , and is averaged over systems with a wide range in merger timescales . nevertheless , we will take @xmath45 = 0.5 gyr as being representative of the merger timescale for the pairs in our sample . now that we have estimated the present epoch merger fraction and the merger timescale , we will attempt to ascertain what fraction of present galaxies have undergone mergers in the past . these galaxies can be classified as merger remnants ; hence , we will refer to this fraction as the remnant fraction ( @xmath299 ) . we begin by imagining the state of affairs at a lookback time of @xmath300 . suppose the merger fraction at the corresponding redshift is given by @xmath301 . in the time interval between then and the present , a fraction @xmath301 of galaxies will undergo mergers , yielding @xmath302 merger remnants . therefore , the remnant fraction at the present epoch is given by @xmath303 similarly , if we extend this to a lookback time of @xmath304 , where @xmath155 is an integer , then the remnant fraction is given by @xmath305 where @xmath109 corresponds to a lookback time of @xmath306 . we now make the simple assumption that the merger rate does not change with time . in this case , our present epoch estimate of the merger fraction holds at all redshifts , giving @xmath301=0.011 . in order to convert between redshift and lookback time , we must specify a cosmological model . we assume a hubble constant of @xmath20=0.7 . for simplicity , we assume @xmath21=0.5 . therefore , @xmath307 . using our merger timescale estimate of @xmath45=0.5 gyr , we can now investigate the cumulative effect of mergers . with the chosen cosmology , @xmath4=1 corresponds to a lookback time of @xmath10 6 gyr , or 12@xmath45 ( @xmath155=12 ) . with this lookback time , equation [ ssrs2mr : eqnremz ] yields @xmath299=0.066 . this implies that @xmath10 6.6% of galaxies with @xmath11 have undergone mergers since @xmath308 . if the mergers taking place in our sample produce elliptical galaxies , it is worthwhile comparing the remnant fraction to the elliptical fraction ( cf . toomre 1977 ) . the elliptical fraction for bright field galaxies is generally found to be about 10% ( e.g. , dressler 1980 , postman and geller 1984 ) . this result is broadly consistent with the remnant fraction found in this study . while our estimate of the remnant fraction is based on our statistically secure measurement of @xmath271 , it also relies on fairly crude assumptions regarding the merger fraction and merger timescale . in particular , the merger rate has been assumed to be constant . there is no physical basis for this assumption ; in fact , a number of studies have predicted a rise in the merger rate with redshift . if this is true , we will have underestimated the remnant fraction , and the relative importance of mergers . in a future paper ( patton et al . 2000 ) , we will address this issue by investigating how the merger rate changes with redshift . we have introduced two new pair statistics , @xmath0 and @xmath1 , which are shown to be related to the galaxy merger and accretion rates respectively . using monte carlo simulations , these statistics are found to be robust to the redshift - dependent density changes inherent in flux - limited samples ; this represents a very significant improvement over all previous estimators . in addition , we provide a clear prescription for relating @xmath0 and @xmath1 to the galaxy cf and lf , enabling straightforward comparison with measurements on larger scales . these statistics are applied to the ssrs2 survey , providing the first statistically sound measurements of pair statistics at low redshift . for an effective range in absolute magnitude of @xmath11 , we find @xmath246 at @xmath4=0.015 , implying that @xmath10 2.3% of these galaxies have companions within a projected physical separation of 5 @xmath5 kpc @xmath6 20 @xmath5 kpc and 500 km / s along the line of sight . if this pair statistic remains fixed with redshift , simple assumptions imply that @xmath10 6.6% of present day galaxies with @xmath11 have undergone mergers since @xmath4=1 . for our luminosity statistic , we find @xmath309 . this statistic gives the mean luminosity in companions , per galaxy . both of these statistics will serve as local benchmarks in ongoing and future studies aimed at detecting redshift evolution in the galaxy merger and accretion rates . it is our hope that these techniques will be applied to a wide range of future redshift surveys . as we have demonstrated , one must carefully account for differences in sampling effects between pairs and field galaxies . this will be of increased importance when applying pair statistics at higher redshift , as @xmath22-corrections , luminosity evolution , band - shifting effects , and spectroscopic completeness have to be properly accounted for . the general approach outlined in this paper indicates the steps that must be taken to allow for a fair comparison between disparate surveys at low and high redshift . these techniques are applicable to redshift surveys with varying degrees of completeness , and are also adaptable to redshift surveys with additional photometric information , such as photometric redshifts , or even simply photometric identifications . finally , this approach can be used for detailed studies of both major and minor mergers . we wish to thank all members of the ssrs2 colloboration for their work in compiling the ssrs2 survey , and for making these data available in a timely manner . digitized sky survey images used in this research were obtained from the canadian astronomical data centre ( cadc ) , and are based on photographic data of the national geographic society palomar observatory sky survey ( ngs - poss ) . the digitized sky surveys were produced at the space telescope science institute under u.s . government grant nag w-2166 . this work was supported by the natural sciences and engineering research council of canada , through research grants to r.g.c . and c.j.p .. alonso , m. v. , da costa , l. n. , pellegrini , p. s. , & kurtz , m. j. 1993 , , 106 , 676 alonso , m. v. , da costa , l. n. , latham , d. w. , pellegrini , p. s. , & milone , a. e. 1994 , , 108 , 1987 bahcall , s. r. & tremaine , s. 1988 , , 326 , l1 barnes , j. e. 1988 , , 331 , 699 barnes , j. e. & hernquist , l. 1992 , , 30 , 705 barton , e. j. , bromley , b. c. , & geller , m. j. 1998 , , 511 , l25 benoist , c. , maurogordato , s. , da costa , l. n. , cappi , a. , & schaeffer , r. 1996 , , 472 , 452 binney , j. , and tremaine , s. 1987 , in galactic dynamics ( princeton : princeton university press ) burkey , j. m. , keel , w. c. , windhorst , r. a. , & franklin , b. e. 1994 , , 429 , l13 carlberg , r. g. , pritchet , c. j. , & infante , l. 1994 , , 435 , 540 courteau , s. , & van den bergh , s. 1999 , , 118 , 337 da costa , l. n. , willmer , c. n. a. , pellegrini , p. s. , chaves , o. l. , rit , c. , maia , m. a. g. , geller , m. j. , latham , d. w. , kurtz , m. j. , huchra , j. p. , ramella , m. , fairall , a. p. , smith , c. , & lpari , s. 1998 , , 116 , 1 davis , m. , & peebles , p. j. e. 1983 , , 267 , 465 dressler , a. 1980 , , 236 , 351 dubinski , j. , mihos , j. c. , and hernquist , l. 1999 , , submitted efron , b. 1981 , biometrika , 68 , 589 efron , b. , & tibshirani , r. 1986 , statistical science , 1 , 54 ellis , r. s. 1997 , , 35 , 389 hibbard , j. e. , & van gorkom , j. h. 1996 , , 111 , 655 koo , d. , & kron , r. 1992 , , 30 , 613 le fvre , o. , abraham , r. , lilly , s. j. , ellis , r. s. , brinchmann , j. , tresse , l. , colless , m. , crampton , d. , glazebrook , k. , hammer , f. , & broadhurst , t. 1999 , , in press lin , h. , yee , h. k. c. , carlberg , r. g. , morris , s. l. , sawicki , m. , patton , d. r. , wirth , g. d. , & shepherd , c. w. 1999 , , 518 , 533 loveday , j. , maddox , s. j. , efstathiou , g. , & peterson , b. a. 1995 , , 442 , 457 marzke , r. o. , geller , m. j. , da costa , l. n. , & huchra , j. p. 1995 , , 110 , 477 marzke , r. o. , da costa , l. n. , pellegrini , p. s. , willmer , c. n. a. , & geller , m. j. 1998 , , 503 , 617 madau , p. , pozzetti , l. , & dickinson , m. 1998 , , 498 , 106 nilson , p. 1973 , uppsala general catalog of galaxies ( uppsala : royal society of sciences of uppsala ) patton , d. r. , pritchet , c. j. , yee , h. k. c. , ellingson , e. , & carlberg , r. g. 1997 , , 475 , 29 patton , d. r. , pritchet , c. j. , carlberg , r. g. , marzke , r. o. , yee , h. k. c. , ellingson , e. , hall , p. b. , lin , h. , morris , s. l. , sawicki , m. , schade , d. , shepherd , c. w. , & wirth , g. d. 2000 , in preparation peebles , p. j. e. 1980 , _ the large - scale structure of the universe _ ( princeton university press : princeton ) postman , m. , & geller , m. j. 1984 , , 281 , 95 toomre , a. , & toomre , j. 1972 , , 179 , 623 toomre , a. 1977 , in evolution of galaxies and stellar populations , ed . b. m. tinsley , & r. b. larson ( new haven : yale obs . ) , p. 401 vogeley , m. s. 1993 , phd dissertation , harvard university , cambridge , ma usa williams , r. e. et al . 1996 , , 112 , 1335 willmer , c. n. a. , da costa , l. n. , pellegrini , p. s. 1998 , , 115 , 869 woods , d. , fahlman , g. g. , & richer , h. b. 1995 , , 454 , 32 yee , h. k. c. , & ellingson , e. 1995 , , 445 , 37 zepf , s. e. , & koo , d. c. 1989 , , 337 , 34

the galaxy merger and accretion rates , and their evolution with time , provide important tests for models of galaxy formation and evolution . close pairs of galaxies are the best available means of measuring redshift evolution in these quantities . in this study , we introduce two new pair statistics , which relate close pairs to the merger and accretion rates . we demonstrate the importance of correcting these ( and other ) pair statistics for selection effects related to sample depth and completeness . in particular , we highlight the severe bias that can result from the use of a flux - limited survey . the first statistic , denoted @xmath0 , gives the number of companions per galaxy , within a specified range in absolute magnitude . @xmath0 is directly related to the galaxy merger rate . the second statistic , called @xmath1 , gives the total luminosity in companions , per galaxy . this quantity can be used to investigate the mass accretion rate . both @xmath0 and @xmath1 are related to the galaxy correlation function @xmath2 and luminosity function @xmath3 in a straightforward manner . both statistics have been designed with selection effects in mind . we outline techniques which account for various selection effects , and demonstrate the success of this approach using monte carlo simulations . if one assumes that clustering is independent of luminosity ( which is appropriate for reasonable ranges in luminosity ) , then these statistics may be applied to flux - limited surveys . these techniques are applied to a sample of 5426 galaxies in the ssrs2 redshift survey . this is the first large , well - defined low-@xmath4 survey to be used for pair statistics . using close ( 5 @xmath5 kpc @xmath6 20 @xmath5 kpc ) dynamical ( @xmath7 km / s ) pairs , we find @xmath8 and @xmath9 at @xmath4=0.015 . these are the first secure estimates of low - redshift pair statistics , and they will provide local benchmarks for ongoing and future pair studies . if @xmath0 remains fixed with redshift , simple assumptions imply that @xmath10 6.6% of present day galaxies with @xmath11 have undergone mergers since @xmath4=1 . when applied to redshift surveys of more distant galaxies , these techniques will yield the first robust estimates of evolution in the galaxy merger and accretion rates . @#1 @@size@false # 1 @@size@false # 1 @@size@false # 1 @mathfonts @#1#2#3 # 1#2#1@xmath12roman#2 2#1#1@xmath12 = =

we study a kazhdan lusztig - like correspondence between a vertex - operator algebra and a quantum group in the case where the conformal field theory associated with the vertex - operator algebra is logarithmic . in its full extent , the kazhdan lusztig correspondence comprises the following claims : 1 . [ item : equiv - cat ] a suitable representation category of the vertex - operator algebra is equivalent to the category of finite - dimensional quantum group representations . [ item : grring ] the fusion algebra associated with the conformal field theory coincides with the quantum - group grothendieck ring . [ item : sliiz ] the modular group representation associated with conformal blocks on a torus is equivalent to the modular group representation on the center of the quantum group . such full - fledged claims of the kazhdan lusztig correspondence @xcite have been established for affine lie algebras at a negative integer level and for some other algebras `` in the negative zone . '' but in the positive zone , the correspondence holds for rational conformal field models @xcite ( such as @xmath6-minimal virasoro models and @xmath7 models with @xmath8 ) with certain `` corrections . '' notably , the semisimple fusion in rational models corresponds to a semisimple quasitensor category obtained as the quotient of the representation category of a quantum group by the tensor ideal of indecomposable tilting modules . taking the quotient ( `` neglecting the negligible '' in @xcite , cf . @xcite ) makes the correspondence somewhat indirect ; in principle , a given semisimple category can thus correspond to different quantum groups . remarkably , the situation is greatly improved for the class of logarithmic ( nonsemisimple ) models considered in this paper , where the quantum group itself ( not only a quasitensor category ) can be reconstructed from the conformal field theory data . in this paper , we are mostly interested in claims [ item : sliiz ] and [ item : grring ] . claim [ item : sliiz ] of the kazhdan lusztig correspondence involves the statement that the counterpart of the quantum group center on the vertex - operator algebra side is given by the endomorphisms of the identity functor in the category of vertex - operator algebra representations . this object morally , the `` center '' of the associated conformal field theory can be identified with the finite - dimensional space @xmath9 of conformal blocks on a torus . in the semisimple case , @xmath9 coincides with the space of conformal field theory characters , but in the nonsemisimple case , it is not exhausted by the characters , although we conveniently call it the ( space of ) extended characters ( all these are functions on the upper complex half - plane ) . the space @xmath9 carries a modular group representation , and the kazhdan lusztig correspondence suggests looking for its relation to the modular group representation on the quantum group center . we recall that an @xmath0-representation can be defined for a class of quantum groups ( in fact , for ribbon quasitriangular categories ) @xcite . remarkably , the two @xmath0-representations ( on @xmath9 and on the quantum group center @xmath10 ) are indeed equivalent for the logarithmic conformal field theory models studied here . the details of our study and the main results are as follows . on the vertex - operator algebra side , we consider the `` triplet '' w - algebra @xmath11 that was studied in @xcite in relation to the logarithmic @xmath4 models of conformal field theory with @xmath12 . the algebra @xmath11 has @xmath3 irreducible highest - weight representations @xmath13 , @xmath14 , which ( in contrast to the case of rational conformal field models ) admit nontrivial extensions among themselves ( @xmath15 is nondiagonalizable on some of extensions , which makes the theory logarithmic ) . the space @xmath9 in the @xmath4-model is @xmath16-dimensional ( cf . @xcite ) . on the quantum - group side , we consider the _ restricted _ ( `` baby '' in a different nomenclature ) quantum group @xmath2 at the primitive @xmath3th root of unity @xmath17 . we define it in * [ sec : cas ] * below , and here only note the key relations @xmath18 , @xmath19 ( with @xmath20 then being central ) . it has @xmath3 irreducible representations and a @xmath16-dimensional center ( prop . * [ prop - center ] * below ) . the center @xmath10 of @xmath2 is endowed with an @xmath0-representation constructed as in @xcite , even though @xmath2 is not quasitriangular @xcite ( the last fact may partly explain why @xmath2 is not as popular as the _ small _ quantum group ) . [ thm:1.1 ] the @xmath0-representations on @xmath9 and on @xmath10 are equivalent . thus , claim [ item : sliiz ] of the kazhdan lusztig correspondence is fully valid for @xmath11 and @xmath2 at @xmath21 . we let @xmath1 denote the @xmath0-representation in the theorem . regarding claim [ item : grring ] , we first note that , strictly speaking , the fusion for @xmath11 , understood in its `` primary '' sense of calculation of the coinvariants , has been derived only for @xmath22 @xcite . in rational conformal field theories , the verlinde formula @xcite allows recovering fusion from the modular group action on characters . in the @xmath4 logarithmic models , the procedure proposed in @xcite as a nonsemisimple generalization of the verlinde formula allows constructing a commutative associative algebra from the @xmath0-action on the @xmath11-characters . this algebra @xmath23 on @xmath3 elements @xmath24 ( @xmath25 , @xmath14 ) is given by @xmath26 where @xmath27 for @xmath22 , this algebra coincides with the fusion in @xcite , and we believe that it is indeed the fusion for all @xmath28 . our next result in this paper strongly supports this claim , setting it in the framework of the kazhdan lusztig correspondence between @xmath11 and @xmath2 at @xmath29 . [ thm:1.2 ] let @xmath29 . under the identification of @xmath24 , @xmath30 , @xmath31 , with the @xmath3 irreducible @xmath2-representations , the algebra @xmath23 in is the grothendieck ring of @xmath32 . we emphasize that the algebras are isomorphic as fusion algebras , i.e. , including the identification of the respective preferred bases given by the irreducible representations . the procedure in @xcite leading to fusion is based on the following structure of the @xmath0-representation @xmath1 on @xmath9 in the @xmath4 model : @xmath33 here , @xmath34 is a @xmath35-dimensional @xmath0-representation ( actually , on characters of a lattice vertex - operator algebra ) , @xmath36 is a @xmath37-dimensional @xmath0-representation ( actually , the representation on the unitary @xmath38-characters at the level @xmath39 ) , and @xmath40 is the standard two - dimensional @xmath0-representation . equivalently , is reformulated as follows . we have two @xmath0-representations @xmath41 and @xmath42 on @xmath9 in terms of which @xmath1 factors as @xmath43 @xmath44 and which commute with each other , @xmath45 ; moreover , @xmath41 restricts to the @xmath3-dimensional space of the @xmath11-characters . in view of theorem * [ thm:1.1 ] * , this structure of the @xmath0-representation is reproduced on the quantum - group side : there exist @xmath0-representations @xmath41 and @xmath42on the center @xmath10 of @xmath2 in terms of which the representation in @xcite factors . remarkably , these representations @xmath41 and @xmath42 on @xmath10 can be constructed in intrinsic quantum - group terms , by modifying the construction in @xcite . we recall that the @xmath46 generator of @xmath0 is essentially given by the ribbon element @xmath47 , and the @xmath48 generator is constructed as the composition of the radford and drinfeld mappings . that @xmath41 and @xmath42 exist is related to the multiplicative jordan decomposition of the ribbon element @xmath49 , where @xmath50 is the semisimple part and @xmath51 is the unipotent ( one - plus - nilpotent ) part . then @xmath50 and @xmath51 yield the respective `` @xmath52''-generators @xmath53 and @xmath54 . the corresponding `` @xmath55''-generators @xmath56 and @xmath57 are constructed by deforming the radford and drinfeld mappings _ respectively _ , as we describe in sec . * [ two - rep - on - z ] * below . we temporarily call the @xmath0-representations @xmath41 and @xmath42 the representations _ associated with _ @xmath50 and @xmath51 . [ thm : modular-2 ] let @xmath49 be the multiplicative jordan decomposition of the @xmath2 ribbon element ( with @xmath50 being the semisimple part ) and let @xmath41 and @xmath42 be the respective @xmath0-representations on @xmath10 associated with @xmath50 and @xmath51 . then 1 . @xmath58 for all @xmath59 , 2 . @xmath60 for all @xmath61 , and 3 . the representation @xmath41 restricts to the image of the grothendieck ring in the center . the image of the grothendieck ring in this theorem is under the drinfeld mapping . the construction showing how the representations @xmath41 and @xmath42 on the center are derived from the jordan decomposition of the ribbon element is developed in sec . * [ two - rep - on - z ] * only for @xmath2 , but we expect it to be valid in general . the multiplicative jordan decomposition of the ribbon element gives rise to @xmath0-representations @xmath41 and @xmath42 with the properties as in theorem * [ thm : modular-2 ] * for any factorizable ribbon quantum group . regarding claim [ item : equiv - cat ] of the kazhdan lusztig correspondence associated with the @xmath4 logarithmic models , we only formulate a conjecture ; we expect to address this issue in the future , beginning with @xcite , where , in particular , the representation category is studied in great detail . in a sense , the expected result is more natural than in the semisimple@xmath62rational case because ( as in theorem * [ thm:1.2 ] * ) it requires no `` semisimplification '' on the quantum - group side . the category of @xmath11-representations is equivalent to the category of finite - dimensional @xmath2-representations with @xmath21 . from the reformulation of fusion in quantum - group terms ( explicit evaluation of the product in the image of the grothendieck ring in the center under the drinfeld mapping ) , we obtain a combinatorial corollary of theorem * [ thm:1.2 ] * ( see for the notation regarding @xmath5-binomial coefficients ) : [ lemma : the - identity ] for @xmath63 , there is the @xmath5-binomial identity @xmath64}{0pt}{}{n - i}{j}$}}}{{\genfrac{[}{]}{0pt}{}{n - i}{j } } } { { \genfrac{[}{]}{0pt}{}{n - i}{j}}}{{\genfrac{[}{]}{0pt}{}{n - i}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{i}{m - j}$}}}{{\genfrac{[}{]}{0pt}{}{i}{m - j } } } { { \genfrac{[}{]}{0pt}{}{i}{m - j}}}{{\genfrac{[}{]}{0pt}{}{i}{m - j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{i + j + s - n}{j}$}}}{{\genfrac{[}{]}{0pt}{}{i + j + s - n}{j } } } { { \genfrac{[}{]}{0pt}{}{i + j + s - n}{j}}}{{\genfrac{[}{]}{0pt}{}{i + j + s - n}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m - i - j + s'}{m - j}$}}}{{\genfrac{[}{]}{0pt}{}{m - i - j + s'}{m - j } } } { { \genfrac{[}{]}{0pt}{}{m - i - j + s'}{m - j}}}{{\genfrac{[}{]}{0pt}{}{m - i - j + s'}{m - j}}}}={}\\ { } = q^{2 m n } \sum_{\ell=0}^{\min(s , s ' ) } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n - \ell}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n - \ell}{m } } } { { \genfrac{[}{]}{0pt}{}{n - \ell}{m}}}{{\genfrac{[}{]}{0pt}{}{n - \ell}{m } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m + s + s ' - \ell - n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{m + s + s ' - \ell - n}{m } } } { { \genfrac{[}{]}{0pt}{}{m + s + s ' - \ell - n}{m}}}{{\genfrac{[}{]}{0pt}{}{m + s + s ' - \ell - n}{m}}}}. \end{gathered}\ ] ] the multiplication in algebra , which underlies this identity , is alternatively characterized in terms of chebyshev polynomials , see * [ prop : quotient ] * below . there are numerous relations to the previous work . the fundamental results in @xcite regarding the modular group action on the center of a drinfeld double can be `` pushed forward '' to @xmath2 , which is a ribbon quantum group . we note that in the standard setting @xcite , a ribbon hopf algebra is assumed to be quasitriangular . this is not the case with @xmath2 , but we keep the term `` ribbon '' with the understanding that @xmath2 is a subalgebra in a quasitriangular hopf algebra from which it inherits the ribbon structure , as is detailed in what follows . the structure , already implicit in @xcite , is parallel to the property conjectured in @xcite for the @xmath0-representation on the center of the _ small _ quantum group @xmath65 . albeit for a different quantum group , we extend the argument in @xcite by choosing the bases in the center that lead to a simple proof and by giving the underlying jordan decomposition of the ribbon element and the corresponding deformations of the radford and drinfeld mappings . the @xmath16-dimensional center of @xmath2 at @xmath17 the primitive @xmath3th root of unity is twice as big as the center of @xmath65 for @xmath17 the primitive @xmath28th root of unity ( for odd @xmath28 ) @xcite . we actually find the center of @xmath2 by studying the bimodule decomposition of the regular representation ( the decomposition of @xmath65 under the _ adjoint _ action has been the subject of some interest ; see @xcite and the references therein ) . there naturally occur indecomposable @xmath3-dimensional @xmath2-representations ( projective modules ) , which have also appeared in @xcite . on the conformal field theory side , the @xmath11 algebra was originally studied in @xcite , also see @xcite . this paper can be considered a continuation ( or a quantum - group counterpart ) of @xcite and is partly motivated by remarks already made there . that the quantum dimensions of the irreducible @xmath11-representations are dimensions of quantum - group representations was noted in @xcite as an indication of a quantum group underlying the fusion algebra derived there . for the convenience of the reader , we give most of the necessary reference to @xcite in sec . [ sec : cftetc ] and recall the crucial conformal field theory formulas there . in sec . [ sec : all - usl2 ] , we define the restricted quantum group @xmath2 , describe some classes of its representations ( most importantly , irreducible ) , and find its grothendieck ring . in sec . [ sec : new ] , we collect the facts pertaining to the ribbon structure and the structure of a factorizable hopf algebra on @xmath2 . there , we also find the center of @xmath2 in rather explicit terms . in sec . [ sec : sliiz - restr ] , we study @xmath0-representations on the center of @xmath2 and establish the equivalence to the representation in sec . [ sec : cftetc ] and the factorization associated with the jordan decomposition of the ribbon element . the appendices contain auxiliary or bulky material . in appendix [ app : hopf ] , we collect a number of standard facts about hopf algebras that we use in the paper . in appendix [ sec : double ] , we construct a drinfeld double that we use to derive the @xmath66-matrix and the ribbon element for @xmath2 . in appendix [ verma - proj - mod - base ] , we give the necessary details about indecomposable @xmath2-modules . the `` canonical '' basis in the center of @xmath2 is explicitly constructed in appendix [ app : center ] . as an elegant corollary of the description of the grothendieck ring in terms of chebyshev polynomials , we reproduce the formulas for the eigenmatrix in @xcite . appendix [ app : derivation ] is just a calculation leading to identity . we use the standard notation @xmath67 = { \mbox{\footnotesize$\displaystyle\frac{q^n - q^{-n}}{q - q^{-1}}$}},\quad n\in{\mathbb{z}},\quad [ n ] ! = [ 1][2]\dots[n],\quad n\in{\mathbb{n}},\quad[0]!=1\ ] ] ( without indicating the `` base '' @xmath5 explicitly ) and set @xmath68}{0pt}{}{m}{n}$}}}{{\genfrac{[}{]}{0pt}{}{m}{n } } } { { \genfrac{[}{]}{0pt}{}{m}{n}}}{{\genfrac{[}{]}{0pt}{}{m}{n}}}}= \begin{cases } 0 , & n<0\quad\text{or}\quad m - n<0,\\ { \mbox{\footnotesize$\displaystyle\frac{[m]!}{[n]!\,[m - n]!}$}}&\text{otherwise}. \end{cases}\ ] ] in referring to the root - of - unity case , we set @xmath69 for an integer @xmath70 . the @xmath28 parameter is as in sec . * [ sec : cftetc]*. for hopf algebras in general ( in the appendices ) and for @xmath2 specifically , we write @xmath71 , @xmath72 , and @xmath55 for the comultiplication , counit , and antipode respectively . some other conventions are as follows : * the quantum group center , * the space of @xmath5-characters ( see * [ sec : q - chars ] * ) , * the integral ( see * [ app : int ] * ) , * the cointegral ( see * [ app : int ] * ) , * the balancing element ( see * [ app : int ] * ) , * the ribbon element ( see * [ sec : ribbon ] * ) , * the @xmath66-matrix ( see * [ app : m ] * ; @xmath73 is used for @xmath2 and @xmath66 in general ) , * the drinfeld mapping @xmath74 ( see * [ sec : drpdef ] * ) , * the image of the irreducible @xmath2-representation @xmath13 in the center under the drinfeld mapping ( see * [ fusion - center ] * ) , * the radford mapping @xmath74 ( see * [ sec : radford - all ] * ) , * the image of the irreducible @xmath2-representation @xmath13 in the center under the radford mapping ( see * [ sec : radford - sl2 ] * ) , * irreducible @xmath2-representations ( see * [ subsec : irrep ] * ) ; in * [ sec : voa ] * , irreducible @xmath11-rerpesentations . * verma modules ( see * [ subsec : verma - mod ] * and * [ verma - mod - base ] * ) , * contragredient verma modules ( see * [ verma - mod - base ] * ) , * projective @xmath2-modules ( see * [ subsec : proj - mod ] * and * [ proj - mod - base ] * ) , * the @xmath5-character of a @xmath2-representation @xmath75 ( see * [ app : qch ] * ) , * the @xmath2 grothendieck ring ; @xmath76 is the grothendieck ring of a hopf algebra @xmath77 , * the grothendieck ring image in the center under the drinfeld mapping , * the grothendieck ring image in the center under the radford mapping . we write @xmath78 , @xmath79 , @xmath80 , etc . ( sweedler s notation ) in constructions like @xmath81 for a linear function @xmath82 , we use the notation @xmath83 , where @xmath84 indicates the position of its argument in more complicated constructions . we choose two elements generating @xmath0 as @xmath85 and @xmath86 and use the notation of the type @xmath48 , @xmath57 , @xmath56 , and @xmath46 , @xmath54 , @xmath53 , for these elements in various representations . logarithmic models of conformal field theory , of which the @xmath4-models are an example , were introduced in @xcite and were considered , in particular , in @xcite ( also see the references therein ) . such models are typically defined as kernels of certain screening operators . the actual symmetry of the theory is the maximal local algebra in this kernel . in the @xmath4-model , which is the kernel of the `` short '' screening operator , see @xcite , this is the w - algebra @xmath11 studied in @xcite . we briefly recall it in * [ sec : voa]*. in * [ mod - on - char ] * , we give the modular transformation properties of the @xmath11-characters and identify the @xmath16-dimensional @xmath0-representation on @xmath9 ( the space of extended characters ) . in * [ thm : r - decomp ] * , we describe the structure of this representation . following @xcite , we consider the vertex - operator algebra @xmath11the w - algebra studied in @xcite , which can be described in terms of a single free field @xmath87 with the operator product expansion @xmath88 . for this , we introduce the energy - momentum tensor @xmath89 with central charge @xmath90 , and the set of vertex operators @xmath91 with @xmath92 . let @xmath93 be the sum of fock spaces corresponding to @xmath94 for @xmath95 and @xmath96 ( see the details in @xcite ) . there exist two screening operators @xmath97 satisfying @xmath98\,{=}\,0 $ ] . we define @xmath11 as a maximal local subalgebra in the kernel of the `` short '' screening @xmath99 . the algebra @xmath11 is generated by the currents @xmath100,\quad\ ; w^+(z)=[s_+,w^0(z)]\ ] ] ( which are primary fields of dimension @xmath101 with respect to energy - momentum tensor ) . the algebra @xmath11 has @xmath3 irreducible highest - weight representations , denoted as @xmath102 and @xmath103 , @xmath96 ( the respective representations @xmath104 and @xmath105 in @xcite ) . the highest - weight vectors in @xmath102 and @xmath103 can be chosen as @xmath106 and @xmath107 respectively . it turns out that @xmath108 we now recall @xcite the modular transformation properties of the @xmath11-characters @xmath109 ( the respective characters @xmath110 and @xmath111 in @xcite ) , where @xmath15 is a virasoro generator , the zero mode of energy - momentum tensor . under the @xmath48-transformation of @xmath112 , these characters transform as @xmath113 - \sum_{s'=1}^{p-1}(-1)^{p+s+s'}{\mathfrak{q}}^{ss'}_{- } \varphi_{s'{\relax}}(\tau ) \biggr)\end{gathered}\ ] ] and @xmath114 + \smash[b]{\sum_{s'=1}^{p-1}}(-1)^{s+1 } { \mathfrak{q}}^{s ' s}_{- } \varphi_{s'{\relax}}(\tau)\biggr),\end{gathered}\ ] ] where @xmath115 , @xmath116 , and we introduce the notation @xmath117 the @xmath11-characters are in fact combinations of modular forms of different weights , and hence their modular transformations involve explicit occurrences of @xmath112 ; in the formulas above , @xmath112 enters only linearly , but much more complicated functions of @xmath112 ( and other arguments of the characters ) can be involved in nonrational theories , cf . @xcite . in the present case , because of the explicit occurrences of @xmath112 , the @xmath0-representation space turns out to be @xmath16-dimensional , spanned by @xmath118 , @xmath96 , and @xmath119 , @xmath120 . indeed , we have @xmath121 where for the future convenience we introduce a special notation for certain linear combinations of the characters : @xmath122 under the @xmath46-transformation of @xmath112 , the @xmath11-characters transform as @xmath123 where @xmath124 and hence @xmath125 we let @xmath9 denote this @xmath16-dimensional space spanned by @xmath118 , @xmath96 , and @xmath119 , @xmath120 . as noted in the introduction , @xmath9 is the space of conformal blocks on the torus , which is in turn isomorphic to the endomorphisms of the identity functor . let @xmath1 be the @xmath0-representation on @xmath9 defined by the above formulas . [ thm : r - decomp ] the @xmath0-representation on @xmath9 has the structure @xmath126 where @xmath34 and @xmath36 are @xmath0-representations of the respective dimensions @xmath127 and @xmath128 , and @xmath40 is the two - dimensional representation . this implies that there exist @xmath0-representations @xmath41 and @xmath42 on @xmath9 such that @xmath129 let @xmath34 be spanned by @xmath130 ( these are the characters of _ verma _ modules over @xmath11 ) . the formulas in * [ mod - on - char ] * show that @xmath34 is an @xmath0-representation ; namely , it follows that @xmath131 and @xmath132 where @xmath133 is another basis in @xmath34 . next , let @xmath134 be the space spanned by @xmath119 in ; another basis in @xmath134 is @xmath135 finally , let another @xmath136-dimensional space @xmath137 be spanned by @xmath138 in ; another basis in @xmath137 is given by @xmath139 equations then imply that @xmath140 and the @xmath46-transformations in eqs . are expressed as @xmath141 therefore , the representation @xmath1 has the structure @xmath142 , where @xmath143 is spanned by @xmath144 , @xmath120 . we now let @xmath145 and @xmath146 act on @xmath147 as @xmath148 and let @xmath149 and @xmath150 act as @xmath151 and @xmath152 it follows that under @xmath42 , we have the decomposition @xmath153 ( where @xmath154 is the trivial representation ) and under @xmath41 , the decomposition @xmath155 it is now straightforward to verify that @xmath41 and @xmath42 satisfy the required relations . 1 . up to some simple multipliers , @xmath42 is just the inverse matrix automorphy factor in @xcite and the restriction of @xmath41 to @xmath156 is the @xmath0-representation in @xcite that leads to the fusion algebra via a nonsemisimple generalization of the verlinde formula . @xmath36 is the @xmath0-representation realized in the @xmath157 minimal model @xcite . in sec . [ sec : sliiz - restr ] , the structure described in * [ thm : r - decomp ] * is established for the @xmath0-representation on the quantum group center . the version of the quantum @xmath158 that is kazhdan lusztig - dual to the @xmath4 conformal field theory model is the restricted quantum group @xmath2 at @xmath17 the primitive @xmath3th root of unity . we introduce it in * [ sec : cas ] * , consider its representations in * [ sec : repr ] * , and find its grothendieck ring in * [ sec : grring]*. the hopf algebra @xmath2 ( henceforth , at @xmath29 ) is generated by @xmath159 , @xmath160 , and @xmath161 with the relations @xmath162 and the hopf - algebra structure given by @xmath163={\mbox{\footnotesize$\displaystyle\frac{k - k^{-1}}{{\mathfrak{q}}-{\mathfrak{q}}^{-1}}$}},\\ \delta(e)={\boldsymbol{1}}\otimes e+e\otimes k,\quad \delta(f)=k^{-1}\otimes f+f\otimes{\boldsymbol{1}},\quad \delta(k)=k\otimes k,\\ \epsilon(e)=\epsilon(f)=0,\quad\epsilon(k)=1,\\ s(e)=-ek^{-1},\quad s(f)=-kf,\quad s(k)=k^{-1}.\end{gathered}\ ] ] the elements of the pbw - basis of @xmath2 are enumerated as @xmath164 with @xmath165 , @xmath166 , @xmath167 , and its dimension is therefore @xmath168 . [ [ section ] ] it follows ( e.g. , by induction ) that @xmath169}{0pt}{}{m}{r}$}}}{{\genfrac{[}{]}{0pt}{}{m}{r } } } { { \genfrac{[}{]}{0pt}{}{m}{r}}}{{\genfrac{[}{]}{0pt}{}{m}{r}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{s}$}}}{{\genfrac{[}{]}{0pt}{}{n}{s } } } { { \genfrac{[}{]}{0pt}{}{n}{s}}}{{\genfrac{[}{]}{0pt}{}{n}{s}}}}\\ * { } \times f^r e^{n - s } k^{r - m+j}{\otimes}f^{m - r } e^s k^{n - s+j}.\end{gathered}\ ] ] for @xmath2 , the right integral and the left right cointegral ( see the definitions in * [ app : int ] * ) are given by @xmath170 and @xmath171 where we choose the normalization as @xmath172!)^2}$}}\ ] ] for future convenience . next , simple calculation shows that the comodulus for @xmath2 ( see * [ app : int ] * ) is @xmath173 . this allows us to find the balancing element using . there are _ four _ possibilities for the square root of @xmath174 , two of which are group - like , and we choose @xmath175 this choice determines a ribbon element for @xmath2 , and hence a particular version of the @xmath0-action on the quantum group studied below . the balancing element allows constructing the `` canonical '' @xmath5-characters of @xmath2-representations ( see * [ app : qch ] * ) . let @xmath10 denote the center of @xmath2 . it contains the element @xmath176 called the casimir element . it satisfies the minimal polynomial relation @xmath177 where @xmath178 a proof of is given in * [ fusion - center ] * below as a spin - off of the technology developed for the grothendieck ring ( we do not need before that ) . it follows from the definition of @xmath2 that @xmath179 . in fact , @xmath20 is in the @xmath3-dimensional subalgebra in @xmath10 generated by @xmath180 because of the identity @xmath181 where we set @xmath182 the @xmath2-representation theory at @xmath29 is not difficult to describe ( also see @xcite ) . there turn out to be just @xmath3 irreducible representations . in what follows , we also need verma modules ( all of which except two are extensions of a pair of irreducible representations ) and projective modules ( which are further extensions ) . the category of all finite - dimensional @xmath2-representations at the primitive @xmath3th root of unity is fully described in @xcite . the irreducible @xmath2-representations @xmath183 are labeled by @xmath30 and @xmath96 . the module @xmath13 is linearly spanned by elements @xmath184 , @xmath185 , where @xmath186 is the highest - weight vector and the @xmath2-action is given by @xmath187[s - n]{| s , n - 1\rangle}^{\pm},\\ f { | s , n\rangle}^{\pm } & = { | s , n + 1\rangle}^{\pm},\end{aligned}\ ] ] where we set @xmath188 . @xmath189 is the trivial module . for later use , we list the weights occurring in the module @xmath190 , i.e. , the eigenvalues that @xmath161 has on vectors in @xmath190 , @xmath191 and in the module @xmath192 , @xmath193 we also note the dimensions and quantum dimensions ( see * [ app : qch ] * ) @xmath194 and @xmath195 $ ] . it follows that @xmath196 and @xmath197 . there are @xmath3 verma modules @xmath198 , @xmath96 . first , these are the two steinberg modules @xmath199 next , for each @xmath200 and @xmath30 , the verma module @xmath201 is explicitly described in * [ verma - mod - base ] * as an extension @xmath202 ; for consistency with more complicated extensions considered below , we represent it as @xmath203 with the convention that the arrow is directed to a _ _ sub__module . we note that @xmath204 and @xmath205 ( negligible modules @xcite ) . for @xmath200 , there are nontrivial extensions yielding the projective modules @xmath206 and @xmath207 , @xmath208 their structure can be schematically depicted as @xmath209 \ar@/_/[dr ] & \\ & \stackrel{{\mathscr{x}}^{-\alpha}(p{-}s)}{\bullet}\ar@/^/[dr ] & & \stackrel{{\mathscr{x}}^{-\alpha}(p{-}s)}{\bullet}\ar@/_/[dl ] \\ & & \stackrel{{\mathscr{x}}^{\alpha}(s)}{\bullet } & } \ ] ] it follows that @xmath210 and @xmath211 . the bases and the action of @xmath2 in @xmath206 and @xmath207 are described in * [ module - l ] * and * [ module - p]*. we next find the grothendieck ring of @xmath2 . [ thm : gr - ring ] multiplication in the @xmath2 grothendieck ring @xmath23 is given by @xmath212{\sum_{\substack{s''=|s - s'| + 1\\ \mathrm{step}=2}}^{s + s ' - 1 } } \widetilde{\mathscr{x}}^{\alpha\alpha'}(s''),\ ] ] where @xmath213 to prove this , we use ( i ) a property of the tensor products of any representation with a verma module , ( ii ) an explicit evaluation of the tensor product of any irreducible representation with a two - dimensional one , and ( iii ) the observation that the information gained in ( i ) and ( ii ) suffices for finding the entire grothendieck ring . we first of all note that the trivial representation @xmath214 is the unit in the grothendieck ring and , obviously , @xmath215 for all @xmath216 and @xmath30 . moreover , @xmath217 and it therefore suffices to find all the products @xmath218 and , furthermore , just the products @xmath219 . in the grothendieck ring , the verma module @xmath201 ( with @xmath120 ) is indistinguishable from @xmath220 , and we choose to consider only the @xmath221 verma modules @xmath222 , @xmath223 , given by @xmath224 their highest weights @xmath225 coincide with the respective highest weights of @xmath226 , @xmath227 , @xmath228 . taking the tensor product of a verma module @xmath222 and an irreducible representation gives a module that is filtered by verma modules . in the grothendieck ring , this tensor product therefore evaluates as a sum of verma modules ; moreover , the verma modules that occur in this sum are known , their highest weights being given by @xmath229 , where @xmath230 is the highest weight of @xmath222 and @xmath231 are the weights of vectors in the irreducible representation . with , this readily gives the grothendieck - ring multiplication @xmath232 where we set @xmath233 for @xmath234 and @xmath235 for @xmath236 . for @xmath237 , we have @xmath238 let @xmath239 for @xmath240 and @xmath241 be the respective bases in @xmath183 and in @xmath242 . under the action of @xmath160 , the highest - weight vector @xmath243 with the weight @xmath244 generates the module @xmath245 . the vector @xmath246e_{0}\otimes f_{1}$ ] satisfies the relations @xmath247 under the action of @xmath160 , it generates the module @xmath248 . as regards the product @xmath249 , we already know it from because @xmath250 is a verma module : with the two relevant verma modules replaced by the sum of the corresponding irreducible representations , the resulting four terms can be written as @xmath251 as we have noted , the products @xmath252 are given by the above formulas with the reversed `` @xmath253 '' signs in the right - hand sides . [ [ section-1 ] ] we next evaluate the products @xmath254 as @xmath255 where the products with @xmath242 are already known . by induction on @xmath256 , this allows finding all the products @xmath218 as @xmath257{\sum_{\substack{s''=|s - s'| + 1\\s''\neq p,\ ; \mathrm{step}=2}}^{p - 1 - |p - s - s'| } } { \mathscr{x}}^{\alpha}(s '' ) + \delta_{p , s , s'}{\mathscr{x}}^{\alpha}(p)\\ * { } + \sum_{\substack{s''= 2p - s - s ' + 1\\ \mathrm{step}=2}}^{p - 1 } ( 2{\mathscr{x}}^{\alpha}(s '' ) + 2{\mathscr{x}}^{-\alpha}(p - s'')),\end{gathered}\ ] ] where @xmath258 is equal to @xmath259 if @xmath260 and @xmath261 , and is @xmath262 otherwise . the statement in * [ thm : gr - ring ] * is a mere rewriting of , taken together with the relations @xmath263 . it shows that the @xmath2 grothendieck ring is the @xmath4-model fusion algebra derived in @xcite . this concludes the proof of * [ thm : gr - ring]*. [ cor : quotient ] the @xmath2 grothendieck ring contains the ideal @xmath264 of verma modules generated by @xmath265 the quotient @xmath266 is a _ fusion _ algebra with the basis @xmath267 , @xmath120 the canonical images of the corresponding @xmath102 and multiplication @xmath268 this is a _ semisimple _ fusion algebra , which coincides with the fusion of the unitary @xmath269 representations of level @xmath270 . [ cor : generated ] the @xmath2 grothendieck ring @xmath23 is generated by @xmath271 . this easily follows from theorem [ thm : gr - ring ] ; therefore , @xmath23 can be identified with a quotient of the polynomial ring @xmath272 $ ] . let @xmath273 denote the chebyshev polynomials of the second kind @xmath274 the lower such polynomials are @xmath275 , @xmath276 , @xmath277 , and @xmath278 . [ prop : quotient ] the @xmath2 grothendieck ring is the quotient of the polynomial ring @xmath272 $ ] over the ideal generated by the polynomial @xmath279 moreover , let @xmath280 under the quotient mapping , the image of each polynomial @xmath281 coincides with @xmath282 for @xmath96 and with @xmath283 for @xmath284 . it follows from * [ thm : gr - ring ] * that @xmath285 we recall that the chebyshev polynomials of the second kind satisfy ( and are determined by ) the recursive relation @xmath286 with the initial data @xmath276 , @xmath277 . from , we then obtain that polynomials satisfy relations after the identifications @xmath287 for @xmath96 and @xmath288 for @xmath284 . then , for eq . to be satisfied , we must impose the relation @xmath289 ; this shows that the grothendieck ring is the quotient of @xmath272 $ ] over the ideal generated by polynomial . [ prop : factor ] the polynomial @xmath290 can be factored as @xmath291 this is verified by direct calculation using the representation @xmath292 which follows from . we note that @xmath293 for @xmath294 . the restricted quantum group @xmath2 is not quasitriangular @xcite ; however , it admits a drinfeld mapping , and hence there exists a homomorphic image @xmath295 of the grothendieck ring in the center . in * [ sec : from ] * , we first identify @xmath2 as a subalgebra in a quotient of a drinfeld double . we then obtain the @xmath66-matrix in * [ sec : m - matrix ] * , characterize the subalgebra @xmath296 in * [ fusion - center ] * , and find the center @xmath10 of @xmath2 at @xmath29 in * [ the - center]*. furthermore , we give some explicit results for the radford mapping for @xmath2 in * [ sec : radford - sl2 ] * and we find a ribbon element for @xmath2 in * [ sl2-ribbon]*. the hopf algebra @xmath2 is not quasitriangular , but it can be realized as a hopf subalgebra of a quasitriangular hopf algebra @xmath297 ( which is in turn a quotient of a drinfeld double ) . the @xmath66-matrix ( see * [ app : m ] * ) for @xmath298 is in fact an element of @xmath299 , and hence @xmath2 can be thought of as a factorizable hopf algebra , even though relation required of an @xmath66-matrix is satisfied not in @xmath2 but in @xmath298 ( but on the other hand , holds only with @xmath300 and @xmath301 being bases in @xmath2 ) . the hopf algebra @xmath298 is generated by @xmath302 , @xmath303 , and @xmath304 with the relations @xmath305={\mbox{\footnotesize$\displaystyle\frac{{k}^2-{k}^{-2}}{{\mathfrak{q}}-{\mathfrak{q}}^{-1}}$ } } , \\ { e}^p=0,\quad{\phi}^p=0,\quad{k}^{4p}={\boldsymbol{1 } } , \\ \epsilon({e})=0,\quad\epsilon({\phi})=0,\quad \epsilon({k})=1 , \\ \delta({e})={\boldsymbol{1}}\otimes{e}+{e}\otimes{k}^2,\quad \delta({\phi})={k}^{-2}{\otimes}{\phi}+{\phi}{\otimes}{\boldsymbol{1}},\quad \delta({k})={k}\otimes{k } , \\ s({e})=-{e}{k}^{-2},\quad s({\phi})=-{k}^{2}{\phi},\quad s({k})={k}^{-1}.\ ] ] a hopf algebra embedding @xmath306 is given by @xmath307 in what follows , we often do not distinguish between @xmath159 and @xmath302 , @xmath160 and @xmath303 , and @xmath161 and @xmath308 . [ thm - bar - one ] @xmath298 is a ribbon quasitriangular hopf algebra , with the universal @xmath309-matrix @xmath310!}$}}\,{\mathfrak{q}}^{m(m-1)/2+m(n - j)-nj/2 } { e}^m{k}^{n}\otimes{\phi}^m{k}^{j}\ ] ] and the ribbon element @xmath311!}$}}\ , { \mathfrak{q}}^{-\frac{m}{2}+mj+ { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(j+p+1)^2}{\phi}^m{e}^m{k}^{2j}.\ ] ] equation follows from the realization of @xmath298 as a quotient of the drinfeld double @xmath312 in * [ thm : double]*. the quotient is over the hopf ideal generated by the central element @xmath313 . it follows that @xmath298 inherits a quasitriangular hopf algebra structure from @xmath312 and @xmath309-matrix is the image of under the quotient mapping . using @xmath309-matrix , we calculate the canonical element @xmath314 ( see ) as @xmath315!}$}}\ , { \mathfrak{q}}^{-m(m+3)/2-rn/2}{\phi}^m{k}^{-r}{e}^m{k}^n.\ ] ] we note that actually @xmath316 . indeed , @xmath317!}$}}\ , { \mathfrak{q}}^{-m(m+3)/2-rm - rn/2}{\phi}^m{e}^m{k}^{n - r}={}\\ = { \mbox{\footnotesize$\displaystyle\frac{1}{4p}$}}\sum_{m=0}^{p-1}\sum_{j=0}^{2p-1 } \bigl(\sum_{r=0}^{4p-1}e^{-i\pi\frac{1}{2p}r(r+2m+2j)}\bigr ) ( -1)^m{\mbox{\footnotesize$\displaystyle\frac{({\mathfrak{q}}-{\mathfrak{q}}^{-1})^m}{[m]!}$}}\ , { \mathfrak{q}}^{- { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}m(m+3)}{\phi}^m{e}^m{k}^{2j}\\ + { \mbox{\footnotesize$\displaystyle\frac{1}{4p}$}}\sum_{m=0}^{p-1}\sum_{j=0}^{2p-1 } \bigl(\sum_{r=0}^{4p-1}e^{-i\pi\frac{1}{2p}r(r+2m+2j+1)}\bigr ) ( -1)^m{\mbox{\footnotesize$\displaystyle\frac{({\mathfrak{q}}-{\mathfrak{q}}^{-1})^m}{[m]!}$}}\ , { \mathfrak{q}}^{- { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}m(m+3)}{\phi}^m{e}^m{k}^{2j+1}. \end{gathered}\ ] ] the second gaussian sum vanishes , @xmath318 to evaluate the first gaussian sum , we make the substitution @xmath319 : @xmath320!}$}}\ , { \mathfrak{q}}^{- { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}m(m+3)+ { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(j+m)^2}{\phi}^m{e}^m{k}^{2j}\\ = { \mbox{\footnotesize$\displaystyle\frac{1}{4p}$}}\sum_{m=0}^{p-1}\sum_{j=0}^{2p-1 } \bigl(\sum_{r=0}^{4p-1}e^{-i\pi\frac{1}{2p}r^2}\bigr ) { \mbox{\footnotesize$\displaystyle\frac{({\mathfrak{q}}-{\mathfrak{q}}^{-1})^m}{[m]!}$}}\ , { \mathfrak{q}}^{- { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}m+m(j - p-1)+ { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}j^2}{\phi}^m{e}^m{k}^{2j}. \end{gathered}\ ] ] then evaluating @xmath321 we obtain @xmath322!}$}}\ , { \mathfrak{q}}^{- { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}m+mj+ { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(j+p+1)^2}{\phi}^m{e}^m{k}^{2j+2p+2}.\ ] ] we then find the ribbon element from relation using the balancing element @xmath323[page : balancing ] from , which gives . we next obtain the @xmath66-matrix ( see * [ app : m ] * ) for @xmath2 from the universal @xmath309-matrix for @xmath298 in . because @xmath324 , it follows from that the @xmath66-matrix for @xmath298 , @xmath325 , actually lies in @xmath299 , and does not therefore satisfy condition in @xmath298 ( and hence @xmath298 is not factorizable ) . but this _ is _ an @xmath66-matrix for @xmath326 . a simple calculation shows that @xmath327 is explicitly rewritten in terms of the @xmath2-generators as @xmath328 ! [ n]!}$}}\ , { \mathfrak{q}}^{m(m - 1)/2 + n(n - 1)/2}\\ * \times { \mathfrak{q}}^{- m^2 - m j + 2n j - 2n i - i j + m i } f^{m } e^{n } k^{j}{\otimes}e^{m } f^{n } k^{i}.\end{gathered}\ ] ] given the @xmath66-matrix , we can identify the @xmath2 grothendieck ring with its image in the center using the homomorphism in * [ lemma : dr - hom]*. we evaluate this homomorphism on the preferred basis elements in the grothendieck ring , i.e. , on the irreducible representations . with the balancing element for @xmath2 in and the @xmath66-matrix in , the mapping in * [ lemma : dr - hom ] * is @xmath330 clearly , @xmath331 . we let @xmath296 denote the image of the grothendieck ring under this mapping . [ prop - eval ] for @xmath216 and @xmath30 , @xmath332}{0pt}{}{s - n+m-1}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s - n+m-1}{m } } } { { \genfrac{[}{]}{0pt}{}{s - n+m-1}{m}}}{{\genfrac{[}{]}{0pt}{}{s - n+m-1}{m } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } { { \genfrac{[}{]}{0pt}{}{n}{m}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } } e^m f^m k^{s-1+\beta p - 2n + m } , \end{gathered}\ ] ] where we set @xmath333 if @xmath334 and @xmath335 if @xmath336 . in particular , it follows that @xmath337 the proof of is a straightforward calculation based on the well - known identity ( see , e.g. , @xcite ) @xmath338 which readily implies that @xmath339!)^2\sum_{n=0}^{s-1}{\mathfrak{q}}^{a(s-1 - 2n ) } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s - n+m-1}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s - n+m-1}{m } } } { { \genfrac{[}{]}{0pt}{}{s - n+m-1}{m}}}{{\genfrac{[}{]}{0pt}{}{s - n+m-1}{m}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } { { \genfrac{[}{]}{0pt}{}{n}{m}}}{{\genfrac{[}{]}{0pt}{}{n}{m}}}}.\ ] ] using this in gives . for @xmath340 , we then have @xmath341}{0pt}{}{1-n+m}{m}$}}}{{\genfrac{[}{]}{0pt}{}{1-n+m}{m } } } { { \genfrac{[}{]}{0pt}{}{1-n+m}{m}}}{{\genfrac{[}{]}{0pt}{}{1-n+m}{m}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } { { \genfrac{[}{]}{0pt}{}{n}{m}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } } e^m f^m k^{1 - 2n + m}=\\ = -{\mathfrak{q}}^{-1}k - { \mathfrak{q}}k^{-1 } - ( { \mathfrak{q}}- { \mathfrak{q}}^{-1})^2 e f. \end{gathered}\ ] ] combining * [ prop - eval ] * and * [ cor : generated ] * , we obtain [ dr - alg - cas ] @xmath295 coincides with the algebra generated by the casimir element . the following corollary is now immediate in view of * [ prop : quotient ] * and * [ prop : factor]*. relation holds for the casimir element . identity holds . the derivation of from the algebra of the @xmath342 is given in appendix * [ app : derivation ] * in some detail . we note that although the left - hand side of is not manifestly symmetric in @xmath343 and @xmath256 , the identity shows that it is . [ [ verma - in - center ] ] in what follows , we keep the notation @xmath264 for the verma - module ideal ( more precisely , for its image in the center ) generated by @xmath344 this ideal is the socle ( annihilator of the radical ) of @xmath295 . we now find the center of @xmath2 at the primitive @xmath3th root of unity . for this , we use the isomorphism between the center and the algebra of _ bimodule _ endomorphisms of the regular representation . the results are in * [ prop - center ] * and * [ prop - center - explicit]*. the @xmath168-dimensional regular representation of @xmath2 , viewed as a free left module , decomposes into indecomposable projective modules , each of which enters with the multiplicity given by the dimension of its simple quotient : @xmath345 we now study the regular representation as a @xmath2-bimodule . in what follows , @xmath346 denotes the external tensor product . as a @xmath2-bimodule , the regular representation decomposes as @xmath347{\bigoplus_{s=0}^{p}{\mathscr{q}}(s)},\ ] ] where 1 . the bimodules @xmath348 are simple , 2 . the bimodules @xmath349 , @xmath120 , are indecomposable and admit the filtration @xmath350 where the structure of subquotients is given by @xmath351 and @xmath352 and where @xmath353 is isomorphic to the quotient @xmath354 . the proof given below shows that @xmath355 is in fact the jacobson radical of @xmath349 and @xmath356 , with @xmath357 , and hence @xmath353 is the socle of @xmath349 . for @xmath200 , the left @xmath2-action on @xmath349 and the structure of subquotients can be visualized with the aid of the diagram @xmath358[l]{${\boxtimes}{{\mathscr{x}}^{+}}(s)$ } } \ar[1,-1 ] \ar[1,1 ] & * { } & * { } & * { } * { } & { { { \mathscr{x}}^{-}}(p{-}s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{-}}(p{-}s)$ } } \ar[1,-1 ] \ar[1,1 ] \\ { { { \mathscr{x}}^{-}}(p{-}s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{+}}(s)$ } } \ar[1,1 ] & * { } & { { { \mathscr{x}}^{-}}(p{-}s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{+}}(s)$ } } \ar[1,-1 ] & * { \quad } & { { { \mathscr{x}}^{+}}(s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{-}}(p{-}s)$ } } \ar[1,1 ] & * { } & { { { \mathscr{x}}^{+}}(s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{-}}(p{-}s)$ } } \ar[1,-1 ] \\ * { } & { { { \mathscr{x}}^{+}}(s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{+}}(s)$ } } & * { } & * { } & * { } * { } & { { { \mathscr{x}}^{-}}(p{-}s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{-}}(p{-}s)$ } } } \ ] ] and the right action with the reader may find it convenient to look at these diagrams in reading the proof below . first , the category @xmath359 of finite - dimensional left @xmath2-modules has the decomposition @xcite @xmath360 where each @xmath361 is a full subcategory . the full subcategories @xmath362 and @xmath363 are semisimple and contain precisely one irreducible module each , @xmath228 and @xmath226 respectively . each @xmath361 , @xmath364 , contains precisely two irreducible modules @xmath190 and @xmath365 , and we have the vector - space isomorphisms @xcite @xmath366 where a basis in each @xmath40 can be chosen as the extensions corresponding to the verma module @xmath198 and to the contragredient verma module @xmath367 ( see * [ verma - mod - base ] * ) . in view of , the regular representation viewed as a @xmath2-bimodule has the decomposition @xmath368{\bigoplus_{s=0}^{p}{\mathscr{q}}(s)}\ ] ] into a direct sum of indecomposable two - sided ideals @xmath349 . we now study the structure of subquotients of @xmath349 . let @xmath355 denote the jacobson radical of @xmath349 . by the wedderburn artin theorem , the quotient @xmath354 is a semisimple matrix algebra over @xmath154 , @xmath369 ( where we note that @xmath370 ) . as a bimodule , @xmath354 has the decomposition @xmath371 for @xmath120 , we now consider the quotient @xmath372 , where we set @xmath356 . for brevity , we write @xmath373 , @xmath374 , @xmath375 and @xmath376 , @xmath377 , @xmath378 , and similarly for the contragredient verma modules @xmath379 in view of , there are the natural bimodule homomorphisms @xmath380 the image of @xmath381 has the structure of the lower - triangular matrix @xmath382 clearly , the radical of @xmath383 is the bimodule @xmath384 . it follows that @xmath385 and the bimodule @xmath384 is a subquotient of @xmath386 . in a similar way , we obtain that @xmath387 and @xmath388 . therefore , we have the inclusion @xmath389 next , the radford mapping @xmath390 ( see * [ sec : radford - all ] * ) establishes a bimodule isomorphism between @xmath391 and @xmath392 , and therefore the socle of @xmath393 is isomorphic to @xmath394 . this suffices for finishing the proof : by counting the dimensions of the subquotients given in and , and the dimension of the socle of @xmath393 , we obtain the statement of the proposition . to find the center of @xmath2 , we consider bimodule endomorphisms of the regular representation ; such endomorphisms are in a @xmath395 correspondence with elements in the center . clearly , @xmath396 for each @xmath349 , @xmath397 , there is a bimodule endomorphism @xmath398 that acts as identity on @xmath349 and is zero on @xmath399 with @xmath400 . these endomorphisms give rise to @xmath221 primitive idempotents in the center of @xmath2 . next , for each @xmath349 with @xmath120 , there is a homomorphism @xmath401 ( defined up to a nonzero factor ) whose kernel , as a linear space , is given by @xmath402 ( see ) ; in other words , @xmath403 sends the quotient @xmath404 into the subbimodule @xmath404 at the bottom of @xmath349 and is zero on @xmath399 with @xmath400 . similarly , for each @xmath200 , there is a central element associated with the homomorphism @xmath405 with the kernel @xmath406 , i.e. , the homomorphism sending the quotient @xmath407 into the subbimodule @xmath408 ( and acting by zero on @xmath399 with @xmath400 ) . in total , there are @xmath409 elements @xmath410 , @xmath120 , which are obviously in the radical of the center . by construction , the @xmath411 and @xmath412 have the properties summarized in the following proposition . [ prop - center ] the center @xmath10 of @xmath2 at @xmath29 is @xmath16-dimensional . its associative commutative algebra structure is described as follows : there are two `` special '' primitive idempotents @xmath413 and @xmath414 , @xmath128 other primitive idempotents @xmath411 , @xmath120 , and @xmath415 elements @xmath410 @xmath416 in the radical such that @xmath417 we call @xmath411 , @xmath410 the canonical basis elements in the center , or simply the _ canonical central elements_. they are constructed somewhat more explicitly in * [ prop - center - explicit]*. we note that the choice of a bimodule isomorphism @xmath418 fixes the normalization of the @xmath410 . [ [ rem : coeffs ] ] for any central element @xmath77 and its decomposition @xmath419 with respect to the canonical central elements , _ the coefficient @xmath420 is the eigenvalue of @xmath77 in the irreducible representation @xmath190_. to determine the @xmath421 and @xmath422 coefficients similarly , we fix the normalization of the basis vectors as in * [ proj - mod - base ] * , i.e. , such that @xmath403 and @xmath423 act as @xmath424 in terms of the respective bases in the projective modules @xmath206 and @xmath425 defined in * [ module - l ] * and * [ module - p]*. then _ the coefficient @xmath421 is read off from the relation @xmath426 in @xmath206 , and @xmath422 , similarly , from the relation @xmath427 in @xmath425_. for a hopf algebra @xmath77 with a given cointegral , we recall the radford mapping @xmath428 , see * [ sec : radford - all ] * ( we use the hat for notational consistency in what follows ) . for @xmath429 , with the cointegral @xmath430 in , we are interested in the restriction of the radford mapping to the space of @xmath5-characters @xmath431 and , more specifically , to the image of the grothendieck ring in @xmath431 via the mapping @xmath432 ( see ) . we thus consider the mapping @xmath433 let @xmath434 be the linear span of the @xmath435 ( the image of the grothendieck ring in the center under the radford mapping ) . as we see momentarily , @xmath434 is @xmath3-dimensional and coincides with the algebra generated by the @xmath436 . it follows that @xmath437 in accordance with the fact that @xmath430 furnishes an embedding of the trivial representation @xmath438 into @xmath2 . a general argument based on the properties of the radford mapping ( cf . @xcite ) and on the definition of the canonical nilpotents @xmath412 above implies that for @xmath439 , @xmath440 coincides with @xmath403 up to a factor and @xmath441 coincides with @xmath442 up to a factor . we now give a purely computational proof of this fact , which at the same time fixes the factors ; we describe this in some detail because similar calculations are used in what follows . [ lemma : phi - idem ] for @xmath120 , @xmath443 ^ 2}$}}.\ ] ] also , @xmath444 therefore , the image of the grothendieck ring under the radford mapping is the socle ( annihilator of the radical ) of @xmath10 . first , we recall and use and to evaluate @xmath445!)^2 { \mathfrak{q}}^{j(s - 1 - 2n)}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s - n + i - 1}{i}$}}}{{\genfrac{[}{]}{0pt}{}{s - n + i - 1}{i } } } { { \genfrac{[}{]}{0pt}{}{s - n + i - 1}{i}}}{{\genfrac{[}{]}{0pt}{}{s - n + i - 1}{i } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{i}$}}}{{\genfrac{[}{]}{0pt}{}{n}{i } } } { { \genfrac{[}{]}{0pt}{}{n}{i}}}{{\genfrac{[}{]}{0pt}{}{n}{i } } } } f^{p - 1 - i } e^{p - 1 - i } k^j\ ] ] ( the calculation is very similar to the one in * [ prop - eval ] * ) . next , we decompose @xmath436 with respect to the canonical basis following the strategy in * [ rem : coeffs]*. that is , we use to calculate the action of @xmath440 on the module @xmath446 ( @xmath447 ) . this action is nonzero only on the vectors @xmath448 ( see * [ module - l ] * ) ; because @xmath440 is central , it suffices to evaluate it on any single vector , which we choose as @xmath449 . for @xmath120 , using and , we then have @xmath450!)^2 { \mathfrak{q}}^{j(s + s ' - 2 - 2n ) } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s - n + i - 1}{i}$}}}{{\genfrac{[}{]}{0pt}{}{s - n + i - 1}{i } } } { { \genfrac{[}{]}{0pt}{}{s - n + i - 1}{i}}}{{\genfrac{[}{]}{0pt}{}{s - n + i - 1}{i } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{i}$}}}{{\genfrac{[}{]}{0pt}{}{n}{i } } } { { \genfrac{[}{]}{0pt}{}{n}{i}}}{{\genfrac{[}{]}{0pt}{}{n}{i}}}}\\ * \shoveright{{}\times \prod_{r=0}^{p-2-i } \bigl({\boldsymbol{c}}-{\mbox{\small$\displaystyle\frac{{\mathfrak{q}}^{2r+1}k+{\mathfrak{q}}^{-2r-1}k^{-1}}{({\mathfrak{q}}-{\mathfrak{q}}^{-1})^2}$ } } \bigr ) { \mathsf{b}}^{(+,s')}_0}\\ { } = \zeta \sum_{n=0}^{s - 1}\ ! \sum_{i=0}^{n}\ ! \sum_{j=0}^{2p - 1 } ( -1)^{p+i } ( [ i]!)^2 { \mathfrak{q}}^{j(s + s ' - 2 - 2n ) } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s - n + i - 1}{i}$}}}{{\genfrac{[}{]}{0pt}{}{s - n + i - 1}{i } } } { { \genfrac{[}{]}{0pt}{}{s - n + i - 1}{i}}}{{\genfrac{[}{]}{0pt}{}{s - n + i - 1}{i } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{i}$}}}{{\genfrac{[}{]}{0pt}{}{n}{i } } } { { \genfrac{[}{]}{0pt}{}{n}{i}}}{{\genfrac{[}{]}{0pt}{}{n}{i } } } } \prod_{r=1}^{p-2-i}\ ! [ s'+r][r]\ , { \mathsf{a}}^{(+,s')}_0 , \end{gathered}\ ] ] with the convention that whenever @xmath451 , the product over @xmath452 evaluates as @xmath259 . we simultaneously see that the diagonal part of the action of @xmath440 on @xmath446 vanishes . analyzing the cases where the product over @xmath452 in involves @xmath453=0 $ ] , it is immediate to see that a necessary condition for the right - hand side to be nonzero is @xmath454 . let therefore @xmath455 , where @xmath456 . it is then readily seen that vanishes for odd @xmath457 ; we thus set @xmath458 , which allows us to evaluate @xmath459!)^2 { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m+i}{i}$}}}{{\genfrac{[}{]}{0pt}{}{m+i}{i } } } { { \genfrac{[}{]}{0pt}{}{m+i}{i}}}{{\genfrac{[}{]}{0pt}{}{m+i}{i}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m+s'-1}{i}$}}}{{\genfrac{[}{]}{0pt}{}{m+s'-1}{i } } } { { \genfrac{[}{]}{0pt}{}{m+s'-1}{i}}}{{\genfrac{[}{]}{0pt}{}{m+s'-1}{i}}}}{\mbox{\footnotesize$\displaystyle\frac{[p-2-i+s']!}{[s']!}$}}\ , [ p-2-i]!\,{\mathsf{a}}^{(+,s')}_0 . \end{gathered}\ ] ] but this vanishes for all @xmath460 in view of the identity @xmath461\dots[j+s'+m-1]}{[j]![m - j]!}$ } } = { \mbox{\footnotesize$\displaystyle\frac{1}{[m]}$}}\sum_{j\in{\mathbb{z}}}(-1)^j{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m}{j}$}}}{{\genfrac{[}{]}{0pt}{}{m}{j } } } { { \genfrac{[}{]}{0pt}{}{m}{j}}}{{\genfrac{[}{]}{0pt}{}{m}{j}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m+s'-1+j}{m-1}$}}}{{\genfrac{[}{]}{0pt}{}{m+s'-1+j}{m-1 } } } { { \genfrac{[}{]}{0pt}{}{m+s'-1+j}{m-1}}}{{\genfrac{[}{]}{0pt}{}{m+s'-1+j}{m-1 } } } } = 0 , \quad m{\,{\geqslant}\,}1 . \end{gathered}\ ] ] thus , @xmath440 acts by zero on @xmath446 for all @xmath462 ; it follows similarly that @xmath440 acts by zero on @xmath463 for all @xmath256 and on both steinberg modules @xmath464 . therefore , @xmath440 is necessarily proportional to @xmath403 , with the proportionality coefficient to be found from the action on @xmath206 . but for @xmath465 , the sum over @xmath466 in the right - hand side of is zero unless @xmath467 , and we have @xmath468}$ } } \sum_{i=0}^{s-1 } ( -1)^{p+i } [ i]!{\mbox{\footnotesize$\displaystyle\frac{[p-2-i]![s+p-2-i]!}{[s-1-i]!}$}}\ , { \mathsf{a}}^{(+,s)}_0 , \intertext{where the terms in the sum are readily seen to vanish unless $ i = s-1 $ , and therefore } & = 2p\,\zeta\,(-1)^{p+s+1}{\mbox{\footnotesize$\displaystyle\frac{[p-1]!\,[s-1]!\,[p-1-s]!}{[s]}$}}\ , { \mathsf{a}}^{(+,s)}_0 , \end{aligned}\ ] ] which gives @xmath469 as claimed . the results for @xmath441 ( @xmath120 ) and @xmath470 are established similarly . it follows ( from the expression in terms of the canonical central elements ; cf . @xcite for the small quantum group ) that the two images of the grothendieck ring in the center , @xmath295 and @xmath434 , span the entire center : @xmath471 we next describe the intersection of the two grothendieck ring images in the center ( cf . @xcite for the small quantum group ) . this turns out to be the verma - module ideal ( see * [ verma - in - center ] * ) . [ prop : phi+phi ] @xmath472 . proceeding similarly to the proof of * [ lemma : phi - idem ] * , we establish the formulas @xmath473!)^2\!}{p}$}}\\ * { } \times\bigl((-1)^{p - s}{\pmb{\boldsymbol{\varkappa}}}(0 ) + \sum_{s'=1}^{p-1 } ( -1)^{p+s+s'}\bigl({\mathfrak{q}}^{s s ' } + { \mathfrak{q}}^{-s s'}\bigr ) { \pmb{\boldsymbol{\varkappa}}}(s ' ) + { \pmb{\boldsymbol{\varkappa}}}(p)\bigr ) \end{gathered}\ ] ] for @xmath200 , and @xmath474 which imply the proposition . the derivation may in fact be simplified by noting that as a consequence of and * [ remarks - d]*([item : w ] ) , @xmath475 belongs to the subalgebra generated by the casimir element , which allows using . we finally recall ( see * [ sec : ribbon ] * and @xcite ) that a ribbon element @xmath476 in a hopf algebra @xmath77 is an invertible central element satisfying . for @xmath2 , the ribbon element is actually given in , rewritten as @xmath477!}$}}\ , { \mathfrak{q}}^{-\frac{m}{2}+mj+ { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(j+p+1)^2 } f^m e^m k^{j}\ ] ] in terms of the @xmath2 generators . a calculation similar to the one in the proof of * [ lemma : phi - idem ] * shows the following proposition . [ ribbon - basis ] the @xmath2 ribbon element is decomposed in terms of the canonical central elements as @xmath478\,{\mbox{\footnotesize$\displaystyle\frac{{\mathfrak{q}}- { \mathfrak{q}}^{-1}}{\sqrt{2 p}}$}}\,{{\widehat}{\pmb{\varphi}}}(s ) , \end{aligned}\ ] ] where @xmath479 strictly speaking , expressing @xmath47 through the canonical central elements requires using * [ lemma : phi - idem ] * , but below we need @xmath47 expressed just through @xmath435 . in this section , we first recall the standard @xmath0-action @xcite reformulated for the center @xmath10 of @xmath2 . its definition involves the ribbon element and the drinfeld and radford mappings . from the multiplicative jordan decomposition for the ribbon element , we derive a factorization of the standard @xmath0-representation @xmath1 , @xmath480 , where @xmath41 and @xmath42 are also @xmath0-representations on @xmath10 . we then establish the equivalence to the @xmath0-representation on @xmath9 in * [ mod - on - char]*. let @xmath1 denote the @xmath0-representation on the center @xmath10 of @xmath2 constructed , as a slight modification of the representation in @xcite , as follows . we let @xmath481 and @xmath482 be defined as @xmath483 where @xmath47 is the ribbon element , @xmath484 is the drinfeld mapping , @xmath485 is the radford mapping , and @xmath486 is the normalization factor @xmath487 we call it the _ standard @xmath0-representation _ , to distinguish it from other representations introduced in what follows . we recall that @xmath488 acts via the antipode on the center of the quantum group , and hence acts identically on the center of @xmath2 , @xmath489 [ thm : equiv ] the standard @xmath0-representation on the center @xmath10 of @xmath490 at @xmath116 is equivalent to the @xmath16-dimensional @xmath0-representation on @xmath9 the extended characters of the @xmath4 conformal field theory model in * [ mod - on - char]*. we therefore abuse the notation by letting @xmath1 denote both representations . we introduce a basis in @xmath10 as @xmath491 where @xmath492 @xmath493 are defined in , and @xmath494 ( with @xmath495 defined in ) . that this is a basis in the center follows , e.g. , from the decomposition into the canonical central elements . the mapping @xmath496 between the bases in @xmath147 and in @xmath10 establishes the equivalence . showing this amounts to the following checks . first , we evaluate @xmath497 as @xmath498 and hence , in view of , @xmath499 we also need this formula rewritten in terms of @xmath500 that is , @xmath501 further , we use and to evaluate @xmath502 as @xmath503 where we set @xmath504 . this shows that @xmath48 acts on @xmath505 , @xmath493 , and @xmath506 as on the respective basis elements @xmath507 , @xmath508 , and @xmath509 in @xmath9 . next , it follows from * [ ribbon - basis ] * that @xmath47 acts on @xmath510 as @xmath511 as an immediate consequence , in view of @xmath512 , we have @xmath513 where @xmath514 is defined in . it follows that @xmath46 acts on @xmath505 and @xmath493 as on the respective basis elements @xmath507 and @xmath508 in @xmath9 . finally , we evaluate @xmath515 . recalling * [ ribbon - basis ] * to rewrite @xmath47 as @xmath516 we use and , with the result @xmath517 but ( a simple rewriting of the formulas in * [ eigenp ] * ) @xmath518 ^ 2}$}}\,{\boldsymbol{w}}_s\bigr),\ ] ] and therefore ( also recalling the projector properties to see that only one term survives in the sum over @xmath519 ) @xmath520 ^ 2}$}}\,{\boldsymbol{w}}_s\bigr)={}\\ = -b{\mbox{\footnotesize$\displaystyle\frac{\sqrt{2p}}{{\mathfrak{q}}^{s } - { \mathfrak{q}}^{-s}}$}}\,{\mathscr{s}}\ , ( -1)^{p } { \mathfrak{q}}^ { { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(s^2 - 1)}{\boldsymbol{e}}_s \bigl({\boldsymbol{e}}_s - { \mbox{\footnotesize$\displaystyle\frac{{\mathfrak{q}}^{s } + { \mathfrak{q}}^{-s}}{[s]^2}$}}\,{\boldsymbol{w}}_s - { \boldsymbol{\varphi}}(1){\boldsymbol{e}}_s\bigr)\\ = b(-1)^{s+1}{\mathfrak{q}}^ { { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(s^2 - 1)}\ , { \mathscr{s}}\,{{\widehat}{\boldsymbol{\rho}}}(r ) + b{\mbox{\footnotesize$\displaystyle\frac{(-1)^{p}\sqrt{2p}\,{\mathfrak{q}}^ { { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(s^2 - 1)}}{{\mathfrak{q}}^{s } - { \mathfrak{q}}^{-s}}$}}\ , { \mathscr{s}}\ , { \boldsymbol{\varphi}}(1){\boldsymbol{e}}_s . \end{gathered}\ ] ] here , @xmath521 and @xmath522 , and hence @xmath523 this completes the proof . in view of the equivalence of representations , the @xmath0-representation @xmath1 on the center admits the factorization established in * [ thm : r - decomp]*. remarkably , this factorization can be described in `` intrinsic '' quantum - group terms , as we now show . that is , we construct two more @xmath0-representations on @xmath10 with the properties described in * [ thm : modular-2]*. [ [ section-2 ] ] for the ribbon element @xmath47 , we consider its multiplicative jordan decomposition @xmath524 into the semisimple part @xmath525 and the unipotent part @xmath526 with , we now let @xmath527 and @xmath528 be defined by the corresponding parts of the ribbon element , similarly to : @xmath529 then , evidently , @xmath530 [ [ section-3 ] ] we next define a mapping @xmath531 as @xmath532 where @xmath533 it intertwines the coadjoint and adjoint actions of @xmath2 , and we therefore have the mapping @xmath534 , which is moreover an isomorphism of vector spaces . we set @xmath535 this gives the decomposition @xmath536 [ thm : factorization ] the action of @xmath537 and @xmath538 on the center generates the @xmath0-representation @xmath42 , and the action of @xmath56 and @xmath53 on the center generates the @xmath0-representation @xmath41 , such that 1 . @xmath539 for all @xmath59 , 2 . the representation @xmath41 restricts to the grothendieck ring ( i.e. , to its isomorphic image in the center ) , and 3 . @xmath480 for all @xmath61 , and @xmath1 and @xmath41 are isomorphic to the respective @xmath0-representations on @xmath9 in * [ thm : r - decomp]*. the verification is similar to the proof of * [ thm : equiv ] * , with @xmath540 and @xmath541 ( and similarly for @xmath56 ) , based on the formula @xmath542 [ [ section-4 ] ] the three mappings involved in @xmath485 defined in , @xmath484 defined in , and @xmath543 in can be described in a unified way as follows . let @xmath77 be a ribbon hopf algebra endowed with the standard @xmath0-representation . for @xmath544 , we define @xmath545\]]as @xmath546 where @xmath48 is the standard action of @xmath85 . taking @xmath547 to be the three elements @xmath548 , @xmath47 , and @xmath51 , we have @xmath549 we have shown that the kazhdan lusztig correspondence , understood in a broad sense as a correspondence between conformal field theories and quantum groups , extends into the nonsemisimple realm such that a number of structures on the conformal field theory side and on the quantum group side are actually isomorphic , which signifies an `` improvement '' over the case of rational@xmath62semisimple conformal field theories . although much of the argument in this paper is somewhat too `` calculational , '' and hence apparently `` accidental , '' we hope that a more systematic derivation can be given . in fact , the task to place the structures encountered in the study of nonsemisimple verlinde algebras into the categorical context @xcite was already formulated in @xcite . with the quantum - group counterpart of nonsemisimple verlinde algebras and of the @xmath0-representations on the conformal blocks studied in this paper in the @xmath4 example , this task becomes even more compelling . we plan to address claim [ item : equiv - cat ] of the kazhdan lusztig correspondence ( see page ) between the representation categories of the @xmath11 algebra and of @xmath2 @xcite . this requires constructing vertex - operator analogues of extensions among the irreducible representations ( generalizing the @xmath550 case studied in @xcite ) . another direction where development is welcome is to go over from @xmath4 to @xmath6 models of logarithmic conformal field theories , starting with the simplest such model , @xmath551 , whose content as a minimal theory is trivial , but whose logarithmic version may be quite interesting . we are grateful to a. belavin , e. feigin , m. finkelberg , k. hori , b. khesin , s. loktev , s. parkhomenko , y. soibelman , m.a . soloviev , and b.l . voronov for useful discussions . this paper was supported in part by the rfbr grants 04 - 01 - 00303 ( blf , amg , ams , and iyt ) , lss-1578.2003.2 ( ams and iyt ) , 02 - 01 - 01015 and lss-2044.2003.2 ( blf ) , intas grant 03 - 51 - 3350 ( blf ) . ams is grateful to the fields institute , where a part of this paper was written , for hospitality . we let @xmath77 denote a hopf algebra with comultiplication @xmath71 , counit @xmath72 , and antipode @xmath55 . the general facts summarized here can be found in @xcite . for a hopf algebra @xmath77 , the adjoint and coadjoint actions @xmath552 and @xmath553 ( @xmath554 ) are defined as @xmath555 the center @xmath556 of @xmath77 can be characterized as the set @xmath557 by definition , the space @xmath558 of @xmath5-characters is @xmath559 given an invertible element @xmath560 satisfying @xmath561 for all @xmath544 , we define the linear mapping @xmath562 for any @xmath77-module @xmath75 as @xmath563 [ lemma : qch ] for any @xmath77-module @xmath75 and an element @xmath519 such that @xmath561 , we have 1 . @xmath564 2 . if in addition @xmath519 is group - like , i.e. , @xmath565 , then @xmath566 is a homomorphism of the grothendieck ring to the ring of @xmath5-characters . for a hopf algebra @xmath77 , a _ right integral _ @xmath567 is a linear functional on @xmath77 satisfying @xmath568 for all @xmath544 . whenever such a functional exists , it is unique up to multiplication with a nonzero constant . a _ comodulus _ @xmath174 is an element in @xmath77 such that @xmath569 the left cointegral _ @xmath430 is an element in @xmath77 such that @xmath570 if it exists , this element is unique up to multiplication with a nonzero constant . we also note that the cointegral gives an embedding of the trivial representation of @xmath77 in the bimodule @xmath77 . we use the normalization @xmath571 . whenever a square root of the comodulus @xmath174 can be calculated in a hopf algebra @xmath77 , the algebra admits the _ balancing element _ @xmath572 that satisfies @xmath573 in fact , we have the following lemma . @xmath574 let @xmath77 be a hopf algebra with the right integral @xmath567 and the left right cointegral @xmath430 . the radford mapping @xmath428 and its inverse @xmath575 are given by @xmath576 [ lemma : rad - map ] @xmath485 and @xmath577 are inverse to each other , @xmath578 , @xmath579 , and intertwine the left actions of @xmath77 on @xmath77 and @xmath580 , and similarly for the right actions . here , the left-@xmath77-module structure on @xmath580 is given by @xmath581 ( and on @xmath77 , by the regular action ) . a quasitriangular hopf algebra @xmath77 has an invertible element @xmath582 satisfying @xmath583 for a quasitriangular hopf algebra @xmath77 , the @xmath66-matrix is defined as @xmath584 it satisfies the relations @xmath585 indeed , using , we find @xmath586 and then using , we obtain . next , from , which we write as @xmath587 , it follows that @xmath588 , that is , . if in addition @xmath66 can be represented as @xmath589 where @xmath300 and @xmath301 are two _ bases _ in @xmath77 , the hopf algebra @xmath77 is called _ factorizable_. in any quasitriangular hopf algebra , the square of the antipode is represented by a similarity transformation @xmath590 where the _ canonical element _ @xmath314 is given by @xmath591 ( where @xmath592 ) and satisfies the property @xmath593 any invertible element @xmath519 such that @xmath561 for all @xmath594 can be expressed as @xmath595 , where @xmath596 is an invertible central element . given an @xmath66-matrix ( see * [ app : m ] * ) , we define the drinfeld mapping @xmath597 as @xmath598 [ lemma : dr - map ] in a factorizable hopf algebra @xmath77 , the drinfeld mapping @xmath597 intertwines the adjoint and coadjoint actions of @xmath77 and its restriction to the space @xmath431 of @xmath5-characters gives an isomorphism of associative algebras @xmath599 a _ ribbon hopf algebra _ @xcite is a quasitriangular hopf algebra equipped with an invertible central element @xmath47 , called the _ ribbon element _ , such that @xmath600 in a ribbon hopf algebra , @xmath601 where @xmath572 is the balancing element ( see * [ app : int ] * ) . [ [ app : qch ] ] let @xmath77 be a ribbon hopf algebra and @xmath75 an @xmath77-module . the balancing element @xmath572 allows constructing the `` canonical '' @xmath5-character of @xmath75 : @xmath602 we also define the quantum dimension of a module @xmath75 as @xmath603 it satisfies the relation @xmath604 for any two modules @xmath605 and @xmath606 . let now @xmath77 be a factorizable ribbon hopf algebra and let @xmath76 be its grothendieck ring . we combine the mapping @xmath607 given by @xmath432 and the drinfeld mapping @xmath597 . [ lemma : dr - hom ] in a factorizable ribbon hopf algebra @xmath77 , the mapping @xmath608 is a homomorphism of associative commutative algebras . in this appendix , we construct a double of the hopf algebra @xmath609 associated with the short screening in the logarithmic conformal field theory outlined in * [ sec : voa]*. the main structure resulting from the double is the @xmath309-matrix , which is then used to construct the @xmath66-matrix @xmath73 for @xmath2 . for @xmath21 , we let @xmath609 denote the hopf algebra generated by @xmath302 and @xmath304 with the relations @xmath610 the pbw - basis in @xmath609 is @xmath611 the space @xmath612 of linear functions on @xmath609 is a hopf algebra with the multiplication , comultiplication , unit , counit , and antipode given by @xmath613 for any @xmath614 and @xmath615 . the quantum double @xmath312 is a hopf algebra with the underlying vector space @xmath616 and with the multiplication , comultiplication , unit , counit , and antipode given by eqs . and and by @xmath617 [ thm : double ] @xmath312 is the hopf algebra generated by @xmath302 , @xmath303 , @xmath304 , and @xmath618 with the relations @xmath619={\mbox{\footnotesize$\displaystyle\frac{{k}^2-{\kappa}^2}{{\mathfrak{q}}-{\mathfrak{q}}^{-1}}$}},\label{rel - bb*}\\ \delta({e})={\boldsymbol{1}}\otimes{e}+{e}\otimes{k}^2,\quad \delta({k})={k}\otimes{k},\quad \epsilon({e})=0,\quad \epsilon({k})=1,\label{rel - b-2}\\ \delta({\phi})={\kappa}^2{\otimes}{\phi}+{\phi}{\otimes}{\boldsymbol{1}},\quad \delta({\kappa})={\kappa}{\otimes}{\kappa},\quad \epsilon({\phi})=0,\quad \epsilon({\kappa})=1,\label{rel - b*-2}\\ s({e})=-{e}{k}^{-2},\quad s({k})={k}^{-1},\label{rel - b-3}\\ s({\phi})=-{\kappa}^{-2}{\phi},\quad s({\kappa})={\kappa}^{-1}.\label{rel - b*-3 } \end{gathered}\ ] ] equations , , and are relations in @xmath609 . the unit in @xmath612 is given by the function @xmath548 such that @xmath620 the elements @xmath621 are uniquely defined by @xmath622 for elements of the pbw - basis of @xmath609 , the first relation in becomes @xmath623 where we use the notation @xmath624,\quad { \langlen\rangle } ! = { \langle1\rangle}{\langle2\rangle}\dots { \langlen\rangle},\quad { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{m}{n}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{m}{n } } } { { \genfrac{\langle}{\rangle}{0pt}{}{m}{n}}}{{\genfrac{\langle}{\rangle}{0pt}{}{m}{n}}}}={\mbox{\footnotesize$\displaystyle\frac{{\langlem\rangle}!}{{\langlen\rangle}!\,{\langlem - n\rangle}!}$}}.\ ] ] we then check that the elements @xmath625 with @xmath165 and @xmath626 constitute a basis in @xmath612 and evaluate on the basis elements of @xmath609 as @xmath627 the easiest way to see that holds is to use to calculate @xmath628 and @xmath629 by induction on @xmath466 and then calculate @xmath630 using again , with @xmath631 and @xmath632 . next , we must show that @xmath625 are linearly independent for @xmath165 and @xmath633@xmath634 . possible linear dependences are @xmath635 with some @xmath636 , that is , @xmath637 for all @xmath638 and @xmath639 . using , we obtain the system of @xmath640 linear equations @xmath641{\sum_{i=0}^{p-1}\sum_{j=0}^{4p-1 } \delta_{mi}{\mbox{\footnotesize$\displaystyle\frac{{\langlei\rangle}!}{({\mathfrak{q}}-{\mathfrak{q}}^{-1})^i}$}}}\,{\mathfrak{q}}^{-(j+2i)n/2-ij -i(i-1 ) } \lambda_{ij}={}\\ { } = { \mbox{\footnotesize$\displaystyle\frac{{\langlem\rangle } ! } { ( { \mathfrak{q}}-{\mathfrak{q}}^{-1})^m}$}}\ , { \mathfrak{q}}^{-mn -m(m-1)}\sum_{j=0}^{4p-1 } { \mathfrak{q}}^{-\frac{1}{2 } j(n+2 m ) } \lambda_{mj}=0 \end{gathered}\ ] ] for the @xmath640 variables @xmath642 . the system decomposes into @xmath28 independent systems of @xmath643 linear equations @xmath644 for @xmath643 variables @xmath645 , @xmath626 ( with @xmath646 fixed ) , where @xmath647 . the determinant of the matrix @xmath648 is the vandermonde determinant , which is nonzero because no two numbers among @xmath649 coincide . with established , we verify , , and . next , to verify , we write for @xmath650 and @xmath651 as the respective relations @xmath652 valid for all @xmath653 . the following formulas are obtained by direct calculation using : @xmath654 these relations and imply , which finishes the proof . as any drinfeld double , @xmath312 is a quasitriangular hopf algebra , with the universal @xmath309-matrix given by @xmath655 where @xmath656 are elements of a basis in @xmath609 and @xmath657 are elements of the dual basis , @xmath658 [ lemma : r ] for @xmath312 constructed in * [ subsec : double ] * , the dual basis is expressed in terms of the generators @xmath303 and @xmath618 as @xmath659!}$}}\,{\mathfrak{q}}^{i(i-1)/2 } { \mbox{\footnotesize$\displaystyle\frac{1}{4p}$}}\sum_{r=0}^{4p-1 } { \mathfrak{q}}^{i(j+r)+rj/2}{\phi}^i{\kappa}^r,\ ] ] and therefore the @xmath309-matrix is given by @xmath660!}$}}\,{\mathfrak{q}}^{m(m-1)/2+m(i - j)-ij/2 } { e}^m{k}^{i}\otimes{\phi}^m{\kappa}^{-j}.\ ] ] by a direct calculation using , we verify that eqs . are satisfied with @xmath661 given by . let @xmath343 be an integer @xmath120 and @xmath30 . the verma module @xmath201 has the basis @xmath662 where @xmath663 correspond to the submodule @xmath664 and @xmath665 correspond to the quotient module @xmath183 in @xmath666 with the @xmath2-action given by @xmath667[s - k]{\mathsf{x}}_{k-1 } , & \quad & 0{\,{\leqslant}\,}k{\,{\leqslant}\,}s-1 \quad(\text{with}\quad{\mathsf{x}}_{-1}\equiv0 ) , \kern-60pt\end{aligned}\ ] ] @xmath668[p - s - n]{\mathsf{a}}_{n-1 } , \quad 0{\,{\leqslant}\,}n{\,{\leqslant}\,}p - s-1\quad(\text{with}\quad{\mathsf{a}}_{-1}\equiv0)\notag\\ \intertext{and } f{\mathsf{x}}_k&= \begin{cases } { \mathsf{x}}_{k+1 } , & 0{\,{\leqslant}\,}k{\,{\leqslant}\,}s-2,\\ { \mathsf{a}}_0 , & k = s-1,\\ \end{cases } \\ f{\mathsf{a}}_n&={\mathsf{a}}_{n+1 } , \quad 0{\,{\leqslant}\,}n{\,{\leqslant}\,}p - s-1 \quad(\text{with}\quad{\mathsf{a}}_{p - s}\equiv0).\notag\end{aligned}\ ] ] in addition , there are verma modules @xmath669 . the contragredient verma module @xmath670 is defined in the basis by the same formulas except and , replaced by the respective formulas @xmath671[s - k]{\mathsf{x}}_{k-1 } , & 1{\,{\leqslant}\,}k{\,{\leqslant}\,}s-1,\\ \end{cases } \\ f{\mathsf{x}}_k&={\mathsf{x}}_{k+1 } , \quad 0{\,{\leqslant}\,}k{\,{\leqslant}\,}s-1 \quad(\text{with}\quad{\mathsf{x}}_{s}\equiv0).\end{aligned}\ ] ] the module @xmath672 , @xmath120 , is the projective module whose irreducible quotient is given by @xmath13 . the modules @xmath672 appeared in the literature several times , see @xcite . in explicitly describing their structure , we follow @xcite most closely . let @xmath343 be an integer @xmath120 . the projective module @xmath206 has the basis @xmath674 where @xmath675 is the basis corresponding to the top module in , + @xmath676 to the bottom , @xmath677 to the left , and @xmath678 to the right module , with the @xmath2-action given by @xmath679[p - s - k]{\mathsf{x}}^{(+,s)}_{k-1 } , & \quad 0{\,{\leqslant}\,}k&{\,{\leqslant}\,}p - s-1 \quad(\text{with}\quad{\mathsf{x}}^{(+,s)}_{-1}\equiv0 ) , \kern-60pt\end{aligned}\ ] ] @xmath680[p - s - k]{\mathsf{y}}^{(+,s)}_{k-1 } , & 1{\,{\leqslant}\,}k{\,{\leqslant}\,}p - s-1,\\ { \mathsf{a}}^{(+,s)}_{s-1 } , & k=0,\\ \end{cases } \\ e{\mathsf{a}}^{(+,s)}_n&=[n][s - n]{\mathsf{a}}^{(+,s)}_{n-1 } , \quad 0{\,{\leqslant}\,}n{\,{\leqslant}\,}s-1\quad(\text{with}\quad{\mathsf{a}}^{(+,s)}_{-1}\equiv0),\\ e{\mathsf{b}}^{(+,s)}_n&= \begin{cases } [ n][s - n]{\mathsf{b}}^{(+,s)}_{n-1}+{\mathsf{a}}^{(+,s)}_{n-1 } , & 1{\,{\leqslant}\,}n{\,{\leqslant}\,}s-1,\\ { \mathsf{x}}^{(+,s)}_{p - s-1 } , & n=0,\\ \end{cases } \\ \intertext{and } f{\mathsf{x}}^{(+,s)}_k&= \begin{cases } { \mathsf{x}}^{(+,s)}_{k+1 } , & 0{\,{\leqslant}\,}k{\,{\leqslant}\,}p - s-2,\\ { \mathsf{a}}^{(+,s)}_0 , & k = p - s-1,\\ \end{cases } \\ f{\mathsf{y}}^{(+,s)}_k&={\mathsf{y}}^{(+,s)}_{k+1 } , \quad 0{\,{\leqslant}\,}k{\,{\leqslant}\,}p - s-1 \quad(\text{with}\quad{\mathsf{y}}^{(+,s)}_{p - s}\equiv0),\\ f{\mathsf{a}}^{(+,s)}_n&={\mathsf{a}}^{(+,s)}_{n+1 } , \quad 0{\,{\leqslant}\,}n{\,{\leqslant}\,}s-1 \quad(\text{with}\quad{\mathsf{a}}^{(+,s)}_s\equiv0),\\ f{\mathsf{b}}^{(+,s)}_n&= \begin{cases } { \mathsf{b}}^{(+,s)}_{n+1 } , & 0{\,{\leqslant}\,}n{\,{\leqslant}\,}s-2,\\ { \mathsf{y}}^{(+,s)}_0 , & n = s-1 . \end{cases}\end{aligned}\ ] ] let @xmath343 be an integer @xmath120 . the projective module @xmath425 has the basis @xmath682 where @xmath683 is the basis corresponding to the top module in , + @xmath684 to the bottom , @xmath685 to the left , and @xmath686 to the right module , with the @xmath2-action given by @xmath687[p - s - k]{\mathsf{x}}^{(-,s)}_{k-1 } , & \quad 0{\,{\leqslant}\,}k&{\,{\leqslant}\,}p - s-1\quad(\text{with}\quad{\mathsf{x}}^{(-,s)}_{-1}\equiv0 ) , \kern-60pt\end{aligned}\ ] ] @xmath688[p - s - k]{\mathsf{y}}^{(-,s)}_{k-1}+{\mathsf{x}}^{(-,s)}_{k-1 } , & 1{\,{\leqslant}\,}k{\,{\leqslant}\,}p - s-1,\\ { \mathsf{a}}^{(-,s)}_{s-1 } , & k=0,\\ \end{cases } \\ e{\mathsf{a}}^{(-,s)}_n&=[n][s - n]{\mathsf{a}}^{(-,s)}_{n-1 } , \quad 0{\,{\leqslant}\,}n{\,{\leqslant}\,}s-1\quad(\text{with}\quad { \mathsf{a}}^{(-,s)}_{-1}\equiv0),\\ e{\mathsf{b}}^{(-,s)}_n&= \begin{cases } [ n][s - n]{\mathsf{b}}^{(-,s)}_{n-1 } , & 1{\,{\leqslant}\,}n{\,{\leqslant}\,}s-1,\\ { \mathsf{x}}^{(-,s)}_{p - s-1 } , & n=0,\\ \end{cases } \\ \intertext{and } f{\mathsf{x}}^{(-,s)}_k&={\mathsf{x}}^{(-,s)}_{k+1 } , \quad 0{\,{\leqslant}\,}k{\,{\leqslant}\,}p - s-1 \quad(\text{with}\quad{\mathsf{x}}^{(-,s)}_{p - s}\equiv0),\\ f{\mathsf{y}}^{(-,s)}_k&= \begin{cases } { \mathsf{y}}^{(-,s)}_{k+1 } , & 0{\,{\leqslant}\,}k{\,{\leqslant}\,}p - s-2,\\ { \mathsf{b}}^{(-,s)}_0 , & k = p - s-1,\\ \end{cases } \\ f{\mathsf{a}}^{(-,s)}_n&= \begin{cases } { \mathsf{a}}^{(-,s)}_{n+1 } , & 0{\,{\leqslant}\,}n{\,{\leqslant}\,}s-2,\\ { \mathsf{x}}^{(-,s)}_0 , & n = s-1 , \end{cases } \\ f{\mathsf{b}}^{(-,s)}_n&={\mathsf{b}}^{(-,s)}_{n+1 } , \quad 0{\,{\leqslant}\,}n{\,{\leqslant}\,}s-1 \quad(\text{with}\quad{\mathsf{b}}^{(-,s)}_s\equiv0).\end{aligned}\ ] ] to explicitly construct the canonical central elements in * [ prop - center ] * in terms of the @xmath2 generators , we use the standard formulas in ( * ? ? v.2 ) ( also cf . @xcite ; we are somewhat more explicit about the representation - theory side , based on the analysis in * [ the - center ] * ) . we first introduce projectors @xmath689 and @xmath690 on the direct sums of the eigenspaces of @xmath161 appearing in the respective representations @xmath190 and @xmath192 for @xmath120 , eqs . and . these projectors are @xmath691 it follows that @xmath692 second , we recall polynomial relation for the casimir element and define the polynomials @xmath693 where we recall that @xmath694 , with @xmath695 for @xmath294 . [ prop - center - explicit ] the canonical central elements @xmath411 , @xmath397 , and @xmath696 , @xmath120 , are explicitly given as follows . the elements in the radical of @xmath10 are @xmath697 where @xmath698 the canonical central idempotents are given by @xmath699 where we formally set @xmath700 . first , @xmath701 acts by zero on @xmath702 and @xmath703 . we next consider its action on @xmath349 for @xmath120 . it follows from * [ proj - mod - base ] * that the casimir element acts on the basis of @xmath206 as @xmath704 for all @xmath185 . clearly , @xmath705 annihilates the entire @xmath206 , and therefore @xmath701 acts by zero on each @xmath349 with @xmath706 . on the other hand , for @xmath707 , we have @xmath708 similar formulas describe the action of the casimir element on the module @xmath425 . it thus follows that @xmath709 sends the quotient of the bimodule @xmath710 in , i.e. , @xmath711 , into the subbimodule @xmath711 at the bottom of @xmath710 . therefore , @xmath712 . to obtain @xmath713 and @xmath714 , we multiply @xmath709 with the respective operators projecting on the direct sums of the eigenspaces of @xmath161 occurring in @xmath190 and @xmath192 . this gives ( the reader may verify independently that although the projectors @xmath715 are not central , their products with @xmath709 are ) . the normalization in is chosen such that we have @xmath716 . to obtain the idempotents @xmath717 , we note that @xmath718 annihilates all @xmath349 for @xmath706 , while on @xmath710 , we have @xmath719 , @xmath720 , @xmath721 , and furthermore , by taylor expanding the polynomial , @xmath722 with higher - order terms in @xmath723 annihilating @xmath724 . similar formulas hold for the action on @xmath425 . therefore , @xmath710 is the root space of @xmath725 with eigenvalue @xmath259 , and the second term in is precisely the subtraction of the nondiagonal part . 1 . [ item : w ] we note that @xmath726 . this follows because @xmath727 . 2 . for any polynomial @xmath728 , decomposition takes the form @xmath729 for example , implies that for @xmath730 defined in * [ sec : casimir ] * , we have @xmath731 using and expressions through the chebyshev polynomials in * [ prop : quotient ] * , we recover the eigenmatrix @xmath732 of the fusion algebra . this eigenmatrix was obtained in @xcite by different means , from the matrix of the modular @xmath55-transformation on @xmath11-characters . the eigenmatrix relates the preferred basis ( the basis of irreducible representations ) and the basis of idempotents and nilpotents in the fusion algebra . specifically , if we order the irreducible representations as @xmath733 and the idempotents and nilpotents that form a basis of @xmath734 as @xmath735 then the eigenmatrix @xmath736 is defined as @xmath737 the calculation of the entries of @xmath736 via is remarkably simple : for example , with @xmath738 taken as @xmath739 ( see * [ prop : quotient ] * ) , we have @xmath740 in accordance with . evaluating the other case in similarly and taking the derivatives , we obtain the eigenmatrix @xmath741 with the @xmath742 blocks @xcite corrects a misprint in @xcite , where @xmath743 occurred in a wrong matrix entry . ] @xmath744 0\;&-{\mbox{\footnotesize$\displaystyle\frac{2\lambda_j}{p}$}}\sin{\mbox{\footnotesize$\displaystyle\frac{j\pi}{p}$ } } \end{pmatrix}\ ! , \\ p_{s,0}&= \begin{pmatrix } s\ ; & ( -1)^{s+1}s\\[2pt ] p{-}s\ ; & ( -1)^{s+1}(p{-}s ) \end{pmatrix}\ ! , \end{alignedat}\\ p_{s , j}= ( -1)^s \begin{pmatrix } -{\mbox{\footnotesize$\displaystyle\frac{\sin\frac{sj\pi}{p}}{\sin\frac{j\pi}{p}}$ } } & { \mbox{\footnotesize$\displaystyle\frac{2\lambda_j}{p^2}$ } } \bigl(-s\cos{\mbox{\footnotesize$\displaystyle\frac{sj\pi}{p}$}}\sin{\mbox{\footnotesize$\displaystyle\frac{j\pi}{p}$ } } + \sin{\mbox{\footnotesize$\displaystyle\frac{sj\pi}{p}$}}\cos{\mbox{\footnotesize$\displaystyle\frac{j\pi}{p}$}}\bigr)\\[12pt ] { \mbox{\footnotesize$\displaystyle\frac{\sin\frac{sj\pi}{p}}{\sin\frac{j\pi}{p}}$ } } & { \mbox{\footnotesize$\displaystyle\frac{2\lambda_j}{p^2}$}}\bigl ( -(p{-}s)\cos{\mbox{\footnotesize$\displaystyle\frac{sj\pi}{p}$}}\sin{\mbox{\footnotesize$\displaystyle\frac{j\pi}{p}$ } } -\sin{\mbox{\footnotesize$\displaystyle\frac{sj\pi}{p}$}}\cos{\mbox{\footnotesize$\displaystyle\frac{j\pi}{p}$}}\bigr ) \end{pmatrix}\end{gathered}\ ] ] for @xmath745 , where , for the sake of comparison , we isolated the factor @xmath746 ^ 3\sin\frac{\pi}{p}}$ } } = { \mbox{\small$\displaystyle\frac{p^2\,\bigl(\sin\frac{\pi}{p}\bigr)^2 } { \bigl(\sin\frac{j\pi}{p}\bigr)^3}$}}\ ] ] whereby the normalization of each nilpotent element , and hence of each even column of @xmath732 starting with the fourth , differs from the normalization chosen in @xcite ( both are arbitrary because the nilpotents can not be canonically normalized ) . we derive identity from the fusion algebra realized on the central elements @xmath747 . in view of * [ lemma : dr - hom ] * , the central elements @xmath748 in with @xmath30 , @xmath216 satisfy the algebra@xmath749 where @xmath750 we now equate the coefficients at the respective pbw - basis elements in both sides of . because of , it suffices to do this for the algebra relation for @xmath751 . writing it as in , we have @xmath752{\sum_{\substack{s''= 2p - s - s ' + 1\\ \mathrm{step}=2}}^{p - 1 } } ( 2{\boldsymbol{\chi}}^{+}(s '' ) + 2{\boldsymbol{\chi}}^{-}(p - s'')).\end{gathered}\ ] ] we first calculate the right - hand side . simple manipulations with @xmath5-binomial coefficients show that @xmath753{\sum_{n=0}^{p-1}\sum_{m=0}^{p-1 } } ( { \mathfrak{q}}-{\mathfrak{q}}^{-1})^{2 m } { \mathfrak{q}}^{-(m+1)(m+s-1 - 2n)}\\ * { } \times{{{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s+m - n-1}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s+m - n-1}{m } } } { { \genfrac{[}{]}{0pt}{}{s+m - n-1}{m}}}{{\genfrac{[}{]}{0pt}{}{s+m - n-1}{m}}}}}_*}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } { { \genfrac{[}{]}{0pt}{}{n}{m}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } } e^m f^m k^{s-1 - 2n + m},\end{gathered}\ ] ] where @xmath754}{0pt}{}{m}{n}$}}}{{\genfrac{[}{]}{0pt}{}{m}{n } } } { { \genfrac{[}{]}{0pt}{}{m}{n}}}{{\genfrac{[}{]}{0pt}{}{m}{n}}}}}_*}= \begin{cases } 0 , & n<0,\\ { \mbox{\footnotesize$\displaystyle\frac{[m - n+1]\dots[m]}{[n]!}$}}&\text{otherwise } , \end{cases}\ ] ] which leads to @xmath755}{0pt}{}{s + s ' - 2 - \ell - n + m}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + m}{m } } } { { \genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + m}{m}}}{{\genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + m}{m } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n - \ell}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n - \ell}{m } } } { { \genfrac{[}{]}{0pt}{}{n - \ell}{m}}}{{\genfrac{[}{]}{0pt}{}{n - \ell}{m } } } } e^m f^m k^{s + s ' - 2 - 2n + m}.\end{gathered}\ ] ] changing the order of summations , using that the @xmath5-binomial coefficients vanish in the cases specified in , and summing over even and odd @xmath646 separately , we have @xmath756}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m}{2}}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m}{2}}{m } } } { { \genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m}{2}}{m}}}{{\genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m}{2}}{m } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n + \frac{m}{2 } - \ell}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n + \frac{m}{2 } - \ell}{m } } } { { \genfrac{[}{]}{0pt}{}{n + \frac{m}{2 } - \ell}{m}}}{{\genfrac{[}{]}{0pt}{}{n + \frac{m}{2 } - \ell}{m } } } } e^m f^m k^{s + s ' - 2 - 2n}+{}}\\ { } + \smash[b ] { ( -1)^{s+s'}\sum_{\substack{m=1\\ \text{odd}}}^{p - 1 } } \sum_{n=0}^{2p - 1 } \sum_{\ell=0}^{\min(n+\frac{m-1}{2 } , s - 1 , s ' - 1 ) } ( { \mathfrak{q}}- { \mathfrak{q}}^{-1})^{2m}{\mathfrak{q}}^{-(m + 1)(s + s ' - 2n - 1)}\\ * { } \times { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m+1}{2}}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m+1}{2}}{m } } } { { \genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m+1}{2}}{m}}}{{\genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m+1}{2}}{m } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n + \frac{m-1}{2 } - \ell}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n + \frac{m-1}{2 } - \ell}{m } } } { { \genfrac{[}{]}{0pt}{}{n + \frac{m-1}{2 } - \ell}{m}}}{{\genfrac{[}{]}{0pt}{}{n + \frac{m-1}{2 } - \ell}{m } } } } e^m f^m k^{s + s ' - 2n - 1}.\end{gathered}\ ] ] next , in the left - hand side of , we use that @xmath757 are central and readily calculate @xmath758{(-1)^{s+1}\sum_{n=0}^{s-1 } \sum_{m=0}^{n } } ( { \mathfrak{q}}-{\mathfrak{q}}^{-1})^{2 m } { \mathfrak{q}}^{-(m+1)(m+s-1 - 2n)}\\ * \shoveright{{}\times{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s - n+m-1}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s - n+m-1}{m } } } { { \genfrac{[}{]}{0pt}{}{s - n+m-1}{m}}}{{\genfrac{[}{]}{0pt}{}{s - n+m-1}{m}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } { { \genfrac{[}{]}{0pt}{}{n}{m}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } } e^m { \boldsymbol{\chi}}^{+}(s ' ) f^m k^{s - 1 - 2n + m}={}}\\ \shoveleft { { } = { ( -1)^{s+s'}\sum_{m=0}^{p - 1 } \sum_{n'=0}^{s ' - 1 } \sum_{n = n'}^{s + n ' - 1 } \sum_{j=0}^{p-1 } } ( { \mathfrak{q}}- { \mathfrak{q}}^{-1})^{2 m } { \mathfrak{q}}^{-m(m + s ' - 2n ' ) } { \mathfrak{q}}^{-(j + 1)(s + s ' - 2 - 2n)}}\\ * { } \times { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s { - } n { + } n ' { + } j { - } 1}{j}$}}}{{\genfrac{[}{]}{0pt}{}{s { - } n { + } n ' { + } j { - } 1}{j } } } { { \genfrac{[}{]}{0pt}{}{s { - } n { + } n ' { + } j { - } 1}{j}}}{{\genfrac{[}{]}{0pt}{}{s { - } n { + } n ' { + } j { - } 1}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n { - } n'}{j}$}}}{{\genfrac{[}{]}{0pt}{}{n { - } n'}{j } } } { { \genfrac{[}{]}{0pt}{}{n { - } n'}{j}}}{{\genfrac{[}{]}{0pt}{}{n { - } n'}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}$}}}{{\genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j } } } { { \genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}}}{{\genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n'}{m { - } j}$}}}{{\genfrac{[}{]}{0pt}{}{n'}{m { - } j } } } { { \genfrac{[}{]}{0pt}{}{n'}{m { - } j}}}{{\genfrac{[}{]}{0pt}{}{n'}{m { - } j } } } } e^m f^m k^{s + s ' - 2 - 2n + m}.\end{gathered}\ ] ] changing the order of summations , using that the @xmath5-binomial coefficients vanish in the cases specified in , and summing over even and odd @xmath646 separately , we have @xmath759}{0pt}{}{s { - } n { - } \frac{m}{2}+ n ' + j { - } 1}{j}$}}}{{\genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m}{2}+ n ' + j { - } 1}{j } } } { { \genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m}{2}+ n ' + j { - } 1}{j}}}{{\genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m}{2}+ n ' + j { - } 1}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n { + } \frac{m}{2 } { - } n'}{j}$}}}{{\genfrac{[}{]}{0pt}{}{n { + } \frac{m}{2 } { - } n'}{j } } } { { \genfrac{[}{]}{0pt}{}{n { + } \frac{m}{2 } { - } n'}{j}}}{{\genfrac{[}{]}{0pt}{}{n { + } \frac{m}{2 } { - } n'}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}$}}}{{\genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j } } } { { \genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}}}{{\genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n'}{m { - } j}$}}}{{\genfrac{[}{]}{0pt}{}{n'}{m { - } j } } } { { \genfrac{[}{]}{0pt}{}{n'}{m { - } j}}}{{\genfrac{[}{]}{0pt}{}{n'}{m { - } j } } } } e^m f^m k^{s + s ' - 2 - 2n}}\\ * \shoveleft{{}+(-1)^{s+s'}\sum_{\substack{m=1\\ \text{odd}}}^{p - 1 } \sum_{j=0}^{p - 1 } \sum_{n=0}^{2p - 1 } \sum_{n'=0}^{s ' - 1 } ( { \mathfrak{q}}- { \mathfrak{q}}^{-1})^{2 m } { \mathfrak{q}}^{-m(m + s ' - 2n ' ) } { \mathfrak{q}}^{-(j + 1)(s + s ' - 2 n - m - 1)}}\\ * { } \times { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m-1}{2 } { + } n ' { + } j { - } 1}{j}$}}}{{\genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m-1}{2 } { + } n ' { + } j { - } 1}{j } } } { { \genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m-1}{2 } { + } n ' { + } j { - } 1}{j}}}{{\genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m-1}{2 } { + } n ' { + } j { - } 1}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n { + } \frac{m-1}{2 } { - } n'}{j}$}}}{{\genfrac{[}{]}{0pt}{}{n { + } \frac{m-1}{2 } { - } n'}{j } } } { { \genfrac{[}{]}{0pt}{}{n { + } \frac{m-1}{2 } { - } n'}{j}}}{{\genfrac{[}{]}{0pt}{}{n { + } \frac{m-1}{2 } { - } n'}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}$}}}{{\genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j } } } { { \genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}}}{{\genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n'}{m { - } j}$}}}{{\genfrac{[}{]}{0pt}{}{n'}{m { - } j } } } { { \genfrac{[}{]}{0pt}{}{n'}{m { - } j}}}{{\genfrac{[}{]}{0pt}{}{n'}{m { - } j } } } } e^m f^m k^{s + s ' - 2n - 1}.\end{gathered}\ ] ] equating the respective coefficients at the pbw - basis elements in and , we obtain @xmath760}{0pt}{}{n - i}{j}$}}}{{\genfrac{[}{]}{0pt}{}{n - i}{j } } } { { \genfrac{[}{]}{0pt}{}{n - i}{j}}}{{\genfrac{[}{]}{0pt}{}{n - i}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{i}{m - j}$}}}{{\genfrac{[}{]}{0pt}{}{i}{m - j } } } { { \genfrac{[}{]}{0pt}{}{i}{m - j}}}{{\genfrac{[}{]}{0pt}{}{i}{m - j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{i + j + s - 1 - n}{j}$}}}{{\genfrac{[}{]}{0pt}{}{i + j + s - 1 - n}{j } } } { { \genfrac{[}{]}{0pt}{}{i + j + s - 1 - n}{j}}}{{\genfrac{[}{]}{0pt}{}{i + j + s - 1 - n}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m - i - j - 1 + s'}{m - j}$}}}{{\genfrac{[}{]}{0pt}{}{m - i - j - 1 + s'}{m - j } } } { { \genfrac{[}{]}{0pt}{}{m - i - j - 1 + s'}{m - j}}}{{\genfrac{[}{]}{0pt}{}{m - i - j - 1 + s'}{m - j}}}}={}\\ { } = { \mathfrak{q}}^{m(2 n + 1 - s ) } \sum_{\ell=0}^{\min(s - 1 , s ' - 1 ) } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n - \ell}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n - \ell}{m } } } { { \genfrac{[}{]}{0pt}{}{n - \ell}{m}}}{{\genfrac{[}{]}{0pt}{}{n - \ell}{m}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m + s + s ' - 2 - \ell - n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{m + s + s ' - 2 - \ell - n}{m } } } { { \genfrac{[}{]}{0pt}{}{m + s + s ' - 2 - \ell - n}{m}}}{{\genfrac{[}{]}{0pt}{}{m + s + s ' - 2 - \ell - n}{m}}}},\end{gathered}\ ] ] where @xmath761 , @xmath762 , @xmath763 . because of the vanishing of @xmath5-binomial coefficients ( see ) , the summations over @xmath466 and @xmath764 in the left - hand side can be extended to @xmath765 , which gives after the shifts @xmath766 , @xmath767 . in the above derivation , @xmath17 was the @xmath3th primitive root of unity , but because @xmath28 does not explicitly enter the resultant identity and because @xmath5-binomial coefficients are ( laurent ) polynomials in @xmath5 , we conclude that is valid for all @xmath5 . 99 d. kazhdan and g. lusztig , _ tensor structures arising from affine lie algebras , _ i , j. amer . soc . 6 ( 1993 ) 905947 ; ii , j. amer . math . soc . 6 ( 1993 ) 9491011 ; iii , j. amer . 7 ( 1994 ) 335381 ; iv , j. amer . soc . 7 ( 1994 ) 383453 . v. lyubashenko , _ invariants of @xmath768-manifolds and projective representations of mapping class groups via quantum groups at roots of unity _ , commun . ( 1995 ) 467516 [ hep - th/9405167 ] ; _ modular properties of ribbon abelian categories _ , symposia gaussiana , proc . of the 2nd gauss symposium , munich , 1993 , conf . a ( berlin , new york ) , walter de gruyter , ( 1995 ) 529579 [ hep - th/9405168 ] ; _ modular transformations for tensor categories _ , j. pure applied algebra 98 ( 1995 ) 279327 . feigin , a.m. gainutdinov , a.m. semikhatov , and i.yu . tipunin , _ kazhdan lusztig correspondence for the representation category of the triplet @xmath770-algebra in logarithmic cft _ , math.qa/0512621 . reshetikhin and v.g . turaev , _ ribbon graphs and their invariants derived from quantum groups _ , comm . phys . , 127 ( 1990 ) 126 . a. lachowska , _ on the center of the small quantum group _ , math.qa/0107098 .

the @xmath0-representation @xmath1 on the center of the restricted quantum group @xmath2 at the primitive @xmath3th root of unity is shown to be equivalent to the @xmath0-representation on the _ extended _ characters of the logarithmic @xmath4 conformal field theory model . the multiplicative jordan decomposition of the @xmath2 ribbon element determines the decomposition of @xmath1 into a `` pointwise '' product of two commuting @xmath0-representations , one of which restricts to the grothendieck ring ; this restriction is equivalent to the @xmath0-representation on the @xmath4-characters , related to the fusion algebra via a nonsemisimple verlinde formula . the grothendieck ring of @xmath2 at the primitive @xmath3th root of unity is shown to coincide with the fusion algebra of the @xmath4 logarithmic conformal field theory model . as a by - product , we derive @xmath5-binomial identities implied by the fusion algebra realized in the center of @xmath2 .

in recent years , there have been a considerable number of important developments in the extension of ( classical ) information - theoretic concepts to a quantum - mechanical setting . bennett and shor @xcite have surveyed this progress in the outstanding commemorative issue 19481998 of the _ ieee transactions on information theory_. in particular , they pointed out in strict analogy to the classical case , successfully studied some fifty years ago by shannon in famous landmark work @xcite that quantum data compression allows signals from a redundant quantum source to be compressed into a bulk approaching the source s ( quantum ) entropy . bennett and shor did not , however , discuss the intriguing case which arises when the specific nature of the quantum source is _ unknown_. this , of course , corresponds to the classical question of _ universal _ coding or data compression ( see @xcite , ( * ? ? ? ii.e ) ) . we do address this interesting issue here , by investigating whether or not it is possible to extend to the quantum domain , recent ( classical ) seminal results of clarke and barron @xcite . they , in fact , derived various forms of asymptotic redundancy of universal data compression for parameterized families of probability distributions . their analyses provide a rigorous basis for the reference prior method in bayesian statistical analysis . for an extensive commentary on the results of clarke and barron , see @xcite . also see @xcite , for some recent related research , as well as a discussion of various rationales that have been employed for using the ( classical ) jeffreys prior a possible quantum counterpart of which will be of interest here for bayesian purposes , cf . let us also bring to the attention of the reader that in a brief review @xcite of @xcite , the noted statistician , i. j. good , commented that clarke and barron `` have presumably overlooked the reviewer s work '' and cited , in this regard @xcite . let us briefly recall the basic setup and the results of clarke and barron that are relevant to the analyses of our paper . clarke and barron work in a noninformative bayesian framework , in which we are given a parametric family of probability densities @xmath7 on a space @xmath8 . these probability densities generate independent identically distributed random variables @xmath9 , which , for a fixed @xmath10 , we consider as producing strings of length @xmath4 according to the probability density @xmath11 of the @xmath4-fold product of probability distributions . now suppose that nature picks a @xmath10 from @xmath12 , that is a joint density @xmath11 on the product space @xmath13 , the space of strings of length @xmath4 . on the other hand , a statistician chooses a distribution @xmath14 on @xmath15 as his best guess of @xmath11 . of course , there is a loss of information , which is measured by the total relative entropy @xmath16 , where @xmath17 is the _ kullback leibler divergence _ of @xmath18 and @xmath19 ( the _ relative entropy _ of @xmath18 with respect to @xmath19 ) . for finite @xmath4 , and for a given _ prior _ @xmath20 on @xmath12 , by a result of aitchison @xcite , the best strategy @xmath14 to minimize the average risk @xmath21 is to choose for @xmath14 the mixture density @xmath22 . this is called a _ bayes procedure _ or a _ bayes strategy_. the quantities corresponding to such a procedure that must be investigated are the _ risk _ ( _ redundancy _ ) _ of the bayes strategy _ @xmath23 and the _ bayes risk _ , the average of risks , @xmath24 . the bayes risk equals shannon s mutual information @xmath25 ( see @xcite ) . moreover , the bayes risk is bounded above by the _ minimax redundancy _ @xmath26 . in fact , by a result of gallager @xcite and davisson and leon garcia @xcite ( see @xcite for a generalization ) , for each fixed @xmath4 there is a prior @xmath27 which realizes this upper bound , i.e. , the _ maximin redundancy _ @xmath28 and the minimax redundancy are the same . such a prior @xmath27 is called _ capacity achieving _ or _ least favorable_. clarke and barron investigate the above - mentioned quantities _ asymptotically _ , that is , for @xmath4 tending to infinity . first of all , in ( * ? ? ? * ( 1.4 ) ) , ( * ? ? ? * ( 2.1b ) ) , they show that the redundancy @xmath29 of the bayes strategy is asymptotically @xmath30 as @xmath4 tends to infinity . here , @xmath31 is the @xmath32 fisher information matrix the negative of the expected value of the hessian of the logarithm of the density function . ( although the binary logarithm is usually used in the quantum coding literature , we employ the natural logarithm throughout this paper , chiefly to facilitate comparisons of our results with those of clarke and barron @xcite . ) for priors supported on a compact subset @xmath33 in the interior of the domain @xmath12 of parameters , the asymptotic minimax redundancy @xmath26 was shown to be ( * ? ? ? * ( 2.4 ) ) , @xcite , @xmath34 moreover ( * ? ? ? * ( 2.6 ) ) , it is _ jeffreys prior _ @xmath35 ( with @xmath36 a normalizing constant ; see also @xcite ) which is the unique continuous and positive prior on @xmath33 which is asymptotically least favorable , i.e. , for which the asymptotic maximin redundancy achieves the value ( [ eq:3 ] ) . in particular , asymptotically the maximin and minimax redundancies are the same . in obvious contrast to classical information theory , quantum information theory directly relies upon the fundamental principles of quantum mechanics . this is due to the fact that the basic unit of quantum computing , the quantum bit " or `` qubit , '' is typically a ( two - state ) microscopic system , possibly an atom or nuclear spin or polarized photon , the behavior of which ( e.g. entanglement , interference , superposition , stochasticity , ) can only be accurately explained using the rules of quantum theory @xcite . we refer the reader to @xcite for a comprehensive introduction to these matters ( including the subjects of quantum error - correcting codes and quantum cryptography ) . here , we shall restrict ourselves to describing , in mathematical terms , the basic notions of quantum information theory , how they pertain to data compression , and in what manner they parallel the corresponding notions from classical information theory . in quantum information theory , the role of probability densities is played by _ density matrices _ , which are , by definition , nonnegative definite hermitian matrices of unit trace , and which can be considered as operators acting on a ( finite - dimensional ) hilbert space . any probability density on a ( finite ) set @xmath37 , where the probability of @xmath38 equals @xmath39 , is representable in this framework by a diagonal matrix @xmath40 ( which is quite clearly itself , a nonnegative definite hermitian matrix with unit trace ) . given two density matrices @xmath41 and @xmath42 , the quantum counterpart of the relative entropy , that is , the _ relative entropy _ of @xmath43 with respect to @xmath44 , is @xcite ( cf . @xcite ) , @xmath45 where the logarithm of a matrix @xmath46 is defined as @xmath47 , with @xmath48 the appropriate identity matrix . ( alternatively , if @xmath49 acts diagonally on a basis @xmath50 of the hilbert space by @xmath51 , then @xmath52 acts by @xmath53 , @xmath54 . ) clearly , if @xmath41 and @xmath42 are diagonal matrices , corresponding to classical probability densities , then ( [ eq:5 ] ) reduces to the usual kullback leibler divergence . as we said earlier , our goal is to examine the possibility of extending the results of clarke and barron to quantum theory . that is , first of all we have to replace the ( classical ) probability densities @xmath55 by density matrices . we are not able to proceed in complete generality , but rather we will restrict ourselves to considering the first nontrivial case , that is , we will replace @xmath55 by @xmath56 density matrices . such matrices can be written in the form , @xmath57 where , in order to guarantee nonnegative definiteness , the points @xmath58 must lie within the unit ball ( `` bloch sphere '' @xcite ) , @xmath59 . ( the points on the bounding spherical surface , @xmath60 , corresponding to the _ pure states _ , will be shown to exhibit nongeneric behavior , see ( [ a5 ] ) and the respective comments in sec . [ s3 ] ( cf . @xcite ) . ) such @xmath56 density matrices correspond , in a one - to - one fashion , to the standard ( complex ) two - level quantum systems notably , those of spin-@xmath61 ( electrons , protons , ) and massless spin-@xmath62 particles ( photons ) . these systems carry the basic units of quantum computing , the _ quantum bits_. ( if we set @xmath63 in ( [ eq:6 ] ) , we recover a classical binomial distribution , with the probability of `` success '' , say , being @xmath64 and of `` failure '' , @xmath65 . setting either @xmath66 or @xmath67 to zero , puts us in the framework of real as opposed to complex quantum mechanics . ) the quantum analogue of the product of ( classical ) probability distributions is the _ tensor product _ of density matrices . ( again , it is easily seen that , for diagonal matrices , this reduces to the classical product . ) hence , we will replace @xmath11 by the tensor products @xmath68 , where @xmath49 is a @xmath56 density matrix ( [ eq:6 ] ) . these tensor products are @xmath69 matrices , and can be used to compute ( _ via _ the fundamental rule that the expected value of an observable is the trace of the matrix product of the observable and the density matrix ; see @xcite ) the probability of strings of quantum bits of length @xmath4 . in @xcite it was argued that the quantum fisher information matrix ( requiring due to noncommutativity the computation of symmetric logarithmic derivatives @xcite ) , one must find the symmetric logarithmic derivatives ( @xmath70 ) satisfying @xmath71 and then compute the entries of ( [ eq:8 ] ) in the form ( * ? ? ? ( 2 ) , ( 3 ) ) @xmath72 , \quad \beta , \gamma = x , y , z.\ ] ] for a well - motivated discussion of these formulas and the manner in which classical and quantum fisher information are related , see @xcite . ] ) for the density matrices ( [ eq:6 ] ) should be taken to be of the form @xmath73 the quantum counterpart of the jeffreys prior was , then , taken to be the normalized form ( dividing by @xmath74 ) of the square root of the determinant of ( [ eq:8 ] ) , that is , @xmath75 on the basis of the above - mentioned result of clarke and barron that the jeffreys prior yields the asymptotic common minimax and maximin redundancy , it was conjectured @xcite that its assumed quantum counterpart ( [ eq:9 ] ) would have similar properties , as well . to examine this possibility , ( [ eq:9 ] ) was embedded as a specific member ( @xmath76 ) of a one - parameter family of spherically - symmetric / unitarily - invariant probability densities ( i.e. , under unitary transformations of @xmath49 , the assigned probability is invariant ) , @xmath77 ( embeddings of ( [ eq:9 ] ) in other ( possibly , multiparameter ) families are , of course , possible and may be pursued in further research . in this regard , see theorem [ t15 ] in sec . [ s3 ] . ) for @xmath78 , we obtain a uniform distribution over the unit ball . ( this has been used as a prior over the two - level quantum systems , at least , in one study @xcite . ) for @xmath79 , the uniform distribution over the spherical boundary ( the locus of the pure states ) is approached . ( this is often employed as a prior , for example @xcite . ) for @xmath80 , a dirac distribution concentrated at the origin ( corresponding to the fully mixed state ) is approached . for a treatment in our setting that is analogous to that of clarke and barron , we average @xmath81 with respect to @xmath0 . doing so yields a one - parameter family of @xmath2 _ bayesian density matrices _ @xcite , @xmath82 @xmath1 , which are the analogues of the mixtures @xmath83 , and which exhibit highly interesting properties . now , still following clarke and barron , we have to compute the analogue of the risk @xmath29 , i.e. , the relative entropy @xmath84 . keeping the definition ( [ eq:5 ] ) in mind , this requires us to explicitly find the eigenvalues and eigenvectors of the matrices @xmath6 , which we do in sec . subsequently , in sec . [ s2.3 ] , we determine explicitly the relative entropy of @xmath81 with respect to @xmath6 . we do this by using identities for hypergeometric series and some combinatorics . ( it is also possible to obtain some of our results by making use of representation theory of @xmath85 . an even more general result was derived by combining these two approaches . we comment on this issue at the end of sec . [ s3 ] . ) on the basis of these results , we then address the question of finding asymptotic estimations in sec . [ s2.4 ] and [ s2.5 ] . these , in turn , form the basis of examining to what degree the results of clarke and barron are capable of extension to the quantum domain . let us ( naively ) attempt to apply the formulas of clarke and barron @xcite ( [ eq:4 ] ) and ( [ eq:3 ] ) above to the quantum context under investigation here . we do this by setting @xmath86 to 3 ( the dimensionality of the unit ball which we take as @xmath33 ) , @xmath87 to @xmath88 ( the determinant of the quantum fisher information matrix ( [ eq:8 ] ) ) , so that @xmath89 is @xmath74 , and @xmath90 to @xmath91 . then , we obtain from the expression for the asymptotic redundancy ( [ eq:4 ] ) , @xmath92 where @xmath93 , and from the expression for the asymptotic minimax redundancy ( [ eq:3 ] ) , @xmath94 we shall ( in sec . [ s3 ] ) compare these two formulas , ( [ eq:12 ] ) and ( [ eq:11 ] ) , with the results of sec . [ s2 ] and find some striking similarities and coincidences , particularly associated with the fully mixed state ( @xmath95 ) . these findings will help to support the working hypothesis of this study that there are meaningful extensions to the quantum domain of the ( commutative probabilistic ) theorems of clarke and barron . however , we find that the minimax and maximin properties of the jeffreys prior do not strictly carry over , but transfer only in an approximate sense , which is , nevertheless , still quite remarkable . in any case , we can not formally rule out the possibility that the actual global ( perhaps common ) minimax and maximin are achieved for probability distributions not belonging to the one - parameter family @xmath0 . in analogy to ( * ? ? ? 5.2 ) , the matrices @xmath6 should prove useful for the _ universal _ version of schumacher data compression @xcite . schumacher s result @xcite must be considered as the quantum analogue of shannon s noiseless coding theorem ( see e.g. ( * ? ? ? roughly , _ quantum data compression _ , as proposed by schumacher @xcite , works as follows : a ( quantum ) signal source ( sender " ) generates signal states of a quantum system @xmath96 , the ensemble of possible signals being described by a density operator @xmath97 . the signals are projected down to a dominant " subspace of @xmath96 , the rest is discarded . the information in this dominant subspace is transmitted through a ( quantum ) channel . the receiver tries to reconstruct the original signal by replacing the discarded information by some typical " state . the quality ( or _ faithfulness _ ) of a coding scheme is measured by the _ fidelity _ , which is by definition the overall probability that a signal from the signal ensemble @xmath96 that is transmitted to the receiver passes a validation test comparing it to its original ( see ( * ? ? ? what schumacher shows is that , for each @xmath98 and @xmath99 , under the above coding scheme a compression rate of @xmath100 qubits per signal is possible , where @xmath101 is the _ von neumann entropy _ of @xmath97 , @xmath102 at a fidelity of at least @xmath103 . ( thus , the von neumann entropy is the quantum analogue of the shannon entropy , which features in shannon s classical noiseless coding theorem . indeed , as is easy to see , for diagonal matrices , corresponding to classical probability densities , the right - hand side of ( [ eq:1 ] ) reduces to the shannon entropy . ) this is achieved by choosing as the dominant subspace that subspace of the quantum system @xmath96 which is the span of the eigenvectors of @xmath97 corresponding to the largest eigenvalues , with the property that the eigenvalues add up to at least @xmath104 . consequently , in a universal compression scheme , we propose to project blocks of @xmath4 signals ( qubits ) onto those `` typical '' subspaces of @xmath105-dimensional hilbert space corresponding to as many of the dominant eigenvalues of @xmath3 as it takes to exceed a sum @xmath106 . for all @xmath107 , the leading one of the @xmath108 distinct eigenvalues has multiplicity @xmath109 , and belongs to the ( @xmath109)-dimensional ( bose einstein ) symmetric subspace @xcite . ( projection onto the symmetric subspace has been proposed as a method for stabilizing quantum computations , including quantum state storage @xcite . ) for @xmath110 , the leading eigenvalue can be obtained by dividing the @xmath111-st catalan number that is , @xmath112 by @xmath113 . ( the catalan numbers `` are probably the most frequently occurring combinatorial numbers after the binomial coefficients '' @xcite . ) let us point out to the reader the quite recent important work of petz and sudr @xcite . they demonstrated that in the quantum case in contrast to the classical situation in which there is , as originally shown by chentsov @xcite , essentially only one monotone metric and , therefore , essentially only one form of the fisher information there exists an infinitude of such metrics . `` the monotonicity of the riemannian metric @xmath114 is crucial when one likes to imitate the geometrical approach of [ chentsov ] . an infinitesimal statistical distance has to be monotone under stochastic mappings . we note that the monotonicity of @xmath114 is a strengthening of the concavity of the von neumann entropy . indeed , positive definiteness of @xmath114 is equivalent to the strict concavity of the von neumann entropy and monotonicity is much more than positivity '' @xcite . the monotone metrics on the space of density matrices are given @xcite by the operator monotone functions @xmath115 , such that @xmath116 and @xmath117 . for the choice @xmath118 , one obtains the minimal metric ( of the symmetric logarithmic derivative ) , which serves as the basis of our analysis here . `` in accordance with the work of braunstein and caves , this seems to be the canonical metric of parameter estimation theory . however , expectation values of certain relevant observables are known to lead to statistical inference theory provided by the maximum entropy principle or the minimum relative entropy principle when _ a priori _ information on the state is available . the best prediction is a kind of generalized gibbs state . on the manifold of those states , the differentiation of the entropy functional yields the kubo - mori / bogoliubov metric , which is different from the metric of the symmetric logarithmic derivative . therefore , more than one privileged metric shows up in quantum mechanics . the exact clarification of this point requires and is worth further studies '' @xcite . it remains a possibility , then , that a monotone metric other than the minimal one ( which corresponds to @xmath119 , that is ( [ eq:9 ] ) ) may yield a common global asymptotic minimax and maximin redundancy , thus , fully paralleling the classical / nonquantum results of clarke and barron @xcite . we intend to investigate such a possibility , in particular , for the kubo - mori / bogoliubov metric @xcite . in this section , we implement the analytical approach described in the introduction to extending the work of clarke and barron @xcite to the realm of quantum mechanics , specifically , the two - level systems . such systems are representable by density matrices @xmath49 of the form ( [ eq:6 ] ) . a composite system of @xmath4 independent ( unentangled ) and identical two - level quantum systems is , then , represented by the @xmath4-fold tensor product @xmath81 . in theorem [ t1 ] of sec . [ s2.1 ] , we average @xmath120 with respect to the one - parameter family of probability densities @xmath0 defined in ( [ eq:10 ] ) , obtaining the bayesian density matrices @xmath6 and formulas for their @xmath121 entries . then , in theorem [ t2 ] of sec . [ s2.2 ] , we are able to explicitly determine the @xmath105 eigenvalues and eigenvectors of @xmath6 . using these results , in sec . [ s2.3 ] , we compute the relative entropy of @xmath122 with respect to @xmath6 . then , in sec . [ s2.4 ] , we obtain the asymptotics of this relative entropy for @xmath123 . in sec . [ s2.5 ] , we compute the asymptotics of the von neumann entropy ( see ( [ eq:1 ] ) ) of @xmath6 . all these results will enable us , in sec . [ s3 ] , to ascertain to what extent the results of clarke and barron could be said to carry over to the quantum domain . the @xmath4-fold tensor product @xmath81 is a @xmath69 matrix . to refer to specific rows and columns of @xmath81 , we index them by subsets of the @xmath4-element set @xmath124 . we choose to employ this notation instead of the more familiar use of binary strings , in order to have a more succinct way of writing our formulas . for convenience , we will subsequently write @xmath125 $ ] for @xmath124 . thus , @xmath81 can be written in the form @xmath126},\ ] ] where @xmath127 with @xmath128 denoting the number of elements of @xmath125 $ ] contained in both @xmath48 and @xmath129 , @xmath130 denoting the number of elements _ not _ in both @xmath48 and @xmath129 , @xmath131 denoting the number of elements not in @xmath48 but in @xmath129 , and @xmath132 denoting the number of elements in @xmath48 but not in @xmath129 . in symbols , @xmath133 \backslash ( i\cup j)},\\ n_{\notin\in}&=\v{j\backslash i},\\ n_{\in\notin}&=\v{i\backslash j}.\end{aligned}\ ] ] we consider the average @xmath6 of @xmath81 with respect to the probability density @xmath134 defined in ( [ eq:10 ] ) taken over the unit sphere @xmath135 . this average can be described explicitly as follows . [ t1 ] the average @xmath6 , @xmath136 equals the matrix @xmath137}$ ] , where @xmath138 here , @xmath139 denotes the kronecker delta , @xmath140 if @xmath141 and @xmath142 otherwise . _ it is important for later considerations to observe that because of the term @xmath143 in ( [ e4 ] ) the entry @xmath144 is nonzero if and only if the sets @xmath48 and @xmath129 have the same cardinality . if @xmath48 and @xmath129 have the same cardinality , @xmath145 say , then @xmath144 only depends on @xmath128 , the number of common elements of @xmath48 and @xmath129 , since in this case @xmath130 is expressible as @xmath146 . _ . to compute @xmath144 , we have to compute the integral @xmath147 for convenience , we treat the case that @xmath148 and @xmath149 . the other four cases are treated similarly . first , we rewrite the matrix entries @xmath150 , @xmath151 of course , in order to compute the integral ( [ e5 ] ) , we transform the cartesian coordinates into polar coordinates , @xmath152 thus , using ( [ e6 ] ) , the integral ( [ e5 ] ) is transformed into @xmath153 to evaluate this triple integral we use the following standard formulas : [ e8 ] @xmath154 for any nonnegative integers @xmath96 and @xmath155 . furthermore , we need the beta integral @xmath156 now we consider the integral over @xmath157 in ( [ e7 ] ) . using ( [ e8b ] ) and ( [ e8c ] ) , we see that each summand in ( [ e7 ] ) vanishes if @xmath131 has a parity different from @xmath132 . on the other hand , if @xmath131 has the same parity as @xmath132 , then we can evaluate the integrals over @xmath157 using ( [ e8a ] ) and ( [ e8d ] ) . discarding for a moment the terms independent of @xmath157 and @xmath158 , we have @xmath159 the last line being due to the binomial theorem . these considerations reduce ( [ e7 ] ) to @xmath160 using ( [ e8b ] ) , ( [ e8d ] ) and ( [ e9 ] ) this can be further simplified to @xmath161 next we interchange sums over @xmath162 and @xmath163 and write the sum over @xmath163 in terms of the standard hypergeometric notation @xmath164=\sum _ { k=0 } ^{\infty}\frac { \po{a_1}{k}\cdots\po{a_r}{k } } { k!\,\po{b_1}{k}\cdots\po{b_s}{k } } z^k\ , \ ] ] where the shifted factorial @xmath165 is given by @xmath166 , @xmath167 , @xmath168 . thus we can write ( [ e10 ] ) in the form @xmath169.\end{gathered}\ ] ] the @xmath170 series can be summed by means of gau @xmath170 summation ( see e.g. ( * ? ? ? * ( 1.7.6 ) ; appendix ( iii.3 ) ) ) @xmath171 = \frac { \ga(c)\,\ga(c - a - b ) } { \ga(c - a)\,\ga(c - b ) } , \ ] ] provided the series terminates or @xmath172 . applying ( [ e12 ] ) to the @xmath170 in ( [ e11 ] ) ( observe that it is terminating ) and writing the sum over @xmath163 as a hypergeometric series , the expression ( [ e11 ] ) becomes @xmath173.\end{gathered}\ ] ] another application of ( [ e12 ] ) gives @xmath174 trivially , we have @xmath175 . since ( [ e13 ] ) vanishes unless @xmath176 , we can substitute @xmath177 for @xmath131 in the arguments of the gamma functions . thus , we see that ( [ e13 ] ) equals ( [ e4 ] ) . this completes the proof of the theorem . with the explicit description of the result @xmath6 of averaging @xmath120 with respect to @xmath0 at our disposal , we now proceed to describe the eigenvalues and eigenspaces of @xmath6 . the eigenvalues are given in theorem [ t2 ] . lemma [ l4 ] gives a complete set of eigenvectors of @xmath6 . the reader should note that , though complete , this is simply a set of linearly independent eigenvectors and not a fully orthogonal set . [ t2 ] the eigenvalues of the @xmath178 matrix @xmath6 , the entries of which are given by _ ( [ e4 ] ) _ , are @xmath179 with respective multiplicities @xmath180 the theorem will follow from a sequence of lemmas . we state the lemmas first , then prove theorem [ t2 ] assuming the truth of the lemmas , and after that provide proofs of the lemmas . in the first lemma some eigenvectors of the matrix @xmath6 are described . clearly , since @xmath6 is a @xmath178 matrix , the eigenvectors are in @xmath105-dimensional space . as we did previously , we index coordinates by subsets of @xmath125 $ ] , so that a generic vector is @xmath181}$ ] . in particular , given a subset @xmath182 of @xmath125 $ ] , the symbol @xmath183 denotes the standard unit vector with a 1 in the @xmath182-th coordinate and 0 elsewhere , i.e. , @xmath184}$ ] . now let @xmath185 be integers with @xmath186 and let @xmath187 and @xmath188 be two disjoint @xmath189-element subsets @xmath187 and @xmath188 of @xmath125 $ ] . then we define the vector @xmath190 by @xmath191\backslash ( a\cup b),\ \v{y}=s - h } { \sum _ { x\subseteq a } ^{}}(-1)^{\v{x}}\ , e_{x\cup x'\cup y},\ ] ] where @xmath192 is the _ complement of @xmath8 in @xmath188 _ " by which we mean that if @xmath8 consists of the @xmath193- , @xmath194- , -largest elements of @xmath187 , @xmath195 , then @xmath192 consists of all elements of @xmath188 _ except for _ the @xmath193- , @xmath194- , -largest elements of @xmath188 . for example , let @xmath196 . then the vector @xmath197 is given by @xmath198 ( in this special case , the possible subsets @xmath8 of @xmath199 in the sum in ( [ e16 ] ) are @xmath200 , @xmath201 , @xmath202 , @xmath203 , with corresponding complements in @xmath204 being @xmath205 , @xmath206 , @xmath207 , @xmath200 , respectively , and the possible sets @xmath208 are @xmath209 , @xmath210 , @xmath211 . ) observe that all sets @xmath212 which occur as indices in ( [ e16 ] ) have the same cardinality @xmath213 . [ l3 ] let @xmath185 be integers with @xmath186 and let @xmath187 and @xmath188 be disjoint @xmath189-element subsets of @xmath125 $ ] . then @xmath190 as defined in _ ( [ e16 ] ) _ is an eigenvector of the matrix @xmath6 , the entries of which are given by _ ( [ e4 ] ) _ , for the eigenvalue @xmath214 , where @xmath214 is given by _ ( [ e14])_. we want to show that the multiplicity of @xmath214 equals the expression in ( [ e15 ] ) . of course , lemma [ l3 ] gives many more eigenvectors for @xmath214 . therefore , in order to describe a basis for the corresponding eigenspace , we have to restrict the collection of vectors in lemma [ l3 ] . we do this in the following way . fix @xmath189 , @xmath215 . let @xmath18 be a lattice path in the plane integer lattice @xmath216 , starting in @xmath217 , consisting of @xmath218 up - steps @xmath219 and @xmath189 down - steps @xmath220 , which never goes below the @xmath66-axis . figure 1 displays an example with @xmath196 and @xmath221 . clearly , the end point of @xmath18 is @xmath222 . we call a lattice path which starts in @xmath217 and never goes below the @xmath66-axes a _ ballot path_. ( this terminology is motivated by its relation to the ( two - candidate ) _ ballot problem _ , see e.g. ( * ? ? ? * ch . 1 , sec . 1 ) . an alternative term for ballot path which is often used is `` dyck path '' , see e.g. @xcite . ) we will use the abbreviation `` b.p . '' for `` ballot path '' in displayed formulas . given such a lattice path @xmath18 , label the steps from @xmath62 to @xmath4 , as is indicated in figure 1 . then define @xmath223 to be set of all labels corresponding to the first @xmath189 up - steps of @xmath18 and @xmath224 to be set of all labels corresponding to the @xmath189 down - steps of @xmath18 . in the example of figure 1 we have for the choice @xmath221 that @xmath225 and @xmath226 . thus , to each @xmath189 and @xmath213 , @xmath227 , and @xmath18 as above we can associate the vector @xmath228 . in our running example of figure 1 the vector @xmath229 would hence be @xmath197 , the vector in ( [ e17 ] ) . to have a more concise form of notation , we will write @xmath230 for @xmath228 from now on . [ l4 ] the set of vectors @xmath231 is linearly independent . the final lemma tells us how many such vectors @xmath230 there are . [ l5 ] the number of ballot paths from @xmath217 to @xmath222 is @xmath232 . the total number of all vectors in the set _ ( [ e18 ] ) _ is @xmath105 . now , let us for a moment assume that lemmas [ l3][l5 ] are already proved . then , theorem [ t2 ] follows immediately , as it turns out . proof of theorem [ t2 ] . consider the set of vectors in ( [ e18 ] ) . by lemma [ l3 ] we know that it consists of eigenvectors for the matrix @xmath6 . in addition , lemma [ l4 ] tells us that this set of vectors is linearly independent . furthermore , by lemma [ l5 ] the number of vectors in this set is exactly @xmath105 , which is the dimension of the space where all these vectors are contained . therefore , they must form a basis of the space . lemma [ l3 ] says more precisely that @xmath230 is an eigenvector for the eigenvalue @xmath214 . from what we already know , this implies that for fixed @xmath189 the set @xmath233 forms a basis for the eigenspace corresponding to @xmath214 . therefore , the dimension of the eigenspace corresponding to @xmath214 equals the number of possible numbers @xmath213 times the number of possible lattice paths @xmath18 . this is exactly @xmath234 the number of possible lattice paths @xmath18 being given by the first statement of lemma [ l5 ] . this expression equals exactly the expression ( [ e15 ] ) . thus , theorem [ t2 ] is proved . now we turn to the proofs of the lemmas . proof of lemma [ l3 ] . let @xmath185 and @xmath235 be fixed , satisfying the restrictions in the statement of the lemma . we have to show that @xmath236 restricting our attention to the @xmath48-th component , we see from the definition ( [ e16 ] ) of @xmath190 that we need to establish @xmath237\backslash ( a\cup b),\ \v{y}=s - h } { \sum _ { x\subseteq a } ^{}}\kern-1 cm z_{i , x\cup x'\cup y}\,(-1)^{\v{x}}= \begin{cases } \la_h ( -1)^{\v{u}}&\text { if $ i$ is of the form $ u\cup u'\cup v$}\\ & \text { for some $ u$ and $ v$ , $ u\subseteq a$,}\\ & \text { $ v\subseteq [ n]\backslash ( a\cup b)$ , $ \v{v}=s - h$}\\ 0&\text { otherwise.}\end{cases}\ ] ] we prove ( [ e19 ] ) by a case by case analysis . the first two cases cover the case otherwise " in ( [ e19 ] ) , the third case treats the first alternative in ( [ e19 ] ) . . as we observed earlier , the cardinality of any set @xmath212 which occurs as index at the left - hand side of ( [ e19 ] ) equals @xmath213 . the cardinality of @xmath48 however is different from @xmath213 . as we observed in the remark after theorem [ t1 ] , this implies that any coefficient @xmath238 on the left - hand side vanishes . thus , ( [ e19 ] ) is proved in this case . . now the sum on the left - hand side of ( [ e19 ] ) contains nonzero contributions . we have to show that they cancel each other . we do this by grouping summands in pairs , the sum of each pair being 0 . consider a set @xmath212 which occurs as index at the left - hand side of ( [ e19 ] ) . let @xmath239 be minimal such that 1 . either : the @xmath239-th largest element of @xmath187 and the @xmath239-th largest element of @xmath188 are both in @xmath48 , 2 . or : the @xmath239-th largest element of @xmath187 and the @xmath239-th largest element of @xmath188 are both not in @xmath48 . that such an @xmath239 must exist is guaranteed by our assumptions about @xmath48 . now consider @xmath8 and @xmath192 . if the @xmath239-th largest element of @xmath187 is contained in @xmath8 then the @xmath239-th largest element of @xmath188 is not contained in @xmath192 , and vice versa . define a new set @xmath240 by adding to @xmath8 the @xmath239-th largest element of @xmath187 if it is not already contained in @xmath8 , respectively by removing it from @xmath8 if it is contained in @xmath8 . then , it is easily checked that @xmath241 on the other hand , we have @xmath242 since the cardinalities of @xmath8 and @xmath240 differ by @xmath243 . both facts combined give @xmath244 hence , we have found two summands on the left - hand side of ( [ e19 ] ) which cancel each other . summarizing , this construction finds for any @xmath245 sets @xmath246 such that the corresponding summands on the left - hand side of ( [ e19 ] ) cancel each other . moreover , this construction applied to @xmath246 gives back @xmath245 . hence , what the construction does is exactly what we claimed , namely it groups the summands into pairs which contribute 0 to the whole sum . therefore the sum is 0 , which establishes ( [ e19 ] ) in this case also . . this assumption implies in particular that the cardinality of @xmath48 is @xmath213 . from the remark after the statement of theorem [ t1 ] we know that in our situation @xmath238 depends only on the number of common elements in @xmath48 and @xmath212 . thus , the left - hand side in ( [ e19 ] ) reduces to @xmath247 where @xmath248 is the number of sets @xmath212 , for some @xmath8 and @xmath208 , @xmath249 , @xmath250\backslash ( a\cup b)$ ] , @xmath251 , which have @xmath252 elements in common with @xmath48 , and which have @xmath253 elements in common with @xmath254 . clearly , we used expression ( [ e4 ] ) with @xmath255 and @xmath256 . to determine @xmath248 , note first that there are @xmath257 possible sets @xmath258 which intersect @xmath259 in exactly @xmath253 elements . next , let us assume that we already made a choice for @xmath258 . in order to determine the number of possible sets @xmath208 such that @xmath212 has @xmath252 elements in common with @xmath48 , we have to choose @xmath260 elements from @xmath261 , for which we have @xmath262 possibilities , and we have to choose @xmath263 elements from @xmath125\backslash ( i\cup a\cup b)$ ] to obtain a total number of @xmath213 elements , for which we have @xmath264 possibilities . hence , @xmath265 so it remains to evaluate the double sum ( [ e20 ] ) , using the expression ( [ e21 ] ) for @xmath248 . we start by writing the sum over @xmath162 in ( [ e20 ] ) in hypergeometric notation , @xmath266 } .\end{gathered}\ ] ] to the @xmath267 series we apply a transformation formula of thomae ( see e.g. ( * ? ? ? * ( 3.1.1 ) ) ) , @xmath268 = { \frac { ( { \textstyle -b + e } ) _ { m } } { ( { \textstyle e } ) _ { m } } } { } _ { 3 } f _ { 2 } \!\left [ \begin{matrix } { -m , b , -a + d}\\ { d , 1 + b - e - m}\end{matrix } ; { \displaystyle 1}\right ] \ ] ] where @xmath269 is a nonnegative integer . we write the resulting @xmath267 again as a sum over @xmath162 , then interchange sums over @xmath163 and @xmath162 , and write the ( now ) inner sum over @xmath163 in hypergeometric notation . thus we obtain @xmath270.\end{gathered}\ ] ] the @xmath170 series in this expression is terminating because @xmath271 is a nonpositive integer . hence , it can be summed by means of gau sum ( [ e12 ] ) . writing the remaining sum over @xmath162 in hypergeometric notation , the above expression becomes @xmath272.\ ] ] again , the @xmath170 series is terminating and so is summable by means of ( [ e12 ] ) . thus , we get @xmath273 which is exactly the expression ( [ e14 ] ) for @xmath214 times @xmath274 . this proves ( [ e19 ] ) in this case . the proof of lemma [ l3 ] is now complete . . we know from lemma [ l3 ] that @xmath230 lies in the eigenspace for the eigenvalue @xmath214 , with @xmath214 being given in ( [ e14 ] ) . the @xmath214 s , @xmath275 , are all distinct , so the corresponding eigenspaces are linearly independent . therefore it suffices to show that for any _ fixed _ @xmath189 the set of vectors @xmath233 is linearly independent . on the other hand , a vector @xmath190 lies in the space spanned by the standard unit vectors @xmath183 with @xmath276 . clearly , as @xmath213 varies , these spaces are linearly independent . therefore , it suffices to show that for any _ fixed _ @xmath189 _ and _ @xmath213 the set of vectors @xmath277 is linearly independent . so , let us fix integers @xmath189 and @xmath213 with @xmath227 , and let us suppose that there is some vanishing linear combination @xmath278 we have to establish that @xmath279 for all ballot paths @xmath18 from @xmath217 to @xmath222 . we prove this fact by induction on the set of ballot paths from @xmath217 to @xmath222 . in order to make this more precise , we need to impose a certain order on the ballot paths . given a ballot path @xmath18 from @xmath217 to @xmath222 , we define its _ front portion _ @xmath280 to be the portion of @xmath18 from the beginning up to and including @xmath18 s @xmath189-th up - step . for example , choosing @xmath221 , the front portion of the ballot path in figure 1 is the subpath from @xmath217 to @xmath281 . note that @xmath280 can be any ballot path starting in @xmath217 with @xmath189 up - steps and less than @xmath189 down - steps . we order such front portions lexicographically , in the sense that @xmath282 is before @xmath283 if and only if @xmath282 and @xmath283 agree up to some point and then @xmath282 continues with an up - step while @xmath283 continues with a down - step . now , here is what we are going to prove : fix any possible front portion @xmath284 . we shall show that @xmath279 for all @xmath18 with front portion @xmath280 equal to @xmath284 , _ given that it is already known that @xmath285 for all @xmath286 with a front portion @xmath287 that is before @xmath284 . _ clearly , by induction , this would prove @xmath279 for _ all _ ballot paths @xmath18 from @xmath217 to @xmath222 . let @xmath284 be a possible front portion , i.e. , a ballot path starting in @xmath217 with exactly @xmath189 up - steps and less than @xmath189 down - steps . as we did earlier , label the steps of @xmath284 by @xmath288 , and denote the set of labels corresponding to the down - steps of @xmath284 by @xmath289 . we write @xmath290 for @xmath291 , the number of all down - steps of @xmath284 . observe that then the total number of steps of @xmath284 is @xmath292 . now , let @xmath182 be a fixed @xmath293-element subset of @xmath294 . furthermore , let @xmath295 be a set of the form @xmath296 , where @xmath297 and @xmath298 , and such that @xmath299 . we consider the coefficient of @xmath300 in the left - hand side of ( [ e23 ] ) . to determine this coefficient , we have to determine the coefficient of @xmath300 in @xmath230 , for all @xmath18 . we may concentrate on those @xmath18 whose front portion @xmath280 is equal to or later than @xmath284 , since our induction hypothesis says that @xmath279 for all @xmath18 with @xmath280 before @xmath284 . so , let @xmath18 be a ballot path from @xmath217 to @xmath222 with front portion equal to or later than @xmath284 . we claim that the coefficient of @xmath300 in @xmath230 is zero unless the set @xmath224 of down - steps of @xmath18 is contained in @xmath295 . let the coefficient of @xmath300 in @xmath230 be nonzero . to establish the claim , we first prove that the front portion @xmath280 of @xmath18 has to equal @xmath284 . suppose that this is not the case . then the front portion of @xmath18 runs in parallel with @xmath284 for some time , say for the first @xmath301 steps , with some @xmath302 , and then @xmath284 continues with an up - step and @xmath280 continues with a down - step ( recall that @xmath280 is equal to or later than @xmath284 ) . by ( [ e16 ] ) we have @xmath303\backslash ( a_p\cup b_p),\ \v{y}=s - h } { \sum _ { x\subseteq a_p } ^{}}(-1)^{\v{x}}\ , e_{x\cup x'\cup y}.\ ] ] we are assuming that the coefficient of @xmath300 in @xmath230 is nonzero , therefore @xmath295 must be of the form @xmath304 , with @xmath245 as described in ( [ e24 ] ) . we are considering the case that the @xmath269-th step of @xmath280 is a down - step , whence @xmath305 , while the @xmath269-th step of @xmath284 is an up - step , whence @xmath306 . by definition of @xmath295 , we have @xmath307 , whence @xmath308 . summarizing so far , we have @xmath305 , @xmath308 , for some @xmath309 , and @xmath304 , for some @xmath245 as described in ( [ e24 ] ) . in particular we have @xmath310 . now recall that @xmath192 is the `` complement of @xmath8 in @xmath224 '' . this says in particular that , if @xmath269 is the @xmath311-th largest element in @xmath224 , then the @xmath311-th largest element of @xmath223 , @xmath312 say , is an element of @xmath8 , and so of @xmath295 . by construction of @xmath223 and @xmath224 , @xmath312 is smaller than @xmath269 , so in particular @xmath313 . as we already observed , there holds @xmath314 , so we have @xmath315 , i.e. , the @xmath312-th step of @xmath284 is a down - step . on the other hand , we assumed that @xmath18 and @xmath284 run in parallel for the first @xmath301 steps . since @xmath316 , the set of up - steps of @xmath18 , the @xmath312-th step of @xmath18 is an up - step . we have @xmath317 , therefore the @xmath312-th step of @xmath284 must be an up - step also . this is absurd . therefore , given that the coefficient of @xmath300 in @xmath230 is nonzero , the front portion @xmath280 of @xmath18 has to equal @xmath284 . now , let @xmath18 be a ballot path from @xmath217 to @xmath222 with front portion equal to @xmath284 , and suppose that @xmath295 has the form @xmath318 , for some @xmath245 as described in ( [ e24 ] ) . by definition of the front portion , the set @xmath223 of up - steps of @xmath18 has the property @xmath319 . since @xmath320 , these are the labels of exactly @xmath189 up - steps . since the cardinality of @xmath223 is exactly @xmath189 by definition , we must have @xmath321 . because of @xmath314 , which we already used a number of times , @xmath223 and @xmath295 are disjoint , which in particular implies that @xmath223 and @xmath8 are disjoint . however , @xmath8 is a subset of @xmath223 by definition , so @xmath8 must be empty . this in turn implies that @xmath322 . this says nothing else but that the set @xmath224 of down - steps of @xmath18 equals @xmath192 and so is contained in @xmath295 . this establishes our claim . in fact , we proved more . we saw that @xmath295 has the form @xmath323 , with @xmath324 . this implies that the coefficient of @xmath300 in @xmath230 , as given by ( [ e24 ] ) , is actually @xmath325 . comparison of coefficients of @xmath300 in ( [ e23 ] ) then gives @xmath326 for any @xmath296 , where @xmath297 and @xmath298 , and such that @xmath299 . now , we sum both sides of ( [ e25 ] ) over all such sets @xmath295 , keeping the cardinality of @xmath327 and @xmath328 fixed , say @xmath329 , enforcing @xmath330 , for a fixed @xmath162 , @xmath331 . for a fixed ballot path @xmath18 from @xmath217 to @xmath222 , with front portion @xmath284 , with @xmath332 down - steps in @xmath182 , and hence with @xmath163 down - steps in @xmath333 , there are @xmath334 such sets @xmath297 containing all the @xmath332 down - steps of @xmath18 in @xmath182 , and there are @xmath335 such sets @xmath298 containing all the @xmath163 down - steps of @xmath18 in @xmath333 . therefore , summing up ( [ e25 ] ) gives @xmath336 denoting the inner sum in ( [ e26 ] ) by @xmath337 , we see that ( [ e26 ] ) represents a non - degenerate triangular system of linear equations for @xmath338 . therefore , all the quantities @xmath338 have to equal 0 . in particular , we have @xmath339 . now , @xmath340 consists of just a single term @xmath341 , with @xmath18 being the ballot path from @xmath217 to @xmath222 , with front portion @xmath284 , and the labels of the @xmath342 down - steps besides those of @xmath284 being exactly the elements of @xmath182 . therefore , we have @xmath279 for this ballot path . the set @xmath182 was an arbitrary @xmath293-subset of @xmath294 . thus , we have proved @xmath279 for any ballot path @xmath18 from @xmath217 to @xmath222 with front portion @xmath284 . this completes our induction proof . . that the number of ballot paths from @xmath217 to @xmath222 equals @xmath232 is a classical combinatorial result ( see e.g. ( * ? ? ? * theorem 1 with @xmath343 ) ) . from this it follows that the total number of vectors in the set ( [ e18 ] ) is @xmath344 to evaluate this sum , note that the summand is invariant under the substitution @xmath345 . therefore , extending the range of summation in ( [ e27 ] ) to @xmath346 and dividing the result by @xmath347 gives the same value . so , the cardinality of the set ( [ e18 ] ) is also given by @xmath348 the reader will not have any difficulty in splitting this sum into three parts so that each part can be summed by means of the binomial theorem . ( computer algebra systems like _ maple _ or _ mathematica _ do this automatically . ) the result is exactly @xmath105 , as was claimed . in fact , theorem [ t2 ] can be generalized to a wider class of matrices . [ t6 ] let @xmath349}$ ] be the @xmath350 matrix defined by @xmath351 where @xmath128 , etc . , have the same meaning as earlier , and where @xmath352 is a function of @xmath66 which is symmetric , i.e. , @xmath353 . then , the eigenvalues of @xmath354 are @xmath355 with respective multiplicities @xmath356 independent of @xmath213 . the above proof of theorem [ t2 ] has to be adjusted only insignificantly to yield a proof of theorem [ t6 ] . in particular , the vector @xmath190 as defined in ( [ e16 ] ) is an eigenvector for @xmath357 , for any two disjoint @xmath189-element subsets @xmath187 and @xmath188 of @xmath125 $ ] , and the set ( [ e18 ] ) is a basis of eigenvectors for @xmath354 . we now apply the preceding results to compute the relative entropy @xmath359 of @xmath81 with respect to @xmath6 . utilizing the definition ( [ eq:5 ] ) of relative entropy and employing the property @xcite that @xmath360 , it is given by @xmath361 for the first term , for the entropy @xmath362 of @xmath49 , @xmath49 being given by ( [ eq:6 ] ) , we have , using spherical coordinates @xmath363 , so that @xmath364 , @xmath365 concerning the second term in ( [ e31 ] ) , we have the following theorem . [ t7 ] let @xmath366}$ ] be the matrix with entries @xmath144 given in _ ( [ e4])_. then , we have @xmath367 with @xmath214 as given in _ ( [ e14 ] ) _ , and with @xmath93 . before we move on to the proof , we note that theorem [ t7 ] gives us the following expression for the relative entropy of @xmath81 with respect to @xmath6 [ c8 ] the relative entropy @xmath359 of @xmath81 with respect to @xmath6 equals @xmath368 with @xmath214 as given in _ ( [ e14 ] ) _ , and with @xmath93 . . one way of determining the trace of a linear operator @xmath369 is to choose a basis of the vector space , @xmath370\}$ ] say , write the action of @xmath369 on the basis elements in the form @xmath371 and then form the sum @xmath372 of the `` diagonal '' coefficients , which gives exactly the trace of @xmath369 . clearly , we choose as a basis our set ( [ e18 ] ) of eigenvectors for @xmath6 . to determine the action of @xmath373 we need only to find the action of @xmath122 on the vectors in the set ( [ e18 ] ) . we claim that this action can be described as @xmath374 for any basis vector @xmath230 in ( [ e18 ] ) . to see this , consider the @xmath48-th component of @xmath375 , i.e. , the coefficient of @xmath376 in @xmath375 , @xmath377 $ ] . by the definition ( [ e16 ] ) of @xmath230 it equals @xmath378\backslash ( a_p\cup b_p),\ \v{y}=s - h } { \sum _ { x\subseteq a_p } ^{}}\kern-1 cm r_{i , x\cup x'\cup y}\,(-1)^{\v{x}},\ ] ] where @xmath150 denotes the @xmath379-entry of @xmath122 . ( recall that @xmath150 is given explicitly in ( [ e2 ] ) . ) now , it should be observed that we did a similar calculation already , namely in the proof of lemma [ l3 ] . in fact , the expression ( [ e34 ] ) is almost identical with the left - hand side of ( [ e19 ] ) . the essential difference is that @xmath144 is replaced by @xmath150 for all @xmath129 ( the nonessential difference is that @xmath235 are replaced by @xmath380 , respectively ) . therefore , we can partially rely upon what was done in the proof of lemma [ l3 ] . we distinguish between the same cases as in the proof of lemma [ l3 ] . . we do not have to worry about this case , since @xmath376 then lies in the span of vectors @xmath381 with @xmath382 , which is taken care of in ( [ e33 ] ) . . essentially the same arguments as those in case 2 in the proof of lemma [ l3 ] show that the term ( [ e34 ] ) vanishes for this choice of @xmath48 . of course , one has to use the explicit expression ( [ e2 ] ) for @xmath150 . . in case 3 in the proof of lemma [ l3 ] we observed that there are @xmath248 sets @xmath212 , for some @xmath8 and @xmath208 , @xmath383 , @xmath250\backslash ( a_p\cup b_p)$ ] , @xmath251 , which have @xmath252 elements in common with @xmath48 , and which have @xmath253 elements in common with @xmath384 , where @xmath248 is given by ( [ e21 ] ) . then , using the explicit expression ( [ e2 ] ) for @xmath150 , it is straightforward to see that the expression ( [ e34 ] ) equals @xmath385 in this case . this establishes ( [ e33 ] ) . now we are in the position to write down an expression for the trace of @xmath386 . by theorem [ t2 ] and by ( [ e33 ] ) we have @xmath387 from what was said at the beginning of this proof , in order to obtain the trace of @xmath386 , we have to form the sum of all the `` diagonal '' coefficients in ( [ e35 ] ) . using the first statement of lemma [ l5 ] and replacing @xmath388 by @xmath389 , we see that it is @xmath390 in order to see that this expression equals ( [ e30 ] ) , we have to prove @xmath391 we start with the left - hand side of ( [ e37 ] ) and write the inner sum in hypergeometric notation , thus obtaining @xmath392.\ ] ] to the @xmath170 series we apply the transformation formula ( ( * ? ? ? * ( 1.8.10 ) , terminating form ) @xmath393 = { \frac { ( { \textstyle c - a } ) _ { m } } { ( { \textstyle c } ) _ { m } } } { } _ { 2 } f _ { 1 } \!\left [ \begin{matrix } { -m , a}\\ { 1 + a - c - m}\end{matrix } ; { \displaystyle 1 - z}\right ] , \ ] ] where @xmath269 is a nonnegative integer . we write the resulting @xmath170 series again as a sum over @xmath163 . in the resulting expression we exchange sums so that the sum over @xmath162 becomes the innermost sum . thus , we obtain @xmath394 clearly , the innermost sum can be evaluated by the binomial theorem . then , we interchange sums over @xmath213 and @xmath163 . the expression that results is @xmath395 again , we can apply the binomial theorem . thus , we reduce our expression on the left - hand side of ( [ e37 ] ) to @xmath396 now , we replace @xmath397 by its binomial expansion @xmath398 , interchange sums over @xmath163 and @xmath158 , and write the ( now ) inner sum over @xmath163 in hypergeometric notation . this gives @xmath399\bigg ) .\end{gathered}\ ] ] finally , this @xmath170 series can be summed by means of gau summation ( [ e12 ] ) . simplifying , we have @xmath400 which is easily seen to equal the right - hand side in ( [ e37 ] ) . this completes the proof of the theorem . in the preceding subsection , we obtained in corollary [ c8 ] the general formula ( [ e32 ] ) for the relative entropy of @xmath122 with respect to the bayesian density matrix @xmath6 . we , now , proceed to find its asymptotics for @xmath123 . we prove the following theorem . [ t9 ] the asymptotics of the relative entropy @xmath359 of @xmath81 with respect to @xmath6 for a fixed @xmath93 with @xmath402 is given by @xmath403 in the case @xmath95 , this means that the asymptotics is given by the expression _ ( [ a3 ] ) _ in the limit @xmath404 , i.e. , by @xmath405 for any fixed @xmath98 , the @xmath406 term in _ ( [ a3 ] ) _ is uniform in @xmath107 and @xmath407 as long as @xmath408 . for @xmath409 the asymptotics is given by @xmath410 also here , the @xmath406 term is uniform in @xmath107 . _ it is instructive to observe that , although a comparison of ( [ a3 ] ) and ( [ a5 ] ) seems to suggest that the asymptotics of the relative entropy of @xmath81 with respect to @xmath6 behaves completely differently for @xmath402 and @xmath409 , the two cases are really quite compatible . in fact , letting @xmath407 tend to @xmath62 in ( [ a3 ] ) shows that ( ignoring the error term ) the asymptotic expression approaches @xmath411 for @xmath412 , @xmath413 for @xmath414 , and it approaches @xmath415 for @xmath110 . this indicates that , for @xmath409 , the order of magnitude of the relative entropy of @xmath81 with respect to @xmath6 should be larger than @xmath416 if @xmath412 , smaller than @xmath417 if @xmath414 , and exactly @xmath416 if @xmath110 . how much larger or smaller is precisely what formula ( [ a5 ] ) tells us : the order of magnitude is @xmath418 , and in the case @xmath110 the asymptotics is , in fact , @xmath419 . _ sketch of proof of theorem [ t9 ] . we have to estimate the expression ( [ e32 ] ) for large @xmath4 . clearly , it suffices to concentrate on the sum in ( [ e32 ] ) . because of @xmath420 , this sum can be also expressed as @xmath421 for @xmath409 this sum reduces to @xmath422 , @xmath423 being given by ( [ e14 ] ) . a straightforward application of stirling s formula then leads to ( [ a5 ] ) . from now on let @xmath402 . we recall that @xmath214 is given by ( [ e14 ] ) . consequently , we expand the logarithm in ( [ a6 ] ) according to the addition rule , and split the sum ( [ a6 ] ) into the corresponding parts . the individual parts can be summed by means of the binomial theorem , except for the parts which involve @xmath424 . ( to be precise , they have to be split appropriately before the binomial theorem can be applied . computer algebra systems like _ maple _ or _ mathematica _ do this automatically . ) in order to handle the terms which contain @xmath424 , we use stirling s formula @xmath425 again , after splitting , all the resulting sums can be evaluated by means of the binomial theorem , except for @xmath426 the asymptotics of this sum can now easily ( if though tediously ) be determined by making use of a taylor expansion of @xmath427 about @xmath428 ( i.e. , at @xmath429 ) with sufficiently many terms . if everything is put together , the result is ( [ a3 ] ) . the main result of this section describes the asymptotics of the von neumann entropy ( [ eq:1 ] ) of @xmath6 . in view of the explicit description of the eigenvalues of @xmath6 and their multiplicities in theorem [ t2 ] , this entropy equals @xmath430 with @xmath214 being given by ( [ e14 ] ) . [ t11 ] the asymptotics of the von neumann entropy @xmath431 of @xmath6 is given by @xmath432 where @xmath433 is the digamma function , @xmath434 sketch of proof . we have to estimate the expression ( [ eq : vonn ] ) for large @xmath4 . we proceed as in the proof of theorem [ t9 ] . first we use the property @xmath420 to rewrite the sum ( [ eq : vonn ] ) as @xmath435 next , while recalling that @xmath214 is given by ( [ e14 ] ) , we expand the logarithm in ( [ b11 ] ) according to the addition rule , and split the sum ( [ b11 ] ) into the corresponding parts . here , the individual parts can be summed by means of gau @xmath170 summation ( [ e12 ] ) , except for the parts which involve @xmath424 . ( again , to be precise , they have to be split appropriately before the gau summation can be applied , which is done automatically by computer algebra systems like _ maple _ or _ mathematica_. ) to handle the terms which contain @xmath424 , we invoke again stirling s formula ( [ eq : stirling ] ) . after splitting , all the resulting sums can be evaluated by means of gau @xmath170 summation ( [ e12 ] ) , except for @xmath436 now , to get an asymptotic estimate for this sum , as @xmath4 tends to infinity , is not as obvious as it was for ( [ a7 ] ) . the essential `` trick '' needed was kindly indicated to us by peter grabner : an asymptotic estimate ( in fact , an exact result ) for ( [ b15 ] ) with @xmath427 replaced by @xmath437 can be obtained without difficulty ( but with some amount of tedious calculation ) by starting with the sum @xmath438 evaluating it by applying gau @xmath170 summation ( [ e12 ] ) , differentiating both sides of the resulting equation with respect to @xmath439 , and by finally setting @xmath440 . finally one relates the result to ( [ b15 ] ) by using the asymptotic expansion @xmath441 . if everything is put together , the right - hand side of ( [ b2 ] ) is obtained . let us , first , compare the formula ( [ eq:4 ] ) for the asymptotic redundancy of clarke and barron to that derived here ( [ a3 ] ) for the two - level quantum systems , in terms of the one - parameter family of probability densities @xmath0 , @xmath442 , given in ( [ eq:10 ] ) . since the unit ball or bloch sphere of such systems is three - dimensional in nature , we are led to set the dimension @xmath86 of the parameter space in ( [ eq:4 ] ) to 3 . the quantum fisher information matrix @xmath31 for that case was taken to be ( [ eq:8 ] ) , while the role of the probability function @xmath90 is played by @xmath0 . under these substitutions , it was seen in the introduction that formula ( [ eq:4 ] ) reduces to ( [ eq:12 ] ) . then , we see that for @xmath443 , formulas ( [ a3 ] ) and ( [ eq:12 ] ) coincide except for the presence of the monotonically increasing ( nonclassical / quantum ) term @xmath444 ( see figure 2 for a plot of this term @xmath445 `` nats '' of information equalling one `` bit '' ) in ( [ a3 ] ) . ( this term would have to be replaced by @xmath446 that is , its limit for @xmath447 to give ( [ eq:12 ] ) . ) in particular , the order of magnitude , @xmath448 , is precisely the same in both formulas . for the particular case @xmath95 , the asymptotic formula ( [ a3 ] ) ( see ( [ a4 ] ) ) precisely coincides with ( [ eq:12 ] ) . in the case @xmath409 , however , i.e. , when we consider the boundary of the parameter space ( represented by the unit sphere ) , the situation is slightly tricky . due to the fact that the formula of clarke and barron holds only for interior points of the parameter space , we can not expect that , in general , our formula will resemble that of clarke and barron . however , if the probability density , @xmath0 , is concentrated on the boundary of the sphere , then we may disregard the interior of the sphere , and consider the boundary of the sphere as the _ true _ parameter space . this parameter space is _ two - dimensional _ and consists of interior points throughout . indeed , the probability density @xmath0 is concentrated on the boundary of the sphere if we choose @xmath449 since , as we remarked in the introduction , in the limit @xmath450 , the distribution determined by @xmath0 tends to the uniform distribution over the boundary of the sphere . let us , again , ( naively ) attempt to apply clarke and barron s formula ( [ eq:4 ] ) to that case . we parameterize the boundary of the sphere by polar coordinates @xmath451 , @xmath452 the probability density induced by @xmath0 in the limit @xmath450 then is @xmath453 , the density of the uniform distribution . using ( * ? ? ? * eq . ( 8) ) ( see footnote 2 ) , the quantum ( symmetric logarithmic derivative ) fisher information matrix turns out to be @xmath454 its determinant equalling , therefore , @xmath455 . so , setting @xmath456 and substituting @xmath453 for @xmath457 and @xmath455 for @xmath458 in ( [ eq:4 ] ) gives @xmath459 . on the other hand , our formula ( [ a5 ] ) , for @xmath449 , gives @xmath460 . so , again , the terms differ only by a constant . in particular , the order of magnitude is again the same . let us now focus our attention on the asymptotic minimax redundancy ( [ eq:3 ] ) of clarke and barron . if in ( [ eq:3 ] ) we again set @xmath86 to 3 , we obtain ( [ eq:11 ] ) , which , numerically , is @xmath461 . clarke and barron prove that this minimax expression is only attained by the ( classical ) jeffreys prior . in order to derive its quantum counterpart at least , a restricted ( to the family @xmath0 ) version we have to determine the behavior of @xmath462 for @xmath463 . by theorem [ t9 ] we know that for large @xmath4 the relative entropy @xmath359 equals @xmath464 up to an error of the order @xmath465 , which is uniform in @xmath107 and @xmath407 as long as @xmath408 for any fixed @xmath98 . let us for the moment ignore the error term . then what we have to do is to determine the minimax of the expression ( [ d6a ] ) , that is @xmath466 where @xmath467 this is an easy task . first of all , if @xmath468 then the function @xmath469 is unbounded at @xmath409 . hence , to determine the minimax , we can ignore that range of @xmath107 . if @xmath76 , then @xmath469 is maximal at @xmath409 , at which it attains the value @xmath470 . on the other hand , if @xmath471 then @xmath469 attains a maximum in the interior of the interval @xmath472 . to determine this maximum , we differentiate @xmath469 with respect to @xmath407 , to obtain @xmath473 equating this to 0 gives @xmath474 now we have to express @xmath407 in terms of @xmath107 , @xmath475 say , substitute in @xmath469 , and determine @xmath476 . however , equivalently , we can express @xmath107 in terms of @xmath407 , @xmath477 say ( as was previously done in ( [ d6d ] ) ) , substitute in @xmath469 , and determine @xmath478 . in order to do so , we differentiate @xmath479 with respect to @xmath407 , equate the result to 0 , and solve for @xmath407 . numerically , the result is @xmath480 . substituting this back into ( [ d6d ] ) , we obtain @xmath481 . the value of @xmath469 at these values of @xmath407 and @xmath107 is @xmath482 . this is smaller than that previously found ( @xmath483 ) for @xmath76 , so that particular value of @xmath107 is not of concern for the minimax , as well . in the beginning , we did ignore the error term . in fact , as is not very difficult to see , since the error term is uniform in @xmath107 and @xmath407 as long as @xmath408 for any fixed @xmath98 , it is legitimate to ignore the error term . to be precise , the asymptotic minimax is the result above , subject to an error of @xmath484 , that is , the value of ( [ d6a ] ) for @xmath480 and @xmath481 . this is @xmath485 . for @xmath76 , on the other hand , asymptotically , the maximum of the redundancy ( [ e32 ] ) ( which , by the considerations above , is ( [ d6a ] ) for @xmath409 ) equals @xmath486 . we must , therefore , conclude that in contrast to the classical case @xcite our trial candidate ( @xmath119 ) for the quantum counterpart of jeffreys prior does not exactly achieve the minimax redundancy , although the prior @xmath487 is remarkably close to @xmath119 , the hypothesized `` quantum jeffreys prior '' from @xcite . we now concern ourselves with the asymptotic _ maximin _ redundancy . clarke and barron @xcite prove that the maximin redundancy is attained asymptotically , again , by the jeffreys prior . to derive the quantum counterpart of the maximin redundancy within our analytical framework , we would have to calculate @xmath488 where @xmath489 varies over the @xmath490-dimensional convex set of @xmath2 density matrices and @xmath491 varies over all probability densities over the unit ball . as we already mentioned in the introduction , in the classical case , due to a result of aitchison @xcite , the minimum is achieved by setting @xmath489 to be the bayes estimator , i.e. , the average of all possible probability densities in the family that is considered with respect to the given probability distribution . in the quantum domain the same assertion is true . for the sake of completeness , we include the proof in the appendix . we can , thus , take the quantum analog of the bayes estimator to be the bayesian density matrix @xmath3 . that is , we set @xmath492 in ( [ d7 ] ) . let us , for the moment , restrict the possible @xmath491 s over which the maximum is to be taken to the family @xmath0 , @xmath1 . thus , we consider @xmath493 by the definition ( [ eq:5 ] ) of relative entropy , we have @xmath494 the second line being due to ( [ eq:7 ] ) . therefore , we get @xmath495 from theorem [ t11 ] , we know the asymptotics of the von neumann entropy @xmath431 . hence , we find that the expression ( [ d7b ] ) is asymptotically equal to @xmath496 we have to , first , perform the maximization required in ( [ d7a ] ) , and then determine the asymptotics of the result . due to the form of the asymptotics in ( [ d8 ] ) , we can , in fact , derive the proper result by proceeding in the reverse order . that is , we first determine the asymptotics of @xmath497 , which we did in ( [ d8 ] ) , and then we maximize the @xmath107-dependent part in ( [ d8 ] ) with respect to @xmath107 ( ignoring the error term ) . ( in figure 3 we display this @xmath107-dependent part over the range @xmath498 $ ] . ) of course , we do the latter step by equating the first derivative of the @xmath107-dependent part in ( [ d8 ] ) with respect to @xmath107 to zero and solving for @xmath107 . it turns out that this equation takes the appealingly simple form @xmath499 numerically , we find this equation to have the solution @xmath500 , at which the asymptotic maximin redundancy assumes the value @xmath501 . for @xmath76 , on the other hand , we have for the asymptotic redundancy ( [ d8 ] ) , @xmath502 . again , we must , therefore , conclude that in contrast to the classical case @xcite our trial candidate ( @xmath119 ) for the quantum counterpart of jeffreys prior can not serve as a `` reference prior , '' in the sense introduced by bernardo @xcite . moreover , again in contrast to the classical situation @xcite we find that the minimax and the maximin are _ not _ identical ( although remarkably close ) . the two distinct priors yielding these values ( @xmath487 , respectively @xmath503 ) are themselves remarkably close , as well . since they are mixtures of product states , the matrices @xmath3 are classically as opposed to epr ( einstein podolsky rosen ) correlated @xcite . therefore , @xmath504 must not be less than the sum of the von neumann entropies of any set of reduced density matrices obtained from it , through computation of partial traces . for positive integers , @xmath505 , the corresponding reduced density matrices are simply @xmath506 , due to the mixing ( * ? ? ? * exercise 7.10 ) . using these reduced density matrices , one can compute _ conditional _ density matrices and quantum entropies @xcite . clarke and barron @xcite have an alternative expression for the redundancy in terms of conditional entropies , and it would be of interest to ascertain whether a quantum analogue of this expression exists . let us note that the theorem of clarke and barron utilized the uniform convergence property of the asymptotic expansion of the relative entropy ( kullback leibler divergence ) . condition 2 in their paper @xcite is , therefore , crucial . it assumes as is typically the case classically that the matrix of second derivatives , @xmath507 , of the relative entropy is identical to the fisher information matrix @xmath31 . in the quantum domain , however , in general , @xmath508 , where @xmath507 is the matrix of second derivatives of the quantum relative entropy ( [ eq:5 ] ) and @xmath31 is the symmetric logarithmic derivative fisher information matrix @xcite . the equality holds only for special cases . for instance , @xmath509 does hold if @xmath510 for the situation considered in this paper . the volume element of the kubo - mori / bogoliubov ( monotone ) metric @xcite is given by @xmath511 . this can be normalized for the two - level quantum systems to be a member ( @xmath110 ) of a one - parameter family of probability densities @xmath512 and similarly studied , it is presumed , in the manner of the family @xmath0 ( cf . ( [ eq:10 ] ) and ( [ e7 ] ) ) analyzed here . these two families can be seen to differ up to the normalization factor by the replacement of @xmath513 in ( [ eq : kubo ] ) by , simply , @xmath407 . ( these two last expressions are , of course , equal for @xmath95 . ) in general , the volume element of a monotone metric over the two - level quantum systems is of the form ( * ? ? ? * eq . 3.17 ) @xmath514 where @xmath515 is an operator monotone function such that @xmath516 and @xmath517 . for @xmath518 , one recovers the volume element ( @xmath519 ) of the metric of the symmetric logarithmic derivative , and for @xmath520 , that ( @xmath521 ) of the kubo - mori / bogoliubov metric @xcite . ( it would appear , then , that the only member of the family @xmath0 proportional to a monotone metric is @xmath119 , that is ( [ eq:9 ] ) . the maximin result we have obtained above corresponding to @xmath500 the solution of ( [ maximin ] ) would appear unlikely , then , to extend globally beyond the family . of course , a similar remark could be made in regard to to the minimax , corresponding to @xmath522 , as shown above . ) while @xmath507 can be generated from the relative entropy ( [ eq:5 ] ) ( which is a limiting case of the @xmath523-entropies @xcite ) , @xmath31 is similarly obtained from ( * ? ? ? * eq . 3.16 ) @xmath524 it might prove of interest to repeat the general line of analysis carried out in this paper , but with the use of ( [ jan ] ) rather than ( [ eq:5 ] ) . also of importance might be an analysis in which the relative entropy ( [ eq:5 ] ) is retained , but the family ( [ eq : kubo ] ) based on the kubo - mori / bogoliubov metric is used instead of @xmath0 . let us also indicate that if one equates the asymptotic redundancy formula of clarke and barron ( [ eq:4 ] ) ( using @xmath525 ) to that derived here ( [ a3 ] ) , neglecting the residual terms , solves for @xmath526 , and takes the square root of the result , one obtains a prior of the form ( [ monotone ] ) based on the monotone function @xmath527 . ( let us note that the reciprocal of the related `` morozova - chentsov '' function @xcite , @xmath528 , in this case , is the _ exponential _ mean @xcite of @xmath66 and @xmath67 , while for the minimal monotone metric , the reciprocal of the morozova - chentsov function is the _ arithmetic _ mean . it is , therefore , quite interesting from an information - theoretic point of view that these are , in fact , the only two means which furnish additive quasiarithmetic average codeword lengths @xcite . also , it appears to be a quite important , challenging question bearing upon the relationship between classical and quantum probability to determine whether or not a family of probability distributions over the bloch sphere exists , which yields as its volume element for the corresponding fisher information matrix , a prior of the form ( [ monotone ] ) with the noted @xmath529 . ) as we said in the introduction , ideally we would like to start with a ( suitably well - behaved ) _ arbitrary _ probability density on the unit ball , determine the relative entropy of @xmath81 with respect to the average of @xmath81 over the probability density , then find its asymptotics , and finally , among all such probability densities , find the one(s ) for which the minimax and maximin are attained . in this regard , we wish to mention that a suitable combination of results and computations from sec . [ s2 ] with basic facts from representation theory of @xmath85 ( cf . @xcite for more information on that topic ) yields the following result . [ t15 ] let @xmath491 be a spherically symmetric probability density on the unit ball , i.e. , @xmath530 depends only on @xmath93 . furthermore , let @xmath531 be the average @xmath532 . then the eigenvalues of @xmath533 are @xmath534 with respective multiplicities @xmath535 and corresponding eigenspaces @xmath536 a ballot path from @xmath217 to @xmath537 , which were described in sec . [ s2.2 ] . the relative entropy of @xmath538 with respect to @xmath533 is given by _ ( [ e32 ] ) _ , with @xmath214 as given in _ ( [ e50])_. we hope that this theorem enables us to determine the asymptotics of the relative entropy and , eventually , to find , at least within the family of spherically symmetric ( that is , unitarily - invariant ) probability densities on the unit ball , the corresponding minimax and maximin redundancies . doing so , would resolve the outstanding question of whether these two redundancies , in fact , coincide , as classical results would suggest @xcite . clarke and barron @xcite ( cf . @xcite ) have derived several forms of asymptotic redundancy for arbitrarily parameterized families of probability distributions . we have been motivated to undertake this study by the possibility that their results may generalize , in some yet not fully understood fashion , to the quantum domain of noncommutative probability . ( thus , rather than probability densities , we have been concerned here with density matrices . ) we have only , so far , been able to examine this possibility in a somewhat restricted manner . by this , we mean that we have limited our consideration to two - level quantum systems ( rather than @xmath4-level ones , @xmath539 ) , and for the case @xmath540 , we have studied ( what has proven to be ) an analytically tractable one - parameter family of possible prior probability densities , @xmath0 , @xmath541 ( rather than the totality of arbitrary probability densities ) . consequently , our results can not be as definitive in nature as those of clarke and barron . nevertheless , the analyses presented here reveal that our trial candidate ( @xmath119 , that is ( [ eq:9 ] ) ) for the quantum counterpart of the jeffreys prior closely approximates those probability distributions which we have , in fact , found to yield the minimax ( @xmath487 ) and maximin ( @xmath503 ) for our one - parameter family ( @xmath542 ) . future research might be devoted to expanding the family of probability distributions used to generate the bayesian density matrices for @xmath540 , as well as similarly studying the @xmath4-level quantum systems ( @xmath543 ) . ( in this regard , we have examined the situation in which @xmath544 , and the only @xmath545 density matrices considered are simply the tensor products of @xmath269 identical @xmath5 density matrices . surprisingly , for @xmath546 , the associated trivariate candidate quantum jeffreys prior , taken , as throughout this study , to be proportional to the volume elements of the metrics of the symmetric logarithmic derivative ( cf . @xcite ) , have been found to be _ improper _ ( nonnormalizable ) over the bloch sphere . the minimality of such metrics is guaranteed , however , only if `` the whole state space of a spin is parameterized '' @xcite . ) in all such cases , it will be of interest to evaluate the characteristics of the relevant candidate quantum jeffreys prior _ vis - - vis _ all other members of the family of probability distributions employed over the @xmath547-dimensional convex set of @xmath545 density matrices . we have also conducted analyses parallel to those reported above , but having , _ ab initio _ , set either @xmath66 or @xmath67 to zero in the @xmath5 density matrices ( [ eq:6 ] ) . this , then , places us in the realm of real as opposed to complex ( standard or conventional ) quantum mechanics . ( of course , setting _ both _ @xmath66 and @xmath67 to zero would return us to a strictly classical situation , in which the results of clarke and barron @xcite , as applied to binomial distributions , would be directly applicable . ) though we have on the basis of detailed computations developed strong conjectures as to the nature of the associated results , we have not , at this stage of our investigation , yet succeeded in formally demonstrating their validity . in conclusion , again in analogy to classical results , we would like to raise the possibility that the quantum asymptotic redundancies derived here might prove of value in deriving formulas for the _ stochastic complexity _ @xcite ( cf . @xcite ) the shortest description length of a string of @xmath4 _ quantum _ bits . the competing possible models for the data string might be taken to be the @xmath5 density matrices ( @xmath49 ) corresponding to different values of @xmath407 , or equivalently , different values of the von neumann entropy , @xmath362 . let @xmath55 , @xmath548 , be a family of density matrices , and let @xmath457 , @xmath548 , be a probability density on @xmath12 . the minimum @xmath549 taken over all density matrices @xmath19 , is achieved by @xmath550 . proof . we look at the difference @xmath551 and show that it is nonnegative . indeed , @xmath552 since relative entropies of density matrices are nonnegative ( * ? ? ? * bottom of p. 17 ) . christian krattenthaler did part of this research at the mathematical sciences research institute , berkeley , during the combinatorics program 1996/97 . paul slater would like to express appreciation to the institute for theoretical physics for computational support . this research was undertaken , in part , to respond to concerns ( regarding the rationale for the presumed quantum jeffreys prior ) conveyed to him by walter kohn and members of the informal seminar group he leads . the co - authors are grateful to : ira gessel for bringing them into initial contact _ via _ the internet ; to helmut prodinger and peter grabner for their hints regarding the asymptotic computations ; to a. r. bishop and an anonymous referee of @xcite ; and to the two anonymous referees of this paper itself , whose comments helped to considerably improve the presentation . j. aczl and z. darczy , _ on measures of information and their characterizations_. academic prss : new york , 1975 . j. aitchison , `` goodness of prediction fit , '' _ biometrika _ , vol . 3 , pp.547554 , 1975 . a. bach and a. srivastav , `` a characterization of the classical states of the quantum harmonic oscillator by means of de finetti s theorem '' _ comm . math . phys . _ , vol . 3 , pp . 453462 , 1989 . a. barenco , a. berthiaume , d. deutsch , a. ekert , r. jozsa , and c. macchiavello , `` stabilisation of quantum computations by symmetrisation , '' _ siam j. comput . 5 , pp . 15411547 , 1997 . h. barnum , c. a. fuchs , r. jozsa , and b. schumacher , `` general fidelity limit for quantum channels , '' _ phys . a _ , vol . 54 , no . 6 , pp . 47074711 , dec 1996 . e. g. beltrametti and g. cassinelli , _ the logic of quantum mechanics _ , addison - wesley : reading , 1981 . c. h. bennett , `` quantum information and computation , '' _ physics today _ , 2430 , oct . 1995 . c. h. bennett and p. w. shor , `` quantum information theory , '' _ ieee trans . 44 , no . 6 , pp . 2724 - 2742 ( 1998 ) . j. m. bernardo , `` reference posterior distributions for bayesian inference , '' , _ b _ , vol . 41 , pp . 113147 , 1979 . j. m. bernardo and a. f. m. smith , _ bayesian theory_. wiley : new york , 1994 . l. c. biedenharn and j. d. louck , _ angular momentum in quantum physics _ , addison wesley : massachusetts , 1981 . s. l. braunstein and g. j. milburn , `` dynamics of statistical distance : quantum limits of two - level clocks , '' _ phys . a _ , vol . 3 , pp . 18201826 , mar . n. j. cerf and c. adami , `` information theory of quantum entanglement and measurement , '' _ physica d _ 1 , pp . 6281 , 1998 . n. n. chentsov , _ statistical decision rules and optimal inference_. amer . soc . : providence , 1982 . b. s. clarke , `` implications of reference priors for prior information and for sample size , '' _ 173184 , march 1996 . b. s. clarke and a. r. barron , information - theoretic asymptotics of bayes methods , " _ ieee trans . inform . theory _ 3 , pp . 453471 , may , 1990 . b. s. clarke and a. r. barron , jeffreys prior is asymptotically least favorable under entropy risk , " _ j. statist . planning and inference _ , vol . 1 , pp . 3761 , aug . 1994 . b. s. clarke and a. r. barron , `` jeffreys prior yields the asymptotic minimax redundancy , '' in _ ieee - ims workshop on information theory and statistics _ , piscataway , nj : ieee , 1995 , p. 14 . r. cleve and d. p. divincenzo , `` schumacher s quantum data compression as a quantum computation , '' _ phys . 4 , pp . 26362650 , oct . 1996 . l. d. davisson , universal noiseless coding , " _ ieee trans . inform . theory _ it-19 , pp . 783795 , 1980 . l. davisson and a. leon garcia , a source matching approach to finding minimax codes , " _ ieee trans . inform . theory _ it-26 , pp . 166174 , 1980 . a. fujiwara and h. nagaoka , quantum fisher metric and estimation for pure state models , _ phys . a _ , vol . 119124 , 1995 . r. gallager , source coding with side information and universal coding , " technical report lids - p-937 , m.i.t . laboratory for information and decision systems , 1979 . g. gasper and m. rahman , _ basic hypergeometric series _ , encyclopedia of mathematics and its applications 35 , cambridge university press , cambridge , 1990 . i. j. good , _ math . _ , 95k:62011 , nov . 1995 . i. j. good , `` utility of a distribution , '' _ nature _ , vol . 219 , no . 5161 , p. 1392 , 28 sept . 1968 . i. j. good , `` what is the use of a distribution , '' in _ multivariate analysis - ii _ ( p. r. krishnaiah , ed . ) . new york : academic press , 1969 , pp . d. haussler , a general minimax result for relative entropy " , _ ieee trans . inform . theory _ 4 , pp . 12761280 , 1997 . k. r. w. jones , `` principles of quantum inference , '' _ ann . 1 , pp . 140170 , 1991 . r. jozsa and b. schumacher , a new proof of the quantum noiseless coding theorem , " _ 41 , no . 12 , pp . 23432349 , 1994 . r. e. kass and l. wasserman , `` the selection of prior distributions by formal rules , '' _ 435 , pp . 13431370 , sept . e. g. larson and p. r. dukes , `` the evolution of our probability image for the spin orientation of a spin-1/2ensemble connection with information theory and bayesian statistics , '' in _ maximum entropy and bayesian methods _ ( w. t. grandy , jr . and l. h. schick , eds . ) . dordrecht : kluwer , 1991 , pp . lo , quantum coding theorem for mixed states , " _ opt . 119 , pp . 552556 , j. d. malley and j. hornstein , `` quantum statistical inference , '' _ statist . _ , vol . 8 , no . 433 - 457 ( 1993 ) . s. massar and s. popescu , `` optimal extraction of information from finite quantum ensembles , '' _ phys . 74 , no . 8 , pp . 12591263 , feb . t. matsushima , h. inazumi , and s. hirasawa , `` a class of distortionless codes designed by bayes decision theory , '' _ ieee trans . 5 , pp . 12881293 , sept . s. g. mohanty , _ lattice path counting and applications _ , academic press , new york , 1979 . m. ohya and d. petz , _ quantum entropy and its use_. berlin : springer - verlag , 1993 . a. peres , _ quantum theory : concepts and methods_. dordrecht : kluwer , 1993 . d. petz , `` geometry of canonical correlation on the state space of a quantum system , '' _ j. math . 780795 , feb . d. petz and h. hasegawa , `` on the riemannian metric of @xmath523-entropies of density matrices , '' _ lett . 38 , pp . 221225 , 1996 . d. petz and c. sudr , geometries of quantum states , " _ j. math . 26622673 , june 1996 . d. petz and g. toth , `` the bogoliubov inner product in quantum statistics , '' _ lett . 27 , pp . 205216 , 1993 . f. qi , `` generalized weighted mean values with two parameters , '' _ proc . a _ , vol . 1978 , pp . 27232732 , 1998 . j. rissanen , fisher information and stochastic complexity , " _ ieee trans . inform . theory _ 1 , pp . 4047 , jan . j. rissanen , _ stochastic complexity in statistical inquiry_. world scientific : singapore , 1989 . b. schumacher , quantum coding , `` _ phys . a _ , vol . 4 , pp . 27382747 , april 1995 . c. e. shannon , ' ' a mathematical theory of communication , " _ bell . 27 , pp . 379423 , 623656 , july oct . 1948 . l. j. slater , _ generalized hypergeometric functions _ , cambridge university press , cambridge , 1966 . p. b. slater , applications of quantum and classical fisher information to two - level complex and quaternionic and three - level complex systems , " _ j. math 6 , pp . 26822693 , june 1996 . p. b. slater , quantum fisher - bures information of two - level systems and a three - level extension , " _ j. phys . a _ , vol . l271l275 , 21 may 1996 . p. b. slater , `` the quantum jeffreys prior / bures metric volume element for squeezed thermal states and a universal coding conjecture , '' _ j. phys . a math l601l605 , 1996 . p. b. slater , _ universal coding of multiple copies of two - level quantum systems _ , march 1996 . n. j. a. sloane and s. plouffe , _ the encyclopaedia of integer sequences _ , academic press , san diego , 1995 . k. svozil , _ quantum algorithmic information theory _ , los alamos preprint archive , quant - ph/9510005 , 5 oct . s. verd , fifty years of shannon theory , " _ ieee trans . inform . theory _ , vol . 44 , no . 6 , pp . 20572078 , 1998 . x. viennot , _ une thorie combinatoire des polynmes orthogonaux generaux _ , uqam : montreal , quebec , 1983 . n. j. vilenkin and a. u. klimyk , _ representation of lie groups and special functions _ , vol . 1 , kluwer : dordrecht , boston , london , 1991 . a. wehrl , `` general properties of entropy , '' _ rev . 2 , pp . 221260 , apr . 1978 . d. welsh , _ codes and cryptography _ , clarendon press , oxford , 1989 . r. f. werner , `` quantum states with einstein - podolsky - rosen correlations admitting a hidden - variable model , '' _ phys . 40 , no . 8 , pp . 42774281 , 15 oct . 1989 .

clarke and barron have recently shown that the jeffreys invariant prior of bayesian theory yields the common asymptotic ( minimax and maximin ) redundancy of universal data compression in a parametric setting . we seek a possible analogue of this result for the two - level _ quantum _ systems . we restrict our considerations to prior probability distributions belonging to a certain one - parameter family , @xmath0 , @xmath1 . within this setting , we are able to compute exact redundancy formulas , for which we find the asymptotic limits . we compare our quantum asymptotic redundancy formulas to those derived by naively applying the classical counterparts of clarke and barron , and find certain common features . our results are based on formulas we obtain for the eigenvalues and eigenvectors of @xmath2 ( bayesian density ) matrices , @xmath3 . these matrices are the weighted averages ( with respect to @xmath0 ) of all possible tensor products of @xmath4 identical @xmath5 density matrices , representing the two - level quantum systems . we propose a form of _ universal _ coding for the situation in which the density matrix describing an ensemble of quantum signal states is unknown . a sequence of @xmath4 signals would be projected onto the dominant eigenspaces of @xmath6 . _ index terms _ quantum information theory , two - level quantum systems , universal data compression , asymptotic redundancy , jeffreys prior , bayes redundancy , schumacher compression , ballot paths , dyck paths , relative entropy , bayesian density matrices , quantum coding , bayes codes , monotone metric , symmetric logarithmic derivative , kubo - mori / bogoliubov metric = 5.9pt plus2pt minus 4pt = 5.9pt plus2pt minus 4pt research supported in part by the msri , berkeley ]

anharmonic oscillators are useful examples of nonlinear phenomena . many vibrating systems found in the real world are nonlinear whether they be macroscopic mechanical oscillators @xcite or microscopic atomic oscillators@xcite . the pendulum at high angles is a classic example of an anharmonic oscillator@xcite . a particular feature is that the restoring force is equivalent to a spring that softens at large amplitudes i.e. @xmath1@xcite in this work we do a detailed analysis of an oscillator materialized by the bobbing cone ; we assume that the movement is a vertical translation and that the top of the solid is always emerged and the base is always submerged in an ideal fluid for which the restoring force is due to archimedes principle . the restoring force being of the type @xmath2 means that the oscillator does not move symmetrically about the origin notwithstanding being periodic . figure([fig : figum ] ) schematically represents a floating cone of radius @xmath3 , height @xmath4 and specific mass @xmath5 , partially immersed in an ideal fluid of specific mass @xmath6 ( @xmath7 ) . the position of the solid is chosen to be the coordinate of the point @xmath8 which coincides with the origin of the reference axis when the buoy is at equilibrium . this origin ( point @xmath9 ) is the intersection of the axis of the cone at equilibrium with the plane of the free liquid . the resultant of the forces acting on the solid is @xmath10 where @xmath11 is the volume of the immersed cone in static equilibrium ( @xmath12 i.e. @xmath13 ) , @xmath14 is the volume of the cone , @xmath15 is the difference between the immersed volume @xmath16 and @xmath11 and @xmath17 is the gravity acceleration . therefore : @xmath18\end{aligned}\ ] ] where @xmath19 then , @xmath20\ ] ] where @xmath21 . introducing the reduced variables @xmath22 and @xmath23 , the reduced restoring force is : @xmath24=-p_3(\overline{y},\alpha)\ ] ] where @xmath25 is a third degree polynomial with coefficients that depend on the parameter @xmath26 . the reduced coordinates of the vertex and of the center of the base are , at equilibrium , respectively : @xmath27{1-\alpha}$ ] and @xmath28{1-\alpha}$ ] . therefore , due to the restrictions of the movement @xmath29 . however , another condition should be imposed on the value of the initial position @xmath30 ( or its reduced form @xmath31 ) as a consequence of the potential energy barrier , i.e. @xmath32 and @xmath33 . the potential energy associated with the restoring force @xmath34 , assuming that @xmath35 at the equilibrium position ( @xmath13 ) is @xmath36dy\\ & = & \rho_{\mbox{\scriptsize f}}\ , g v_{\mbox{\scriptsize c}}\int_0^{\overline{y } } \left[\overline{y}^3 - 3(1-\alpha)^{1/3}\,\overline{y}^2 + 3(1-\alpha)^{2/3}\,\overline{y}\right]d\overline{y}.\end{aligned}\ ] ] in reduced form , @xmath37 , with @xmath38 the choice for the initial values ( @xmath39 and zero velocity ) is subjected to the relations @xmath40{1-\alpha}\leq \overline{y}_0\leq \sqrt[3]{1-\alpha}\\\vspace{-5 mm } \\ p_4(\overline{y}_0)= \min\,\left\{\overline{p}_4(-\overline{y}_{\mbox{\scriptsize c } } ) , \overline{p}_4(-\overline{y}_{\mbox{\scriptsize v}})\right\ } \end{array}\right.\ ] ] these equations express the condition that the oscillation has the maximum energy compatible with the fact that the cone is neither completely immersed nor completely emerged . it is easy to verify that @xmath41{1-\alpha})=\frac{3}{4}\,(1-\alpha)^{4/3}+\alpha -\frac{3}{4 } \quad \mbox{and}\\ p_4(-\overline{y}_{\mbox{\scriptsize v}})=p_4(\sqrt[3]{1-\alpha})=\frac{3}{4}\,(1-\alpha)^{4/3}.\end{aligned}\ ] ] then @xmath42 for @xmath43 @xmath44 @xmath45 . so , if the initial velocity is zero then the initial amplitude @xmath31 should be chosen along with @xmath46{1-\alpha}~ \mbox{and}~ \overline{e}_{total}=\frac{3}{4}\,(1-\alpha)^{4/3}+\alpha -\frac{3}{4}\\\vspace{-5 mm } \\ \alpha > \frac{3}{4 } \longrightarrow \overline{y}_0= -\overline{y}_{\mbox{\scriptsize v}}= \sqrt[3]{1-\alpha}~ \mbox{and}~ \overline{e}_{total}= \frac{3}{4}\,(1-\alpha)^{4/3}. \end{array}\right.\ ] ] the other limit @xmath47 of the interval of @xmath48 can be obtained from the roots of the polynomial @xmath49 . there are four solutions ( two real and two complex conjugate ) ; the real are the relevant solutions : @xmath31 [ eqs.([eqyzero ] ) ] and @xmath47 [ eqs.([eqyum ] ) ] @xmath50^{1/3}}{3}\,\raisebox{-0.1mm}{\large -}\\\vspace{-6mm}\vspace{2.5 mm } \\ & \displaystyle -\frac{2}{3\left[-44 + 54\alpha+6\sqrt{-27 + 81\,(1-\alpha)^2 + 30\,\alpha}\right]^{1/3 } } + ( 1-\alpha)^{1/3 } + { \displaystyle\frac{1}{3}}\\\vspace{-5 mm } \\ \alpha > \frac{3}{4 } \longrightarrow \overline{y}_1&= -(\sqrt[3]{4}-1)\,\sqrt[3]{1-\alpha}. \end{array}\right.\ ] ] for each @xmath26 value there is a definite maximum of total energy and a corresponding interval of amplitude @xmath51\equiv [ -\overline{y}_{\mbox{\scriptsize c } } , \overline{y}_1]$ ] for @xmath52 and @xmath53 $ ] for @xmath54 given by eqs.([eqyzero ] ) and ( [ eqyum ] ) . it means that the procedure to initiate the movement should be as follows : for @xmath55 the base of the cone should be raised near the free surface of the liquid ; for @xmath54 the vertex of the cone should be lowered until near complete immersion ; for @xmath56 the choice of the base or vertex to initiate the movement is irrelevant since the amplitude of the oscillation , that has a maximum ( equal to @xmath4 ) corresponds to a displacement between the base and the vertex i.e. @xmath57 $ ] . this can be seen in fig.([fig : figuredoissemcor ] ) . , and the reduced interval , @xmath58 , of the oscillations as a function of @xmath26 . the maximum of the energy @xmath59 occurs at the crossover @xmath43.,width=4 ] the restoring force ( reduced value ) @xmath60 and the potential energy @xmath61 can be represented in the interval of oscillation with @xmath26 as parameter [ fig.([fig : figuretressemcor ] ) and fig.([fig : figurequatrosemcor ] ) ] . , within the proper interval of oscillation taking @xmath26 as a parameter.,width=4 ] , within the proper interval of oscillation taking @xmath26 as a parameter.,width=4 ] we use the newton equation to find the position of the point @xmath8 of the cone as a function of time . the adoption of reduced values continues to offer simplification in the final equation . let us define the unit of time @xmath62 . therefore @xmath63 . the unit for @xmath64 is @xmath65 which points to @xmath66 . on the other hand @xmath67 , i.e. the equation of motion in terms of the reduced variables is @xmath68 finally @xmath69 the system does not oscillate symmetrically about the origin since the force ( polynomial @xmath70 ) shows odd and even powers . for small displacements @xmath71 , the equation is approximately linear and so the movement is quasi harmonic with period @xmath72{1/(1-\alpha)}$ ] . in other physical situations the oscillator exhibits anharmonic displacements and the period will depend on the amplitude ( or total energy ) . the solutions @xmath73 and @xmath74 were obtained by computational methods and are represented in figs . ( [ fig : figuracinco ] ) , ( [ fig : figuraseis ] ) , ( [ fig : figurasete ] ) and ( [ fig : figuraoito ] ) . .,width=4 ] .,width=4 ] .,width=4 ] .,width=4 ] the conservation of the energy of the oscillator allows the determination of the velocity as a function of displacement : @xmath75 . @xmath76 the phase space representation is shown in fig . ( [ fig : novafiguranove ] ) . from the symmetry of these curves relatively to the horizontal axis , we can conclude that the interval of time between two zeros of the velocity or two extremes of the displacements is equivalent to half of the period of the movement . since @xmath77 , @xmath78 . then , as a function of @xmath48 ) for some values of @xmath26.,width=4 ] @xmath79 the limits of integration and the integrand function have been determined above . the result is only a function of @xmath26 . it is possible to get a close solution for the period in terms of complete elliptic integrals of the first kind@xcite . for @xmath80 , the analytical solution is far too complicated to be considered and so numerical methods were used . however , the solution for @xmath81 was easily obtained : @xmath82{4}-1)\sqrt[3]{1-\alpha}}^{\sqrt[3]{1-\alpha}}\,\frac{d\overline{y}}{\sqrt{3(1-\alpha)^{4/3}-\overline{y}^4 + 4(1-\alpha)^{1/3}\overline{y}^3 - 6(1-\alpha)^{2/3}\overline{y}^2}}\\\vspace{-4 mm } \\\nonumber \overline{t}=&\frac{\sqrt{\alpha}}{\sqrt[3]{1-\alpha}}\,\frac{2^{4/3}}{3^{1/4}}~k\hspace{-1mm}\left(\frac{\sqrt{2}}{2(1+\sqrt{3})}\right)=c\frac{\sqrt{\alpha}}{\sqrt[3]{1-\alpha}},\end{aligned}\ ] ] where @xmath83 is a complete elliptic integral of the first kind of argument @xmath84 . the numerical value of c is : @xmath85 . an interesting result that should be outlined is the fact that the normalized value of the period for @xmath81 , @xmath86 ( @xmath87 , period of small oscillations ) is a constant , i.e. does not depend on @xmath26 : @xmath88{1-\alpha}}\,\frac{\sqrt{6}\sqrt[3]{1-\alpha}}{2\pi\,\sqrt{\alpha}}=\frac{\sqrt{6}}{2\pi}c= 1.192900269.\ ] ] the period of the cone oscillations at the highest energy as a function of @xmath26 is represented in fig . ( [ fig : figuradezpretobranco ] ) together with the normalized values . , calculated for the highest value of its total energy . the derivative @xmath89 is also shown in the inset.,width=4 ] as a parameter . the energy varies from @xmath90 to its maximum value ( eq.([eqyzero])).,width=4 ] so far , we have represented the oscillation with the highest energy or maximum value of the initial amplitude satisfying conditions ( [ conditons ] ) . it is interesting to examine closely the dependence of the period with the energy of the oscillator from zero to the highest value given by eqs.([eqyzero ] ) . this is shown in fig.([fig : figureonzesemcor ] ) taking , as usual @xmath26 as a parameter . it is interesting to observe that the numerically obtained values ( @xmath91 ) fit @xmath92 a second degree polynomial . the fourier analysis is a very useful tool to investigate the harmonic components of a periodic function of time that derives from a non linear equation , as it happens in our study . from figs.([fig : figuracinco ] ) and ( [ fig : figuraseis ] ) one can conclude that the displacement @xmath73 is a periodic symmetric even function with a non null mean value . the fourier decomposition gives for the general case : @xmath93 ; in this case all @xmath94 since @xmath95 is an even function . the coefficients @xmath96 were calculated by numerical methods once the fundamental frequency is known . only the first four coefficients were significant . the fourier components for the particular case @xmath97 that corresponds to the maximum of the highest value of energy are represented in fig.([fig : figuredozesemcor ] ) . it also exhibits the highest coefficients which indicate the highest degree of anharmonicity . the reduced values of all calculated coefficients ( @xmath98 ) are also shown in table[table ] . llllllll & @xmath99&@xmath100&@xmath101&@xmath102&@xmath103&@xmath104&@xmath105 + @xmath106&@xmath107&@xmath108&@xmath109&@xmath110&@xmath111&@xmath112&@xmath113 + @xmath114&@xmath115&@xmath116&@xmath117&@xmath118&@xmath119&@xmath120&@xmath121 + @xmath122&@xmath123&@xmath124&@xmath125&@xmath126&@xmath127&@xmath128&@xmath129 + @xmath130&@xmath131&@xmath132&@xmath133&@xmath134&@xmath135&@xmath136&@xmath137 + @xmath138&@xmath139&@xmath140&@xmath141&@xmath142&@xmath143&@xmath144&@xmath145 + @xmath146&@xmath147&@xmath148&@xmath149&@xmath150&@xmath151&@xmath152&@xmath153 + @xmath154&@xmath155&@xmath156&@xmath157&@xmath158&@xmath159&@xmath160&@xmath161 + @xmath162&@xmath163&@xmath164&@xmath165&@xmath166&@xmath167&@xmath168&@xmath169 + @xmath170&@xmath171&@xmath172&@xmath173&@xmath174&@xmath175&@xmath176&@xmath177 + most real oscillators contain anharmonic components . the present study of a floating cone movement is by all means an interesting case of an anharmonic oscillator . the fact that the restoring force is a polynomial of the third degree with non null coefficients ( except the independent term ) imply an asymmetry of this force and the related potential curve . comparing this to well known based mass - spring systems results that the equivalent spring stiffness is no longer constant . it varies as a sum of two contributions that have opposite sign from a specific position . in solid state physics we encounter these kind of forces such as the cohesive force containing a short range repulsion ( hard sphere interaction ) and a long range attraction . the former varies with the displacement much faster than the latter . 10 filipponi a and cavicchia d r 2011 _ am . j. phys . _ * 79 * 730 arnold t w and case w 1982 _ am . j. phys . _ * 50 * 220 whineray s 1991 _ _ * 12 * 90 pecori b , torzo g and sconza a 1999 _ am . j. phys . _ * 67 * 228 ashcroft n w and mermin n d 1976 _ solid state physics _ ( ny : holt rinehart and winston ) lewowski t and wozniak k 2002 _ _ * 23 * 461 lima f m s and arun p 2006 _ am . j. phys . _ * 74 * 892 boyd j n 1991 _ virginia journal of science _ * 42 * lebedev n n 1972 _ special functions and their applications _ ( ny : dover )

a study of the floating of a circular cone shaped buoy in an ideal fluid has revealed some new interesting results . using reduced variables it is shown , that at a crossover value @xmath0 of the ratio of the specific masses of the fluid and of the buoy , the anharmonicity of the oscillation is the highest and that , unexpectedly , above this crossover value the normalized period is constant .

active galactic nuclei ( agn ) are the most luminous objects in the universe . they emit large amounts of radiation over a wide range of wave - bands ( from radio to @xmath6-rays ) , and sometimes produce relativistic jets . it is a goal of modern astronomy to understand the origin of the extreme activities in agn . one of the key observational clues in the x - ray band to understand agn is the iron line profile in the energy spectrum @xcite . a broad and skewed line feature around 57 kev was clearly detected by asca for the first time in an energy spectrum of the seyfert galaxy mcg6 - 30 - 15 @xcite . since then , a similar feature was detected by asca from several seyfert galaxies ( e.g. @xcite ) . the broad and skewed feature has been interpreted as an iron fluorescent line originating from the innermost region of an accretion disk ( @xcite , so - called `` disk - line '' model ) . in this model , a broad and skewed feature is explained by the combination of doppler broadening due to the relativistic motion of the line emitting matter and gravitational redshift due to the central massive black hole . if the broad and skewed feature in the energy spectra is really a `` disk - line '' , we can determine the inclination angle of the accretion disk through an analysis of the line profile @xcite . however , the thus - obtained inclination angle is sometimes inconsistent with that estimated with other methods ( e.g. @xcite ) . it may also be inconsistent with an expectation from the unified scheme , in which the accretion disk in seyfert 2 galaxies should have an edge - on geometry . asca observations have revealed the presence of a broad and skewed line feature in several seyfert 2 galaxies . the inclination angles have been deduced for these seyfert galaxies from an analysis of the `` disk - line '' , and always found to be around @xmath7 @xcite . this inclination angle is unexpectedly small . this problem may be partially solved if we assume that the line is a composite and consists of a narrow line centered at 6.4 kev and a disk line with intermediate inclination angle @xcite . however , we still need an observational confirmation on such a composite model . ngc 4151 is a bright , nearby seyfert galaxy ( type 1.5 ; @xcite ; for a review see @xcite ) . the presence of the broad and skewed line feature in 4.57.5 kev has been known through asca observations @xcite . thus , ngc 4151 is one of the best targets to precisely study the line feature . a `` disk - line '' analysis of ngc 4151 has resulted in a face - on geometry of the accretion disk @xcite , while observations of the [ o iii ] @xmath85007 image with the hubble space telescope indicate an inclination angle of @xmath9 @xcite . this inconsistency might be partially resolved if we assume two disk lines and an additional narrow peak centered at 6.4 kev to reproduce the profile @xcite . however , it is not clear how the two disk lines with different inclination angles can be produced simultaneously . in this paper , we analyze the broad and skewed feature observed from ngc 4151 using a model - independent method as much as possible , and try to identify the origin of the feature . for this purpose , we analyze the long observations of ngc 4151 made with asca ( in 2000 ) and rxte ( in 1999 ) , while focusing on the spectral variations on various time scales . the results from another set of long asca observations of ngc 4151 have been presented by @xcite . the primary asca data set used for the present analysis was acquired from 2000 may 13 through 25 . concerning the instrumentation on board asca , see the following references : @xcite for a general description of asca , @xcite for the x - ray telescope , @xcite and @xcite for the gas imaging spectrometer ( gis ) , and @xcite for the solid - state imaging spectrometer ( sis ) . in the observations , sis was operated in the 1-ccd faint mode and gis in the standard ph mode . a lower level discriminator was applied to sis at 0.48 kev to avoid telemetry saturation due to flickering pixels . the data selection criteria for sis and gis are summarized in table [ tbl : crit ] . when the elevation angle , i.e. the angle between the field of view and the earth limb , was less than @xmath10 , the data were discarded because they were affected by scattered x - rays from the day earth or by the absorption due to the atmosphere . some of the optical light leakage is known in sis , and the sis data obtained when the elevation from the day earth was less than @xmath11 were discarded . the data with telemetry saturation are also discarded to avoid any deterioration of the detection efficiency . the net exposure of sis was about 350 ks , which spans @xmath12 s , and an average count rate of sis was 0.8 count s@xmath13 after data screening and reduction . an x - ray source was detected with both sis and gis at a position consistent with the optical position of ngc 4151 within the positional accuracy of asca . a bl lac object ( 1207 + 39w4 ; @xcite ) was detected in the field of view of asca about north of ngc 4151 . contamination by this source was found to be negligible above 2 kev , which was used for the current data analysis . according to the chandra observations , a faint source was also present at from ngc 4151 to the southwest @xcite . this source was not resolved with asca . contamination by this source was completely negligible . the energy spectra and light curves of each detector were accumulated from a circular region centered on the source . the extraction radius of sis was set to , which is the maximum radius fit in both chips , on the whole . the background spectra of sis are accumulated from the source free region on the chip . the extraction radius of gis was set to 6 arcmin , because various calibrations of gis had been done with this standard extraction radius . the background data of gis were accumulated from a circular region opposite to the target position against the bore - sight axis , excluding the source region . we usually obtained 4 sets of energy spectra from the asca data , two from sis and the other two from gis , for a single set of observations . to make the analysis simple , we summed up the two energy spectra from sis and the two corresponding response matrices , respectively , which produced a single sis spectrum with the corresponding response . similarly , we summed up the two gis spectra and the corresponding responses , respectively . the summed sis and gis spectra were fitted with a model spectrum simultaneously . the simultaneous fitting made a full utilization of the asca data in the sense that the higher detection efficiency below 5 kev and better energy resolution of sis , and the higher detection efficiency above 5 kev of gis are all reflected in the best - fit spectral parameters . because there is 3% uncertainty of flux normalization between sis and gis , a systematic error of 3% was conservatively added to the sis energy spectra in the simultaneous fitting . although a decrement of sis detection efficiency below 2 kev was reported by the instrument team ( t. yaqoob , private communications ) , we did not include the effect in the analysis because the spectral fittings were carried out only for the energy range above 2 kev . data obtained by the proportional counter array ( pca ) on board the rossi x - ray timing experiment ( rxte ) were also used in the present analysis . details of rxte and pca can be found in @xcite . rxte observations of ngc 4151 were carried out every @xmath45 days from the beginning of 1999 to study the long - term variations of the x - ray flux . we used the data obtained from 1999 january 1 through july 24 , which were publicly available from the archive in heasarc at the time of data analysis . reduction of the pca data was carried out using the standard procedure @xcite . in the analysis , we discarded data obtained when the elevation from the earth limb was less than @xmath14 , data obtained within 30 minute after the satellite s passage of saa , data obtained when the pointing offset was greater than , and data when the count rate of electrons was greater than 0.1 . the background spectra were calculated using a faint - source model according to a method described in the rxte cook book . figure [ fig : ascalc ] shows the x - ray light curve at 0.710 kev from ngc 4151 obtained with asca gis on 2000 may 1325 . here , the data are shown in 5.6 ks binning . this 5.6 ks bin - time corresponds to the orbital period of asca around the earth . we chose this bin time because : ( 1 ) time variations shorter than the orbital period were not significant , ( 2 ) data gaps are usually much shorter than the orbital period and we can get continuous light curve , ( 3 ) possible background variations , which are correlated with the orbital period , could be largely reduced . note that data gaps were produced mainly by the earth occultation of the source , high background regions on the earth , and down - link to the ground stations . x - ray flux variations of a factor of 23 are clearly seen in the figure . we next studied the energy dependence of the flux variations . for this purpose , we calculated the ratios of the time - resolved energy spectra to the average energy spectrum . considering the statistics of the data and the amplitude of the time variations , the time - resolved energy spectra were calculated every @xmath15 s and six spectra were obtained in total . we also calculated the average spectra for gis and sis , each of which correspond to the average of 6 time - resolved spectra , as shown in figure [ fig : avespe ] . the ratios of the six time - resolved spectra to the average spectrum are presented in figure [ fig : pharatio ] . large time variations in both the flux and the shape can be seen at 25 kev , while the ratios are almost flat at 0.71.5 kev and 710 kev . the flat spectral ratios mean that the shape of the energy spectra does not vary at 0.71.5 kev and 710 kev . furthermore , those at 0.71.5 kev are always close to unity this indicates that the energy spectra change neither their shapes nor the fluxes in this energy range . it should be noted that a local structure is seen just around 6.4 kev and the ratios are closer to unity only in this narrow energy band . at least three spectral components may be required to reproduce these time variations of the spectral ratios : a stable soft component below @xmath42 kev , a variable hard component above @xmath42 kev , and a relatively stable component around 67 kev . because we are mostly interested in the feature around 67 kev , we do not discuss the soft stable component hereafter by limiting the energy range in the spectral analyses to above @xmath42 kev . in order to study the nature of the relatively stable component around 67 kev , we need to separate the component from the underlying variable hard component . we first determined the shape of the variable hard component by performing a model fit to the average spectrum at 24 kev and 810 kev after masking the 48 kev range . the results show that a simple model of a power law with a single absorption can not reproduce the spectrum , and that a `` dual absorber '' model is necessary . the need for a dual absorber for the asca spectrum was already pointed out by @xcite . the `` dual absorber '' model is expressed by the following formula : @xmath16 + ( 1 - c_{\rm f } ) \exp[-\sigma(e ) n^{2}_{\rm h } ] \right\ } e^{-\gamma } , \label{eq : dualabs}\ ] ] where @xmath17 is a normalization factor , @xmath18 the photon index , @xmath19 the column density , @xmath20 the photo - electric absorption cross - section , and @xmath21 the source covering fraction ( @xmath22 ) . the superscript to @xmath19 indicates two different column densities . in the following analysis , the photo - electric absorption cross - sections compiled by @xcite are considered . the thus - determined continuum spectrum was subtracted from the average spectrum ; the residual feature at 48 kev is shown in figure [ fig : fe ] . the presence of a narrow peak at 6.4 kev and a broad feature skewed toward the lower energy can be clearly seen . we then studied the time variations of the residual feature at around 6.4 kev . for this purpose , we used the 6 sets of sis spectra , which have a better energy resolution than the gis spectra . the continuum shape was determined separately for the 6 sets of spectra using the same method for the average spectrum . we set @xmath23 in equation ( [ eq : dualabs ] ) to 0 to simplify the model fitting , considering the statistics of the data . we confirmed that setting @xmath23 to zero did not change the continuum shape at 48 kev more than 3% while taking the 4th data set ( those with the largest flux ) as an example . the model was found to be acceptable for all 6 sets of the spectra with the largest value of @xmath24 . when the continuum model was interpolated to the 48 kev band , an excess feature could be clearly seen in all 6 sets of the energy spectra . we show them in figure [ fig : feindiv ] after subtracting the continuum model from the observed spectra . a prominent narrow peak at around 6.4 kev , and broad features at 4.56.0 kev ( hereafter referred to as a red wing ) and at 6.87.5 kev ( a blue wing ) are clearly seen in all 6 data sets . in order to study any profile change of the broad and skewed feature , we made the ratios of each of the 6 excess features to their average . the ratios were calculated by dividing the individual excess feature ( after subtracting the interpolated continuum ) by the excess feature in the average spectrum ( figure [ fig : fe ] ) . the results are shown in figure [ fig : excesspha ] . by performing a @xmath25 test for the hypothesis that the ratios have no energy dependence , we found that the 6 sets of ratios are consistent with being constant in terms of the energy , and that the time variation of the profile of the broad and skewed feature is not significant . at the same time , we calculated the flux ratio of the red wing ( 4.56.0 kev ) to the narrow core ( 6.06.8 kev ) and that of the blue wing ( 6.87.5 kev ) to the narrow core for the 6 sets of sis spectra . no significant time variation was found at the 90% confidence limit in the two sets of ratios ( @xmath26 at most ) . the standard deviation to the mean for the flux ratio of the red wing to the narrow core was calculated to be 20% at most . although the time variation is not significantly found , a variation of less than 20% in amplitude is not rejected . we analyzed the rxte pca data in order to study the spectral variations on a time scale longer than the @xmath12 s covered by asca . we show in figure [ fig : rxtelc ] the light curve calculated from the pca data obtained from 1999 january 1 through july 24 . significant flux variations can be clearly seen . the pca spectrum averaged over the rxte observations is shown in figure [ fig : rxteave ] . as was done for the asca spectra , we fit a power law modified by `` dual absorption '' ( @xmath27 ) to the average spectrum in 2.810 kev after masking the 48 kev energy band . we also checked how it affected the continuum shape to set @xmath23 to zero , and found that there was no noticeable impact if the fit range was restricted to 2.84 and 810 kev . the ratio of the observed spectrum to the continuum model is shown in the lower panel of figure [ fig : rxteave ] . we can clearly see the presence of a broad and skewed feature at 48 kev again . next , in order to search for the time variations of the broad and skewed feature on time scales longer than @xmath12 s , we divided the data into 11 subsets on a time bin of @xmath28 s and calculated an energy spectrum for each subset of data . we then fit a power law modified by `` dual absorption '' to the 11 spectra in 2.810 kev while excluding the 48 kev band again . the fit was acceptable for all of the spectra . the residuals of the 11 spectra after subtracting the best - fit continuum model are shown in figure [ fig : rxteindiv ] . a significant excess flux over the continuum model is always seen at 4.57.5 kev . the excess has a peak at around 6.06.5 kev as well as a significant tail feature on the lower energy side . then , we again investigated the time variations of the spectral shape of the excess by comparing the 11 excess spectra with their average . the spectral ratios of the 11 spectra to their average are shown in figure [ fig : rxteratio ] . we performed @xmath25 tests to check whether or not each of the 11 ratio - spectra is consistent with being flat . a flat model was found to be acceptable for all 11 ratios with @xmath29 , although the flux level of the excess shows a significant variation . this indicates that , although the excess flux at 4.57.5 kev is variable on a time scale longer than @xmath28 s , the profile does not change significantly . in the previous section we discussed the time variability of the flux and the profile of the broad and skewed feature consisting of a narrow peak at 6.4 kev and red / blue wings in a way fairly independent of any spectral models . in this section , we introduce two different model functions to reproduce the profile : one is the so - called `` disk - line '' model , and the other is a `` reflection '' model . it is widely accepted that the broad and skewed feature at around 57 kev often seen from agn is composed of gravitationally redshifted iron - fluorescent lines from matter orbiting in an accretion disk very close to the central black hole @xcite . hence , we first tried to fit the broad line - like feature observed from ngc 4151 with the disk - line model . we adopted a spectrum described by equation ( [ eq : dualabs ] ) as the continuum , where @xmath30 was set to 0 to simplify the model . the outer and inner radii of the accretion disk were set to 1000 @xmath31 and 10 @xmath31 , respectively , where @xmath31 is the schwarzschild radius . the emissivity of the line emission is assumed to have a power - law dependence on the radius with the index of @xmath322 @xcite . the model fitting was carried out in the energy range of 2.210 kev simultaneously to the sis and gis data for each of the 6 sets of the asca energy spectra . the photon index was fixed to 1.55 , which was the average value when we fit a power law with a partial covering absorber to the spectra after masking 48 kev . the model was found to be acceptable for all of the spectra at 2.210.0 kev with @xmath33 . the best - fit parameters are listed in table [ tbl : ascapar ] . the inclination angles and the line center energies obtained from the spectral fits to the 6 spectra are not significantly different from one another . the reduced @xmath25 values for the hypothesis that the 6 values are constant were 1.35 and 0.89 for 5 d.o.f . for the inclination angle and the line center energy , respectively . this means that these two parameters are consistent with being constant . the average inclination angle and the line center energy are @xmath34 and @xmath35 kev , respectively . we applied the same model to the 11 energy spectra of pca at 2.824 kev . however , the fit was not acceptable with @xmath36 . the deviation of the model from the observed spectra became noticeable above 16 kev . hence , we replaced the power law in equation ( [ eq : dualabs ] ) with a broken power law as described below : @xmath37 where @xmath17 is a normalization , @xmath38 and @xmath39 are photon indices , and @xmath40 is a break energy at which the spectral index changes . note that the broken power - law is required due to the apparent change in the spectral slope above @xmath416 kev , not because we set @xmath30 to zero . the 11 energy spectra of pca were all reproduced by the new model with @xmath41 . the best - fit parameters are listed in table [ tbl : rxtepar ] . we checked the time variation of the 11 values of each parameter by performing a @xmath25 test . the photon indices , @xmath38 and @xmath39 , and the break energy , @xmath40 , are found to have not changed significantly during the observational period . we cross - checked the spectral slope above the break energy , which was found to be @xmath42 , using the data from another detector hexte ( high energy x - ray timing experiment ) on board rxte . hexte consists of two clusters of 4 nai / csi phoswich scintillation detectors sensitive to x - rays from 15 to 250 kev . the photon indices obtained from a simultaneous fit of a power - law model to the 11 sets of the pca spectrum ( 1824 kev ) and hexte spectrum ( 18100 kev ) are found to fall around @xmath43 . this is consistent with @xmath44 . it is noteworthy that , although the disk line model can always reproduce the excess feature around 57 kev , the continuum spectrum needs a break at around 17 kev to reproduce the energy spectrum up to 100 kev . in the disk - line model , the peak at 6.4 kev is considered to be a blue part of fluorescent lines coming from matter orbiting in an accretion disk very close to the central black hole . however , the peak energy is just the k@xmath45 line of a neutral , or a lowly ionized iron , in the rest frame ; furthermore , the profile around the peak can be reproduced by a single narrow line . hence , it would be more natural to consider that the peak at 6.4 kev is a single fluorescent iron line coming from a region free from relativistic effects . if the line is emitted through the fluorescent process as a result of x - ray illumination on relatively cold matter far outside the central x - ray source , x - ray reflection at the surface of the matter should take place simultaneously with the fluorescent line emission . the x - rays reflected by the cold matter emitting the fluorescent lines could be a possible origin of the broad red and blue wings accompanied by the narrow line feature on both sides . because x - ray reflection is due to thomson scattering , the reflected x - rays should have the same spectrum as the illuminating x - rays in a typical x - ray band , but suffer from absorption when they pass through the cold mater . thus , the reflected component may be approximated by applying cold - matter absorption to the continuum model . in order to test this possibility , we introduced the following model spectrum : @xmath46 + 1-c_{\rm f}^1\ } \exp[-\sigma(e ) n_{\rm h}^2 ] + i(e_{\rm line } ) + c_{\rm f}^2 p(e,\gamma ) \exp[-\sigma(e ) n_{\rm h}^3 ] . \label{eq : ref}\ ] ] here , the first term is the same as equation ( [ eq : dualabs ] ) , representing continuum x - rays directly coming from the central x - ray source . the second term represents the fluorescent iron line at 6.4 kev and @xmath47 is a gaussian function . the line width is assumed to be zero . the third term is for the reflection component . the efficiency of the x - ray reflection is represented by @xmath48 , which corresponds to a covering fraction of the x - ray reflector optically thick for thomson scattering . the function , @xmath49 , is a power law with a photon index of @xmath18 . the photon index is optimized in the course of the fitting , because it may be affected by the introduction of a highly absorbed component . the column densities and covering fractions are also optimized . model fitting was firstly carried out to the 11 pca spectra in the energy range of 2.824 kev . in this energy range , because the effect of absorption by @xmath30 was negligible , we omitted this term . the fit was acceptable for all energy spectra with @xmath50 . an example of the results of the model fitting is shown in figure [ fig : rxteref ] and the best - fit parameters are listed in table [ tbl : rxteref ] . the results show that 4050% of the direct flux is reflected . we checked the constancy of the 11 values of each spectral parameter with a @xmath25 test . we found that the photon index and the absorption column of the reflected component , @xmath51 , are both consistent with being constant ( @xmath52 , and 1.25/10 , respectively ) . we confirmed that the profile of the broad and skewed feature did not change significantly during the rxte observations . the weighted mean of the 11 photon indices is @xmath53 . this is comparable to @xmath54 , which we obtained from a simultaneous fitting to the 11 sets of the pca energy spectra at 1824 kev and the hexte energy spectra at 18100 kev . if we extrapolate the best - fit model at 2.824 kev to 100 kev , no significant discrepancy is recognized between the model and the observed energy spectra with hexte . this indicates that the intrinsic spectrum emitted from the central source should be a single power law with an index of @xmath41.9 , and should extend up to 100 kev without a break , if the `` reflection '' model explains the broad and skewed feature at around 57 kev . although the best - fit line energies are found to be consistent with being constant , the mean line center energy is @xmath55 kev . this is slightly lower than the center energy of the iron fluorescent line from neutral matter ( 6.4 kev ) . however , it is known that there is a systematic uncertainty of about 23% in the energy scale of the rxte instrument . thus , the line energy can still be interpreted as the iron fluorescent line . the time variation of the line flux is shown together with that of the continuum flux at 810 kev in figure [ fig : rxtefe ] . the time variation of the line flux is clearly seen in a time scale of @xmath56@xmath57 s. next , we also applied the model function ( equation [ eq : ref ] ) to the asca data . because the energy range of the asca detectors is limited below 10 kev , some of the model parameters were not well constrained . hence , we fixed the following parameters in the fitting : @xmath18 ( @xmath58 ) , @xmath51 ( @xmath59 atom @xmath60 ) and @xmath48 ( @xmath61 % ) . other parameters were optimized during the course of fitting . the model fitting was carried out to each of the 6 sets of the asca data , simultaneously to the sis and gis spectra . fittings of this model were all acceptable with @xmath62 . an example of the results of the model fittings to the asca sis spectra is shown in figure [ fig : ascaref ] . the best - fit parameters are listed in table [ tbl : ascaref ] . the time histories of the line flux and the continuum flux at 810 kev for the asca data are plotted in figure [ fig : ascalcfe ] . no significant change of the line flux can be seen on time scales of @xmath63@xmath56 s. as can be seen from the table , the values of @xmath64 , @xmath65 and @xmath30 obtained from the spectral fits to the 6 sets of asca spectra significantly change on time scales of @xmath63@xmath56 s. the relative amplitude of the variations of @xmath64 and @xmath30 are @xmath420% and @xmath412% , respectively . finally , we investigated the parameters of the narrow line component . we optimized the center energy and the intrinsic width of the narrow line component in the `` reflection '' model by fitting it to the average spectra of the asca observations in 2000 . the line center energy and the intrinsic width were found to be @xmath66 kev and less than 92 ev ( gaussian @xmath67 ) , respectively . an analysis of the time - resolved energy spectra showed that the iron line flux was consistent with being constant on time scales of @xmath63@xmath56 s , while significant variations were seen in its flux on time scales of @xmath56@xmath57 s. in order to study the origin of the line emission , the relation between the continuum flux and the line flux was studied . here , the continuum flux was calculated in the energy band of 810 kev , which is higher than the k edge energy of neutral iron . in this energy range , the flux variation should reflect the variation of the intrinsic , power - law component . because the line emission probably results from a reprocessing of the continuum x - rays by matter ambient to the continuum source , the light curve of the line flux might suffer from some amount of smearing and/or time delay to the continuum flux light curve . in fact , if we compare the light curves of the continuum flux and the line flux ( figure [ fig : rxtefe ] ) , the line flux variation seems to follow the smeared variations of the continuum flux . in order to see the effect of smearing , we applied smearing to the continuum light curve using a following simple method . we redistributed the continuum flux of @xmath68-th time bin into @xmath69 bins starting from @xmath68-th bin . thus , the smeared light curve is expressed as @xmath70 where @xmath71 is the flux of @xmath68-th bin in the original light curve , @xmath72 the flux of @xmath68-th bin in the smeared light curve , and @xmath69 the number of bins to smear . we then fitted the smeared light curve to the line flux light curve . we increased @xmath69 from 1 until we obtain an acceptable fit . note that @xmath73 corresponds to the case without smearing . the relation between @xmath69 and @xmath74 is plotted in figure [ fig : rxtesmear ] . we could obtain an acceptable agreement with @xmath75 between the smeared light curve of the continuum flux and the line flux light curve . the smeared light curve in the case of @xmath75 is compared with the light curve of the line flux in figure [ fig : rxtesmcomp ] . the above - mentioned smearing algorithm of the continuum flux also introduces a delay of @xmath76 bins together with smearing of @xmath76 bins to both sides . thus the typical delay and smearing time scale may be regarded as @xmath77 s. we then checked whether or not smearing of @xmath78 s can explain the absence of a correlation between the continuum flux and the line flux on shorter time scales for the asca data . since the asca data is shorter than @xmath78 s , we could not smear the continuum flux light curve directly . hence , we assumed that the observed light curve of the continuum flux in the asca observations repeats periodically . we then smeared the assumed light curve using the method described above . the fractional variation ( i.e. standard deviation divided by the average ) of the continuum flux was originally @xmath40.25 , but was reduced to @xmath40.06 by smearing . the reduced value is consistent with the observed upper limit , 7% , of the fractional variation of the line flux in the asca observations . we analyzed the asca data obtained in 2000 may which covered time intervals of @xmath79 s. the data were divided into 6 sets with an integration time of @xmath80 s each to study the time variabilities of the spectrum . in order to analyze the excess component separately from the continuum , we masked a range of 48 kev in spectral fitting and determined the continuum model . we found that a power law with a photon index of @xmath41.55 modified by two absorbers with different column densities and covering fractions can smoothly connect the two spectral parts at 24 kev and 810 kev . above this continuum , an excess is clearly detected in 4.57.5 kev , which has a broad and skewed feature . the feature has a prominent narrow peak at 6.4 kev , but accompanies a broad red wing at 4.56.0 kev and a blue wing at 6.87.5 kev . the flux and shape of the excess feature obtained from each of the 6 spectra were compared with one another . it is found that the excess flux is consistent with being stable , concerning both the flux and the spectral shape , over the observations of @xmath79 s. we also analyzed data obtained by rxte from january through july in 1999 in order to search for any time variations of the iron line on a time scale of @xmath81 s. we obtained 11 sets of time - resolved energy spectra , each of which covers a time interval of @xmath82 s. if we selected only the energy bands free from the iron structures , i.e. the 2.84.0 kev and 810 kev bands , each energy spectrum could be reproduced again by a power law with a photon index of @xmath41.5 modified by the partial covering absorption . this result is consistent with that of asca . in 4.57.5 kev , an excess flux over the continuum is clearly seen as a broad feature , in which significant flux variations are detected . however , no clear change was noticed in its profile . the absence of a profile change in spite of the significant flux variations in the broad and skewed feature strongly indicates that the feature is produced through a single mechanism . we studied the time scale of the flux variations of the narrow line . it was found that the flux variations are not significant on time scales of @xmath63@xmath56 s , whereas they become significant on time scales of @xmath56@xmath57 s. we also studied whether or not an introduction of a smearing effect can improve the correlation between the line flux and the continuum flux . we redistributed the continuum flux light curve with a simple method approximating a smear and delay . we found that the agreement between the line flux light curve and the smeared light curve of the continuum flux becomes acceptable when we introduce a smear and a delay on a time scale of @xmath77 s. these results concerning the line flux variations and the effect of a smear strongly suggest that the line emitting region should have a size extent of 10@xmath83 cm . we further studied the time scale of the variation in the absorption column density . a column density of the order of @xmath84 atom @xmath60 is necessary to reproduce the continuum spectrum , and is found to vary significantly on a time scale of @xmath63@xmath56 s with a relative amplitude of @xmath420% . taking account of the relative amplitude of the variation , the size of the absorber is indicated to be no larger than @xmath1 cm , which is just the size of the line emitting region . the absorber should be around , or inside , the line emitting region . we tried a continuum plus a disk - line model to reproduce the energy spectra , and found that this model can reproduce the energy spectra at 210 kev for both asca and rxte data . as a result of the previous discussion , the size of the line emitting region should be as large as @xmath1 cm . in the disk - line model , the line is assumed to be emitted from a region with a size of several to ten times the schwarzschild radius . if the size of @xmath1 cm deduced from observational results corresponds to several to ten - times the schwarzschild radius , we need to assume a central mass close to @xmath2 @xmath3 . this central mass is not consistent with @xmath57 @xmath3 estimated for ngc 4151 from various methods , e.g. x - ray variability time scale @xcite and the analysis of the high ionization lines in the broad line region @xcite . this mass is rather close to the typical mass of a galaxy , and would be too large as the mass of the central black hole . the presence of the absorber around or inside the line emitting region would also be difficult to explain in the context of the `` disk - line '' model . if the line emitting region has a size of several to ten - times the schwarzschild radius , the absorber should necessarily be located at a region very close to the central black hole . this may not be consistent with the unified scheme in which the heavy absorption as seen in ngc 4151 is considered to be due to a dusty torus around the central active region . we obtain an inclination angle of the disk to be @xmath85 . this inclination angle is smaller than the generally accepted value @xmath86 , which is supported by the chandra observation @xcite and the optical observations . this is also a disadvantage of the `` disk - line '' model . as discussed above , there exist some serious difficulties in the disk - line model , which are not consistent with a reasonable , common picture of agn , although we can not completely exclude its possibility . we consider here an alternative explanation for the origin of the broad and skewed profile , in which the narrow peak at 6.4 kev should be a narrow line and the red and blue wings should be a part of the continuum . however , according to the constant profile of the broad feature around 57 kev , the narrow line and the continuum should have a strong physical connection . the line energy at 6.4 kev strongly suggests that the line is the fluorescent k - line from neutral or low ionization iron . if the line is really emitted through a fluorescent process , it implies that some amount of matter exists in the vicinity of the x - ray source . this matter should be irradiated by x - rays from the x - ray source and the fluorescent lines should be emitted there . if this is the case , continuum x - rays should also be emitted from the matter through electron scattering of irradiating x - rays . this reflected component should be observed together with the fluorescent line , and could be the red and blue wings . when an x - ray from the source is absorbed by an iron atom in the x - ray reflector through photo - ionization of a k - electron , a fluorescent k - line is emitted with a certain probability called a fluorescent yield . on the other hand , when an x - ray from the source hits an electron in the reflector , it could be re - emitted towards us through thomson ( compton ) scattering . the cross section of thomson scattering is about @xmath87 @xmath88 . hence , if the column density of the x - ray reflector is sufficiently large , an x - ray penetrated into the reflector would , on average , experiences photo - electric absorption by matter with a column density of about @xmath89 @xmath60 before being scattered by an electron . as a result , x - rays reflected by sufficiently thick matter should have a spectrum with a continuum shape being the same as the intrinsic x - rays , but suffering from photo - electric absorption by matter with a column density of about @xmath89 @xmath60 . in fact , the broad and skewed feature can be reproduced by a model consisting of a narrow line at 6.4 kev , and a power law with the same slope as the remaining continuum and with a photo - electric absorption by a column of @xmath89 @xmath60 . the equivalent width of the 6.4 kev line with respect to the heavily absorbed component is about 2 kev . this is roughly consistent with a value expected from a case when the heavily absorbed component is just the reflected component ( see e.g. @xcite ) . if we adopt the above model for the broad and skewed feature around 4.58 kev , the total spectrum becomes to have three differently absorbed components with the same continuum shape . the absorption column densities are about @xmath90 @xmath60 , ( 12)@xmath91 @xmath60 , and about @xmath89 @xmath60 . among the three components , two components absorbed by smaller columns are found to be variable on a time scale of @xmath63@xmath56 s , whereas the component with the absorption by @xmath92 @xmath60 was steady on that time scale . this strongly suggests that the heavily absorbed component has a different origin from the other two components . the time variability of the heavily absorbed component is rather similar to that of the fluorescent iron line . this indicates a strong physical coupling between the fluorescent iron line and the heavily absorbed component , and strongly supports the idea that the heavily absorbed component is a reflected emission by cold matter which is also responsible for the fluorescent iron line . the above model fits both the asca spectra at 210 kev and the rxte spectra at 224 kev well . the model can also fit the data in a higher energy range . actually , if we extrapolate the best - fit model in the energy range of 2.824 kev to 100 kev , no significant discrepancy is recognized between the model and the observed energy spectra . therefore , this model can explain wider range of energy spectra without the spectral break than the model based on the disk - line . if the reflection is really at work , the covering fraction of the absorber may be related to the solid angle of the reflector , @xmath93 , subtending to the x - ray source . the covering fraction that we obtained for an absorber of @xmath89 @xmath60 is @xmath94 , which corresponds to @xmath95 . the reflection structure was detected in the energy spectra of ngc 4151 with the ginga / osse data @xcite and the bepposax data @xcite . the bepposax data indicated that the relative contribution of the reflection changed with time . the covering fraction of the reflector so far reported ranges over @xmath96 @xcite . the covering fraction which we obtained is slightly larger than this range , but is not very different , if we consider the different modeling of the reflection component . based on the reflection model , we may be able to constrain the mass of the central black hole using the iron line parameters . from the model fitting , we found that the iron line is consistent with having no intrinsic width , and its upper limit is 92 ev . we also found that the iron line emitting region has a size of @xmath1 cm . it may be natural to assume that the line emission region is located around the central black hole and is rotating at the kepler velocity around the black hole . then , the line width may be determined by the doppler effect ; the upper limit of the line width can be related to the upper limit of the kepler velocity as @xmath97 . this may be converted to the mass of the black hole using the relation @xmath98 , which yields the upper limit of the black hole mass , @xmath99 @xmath100 . here , we assume @xmath101 cm . this is consistent with the estimated mass of @xmath57 @xmath100 . if the mass of the central black hole is @xmath57 @xmath100 , a few times 10@xmath102 cm corresponds to @xmath103 @xmath31 . at this distance , the x - ray reflector should exist . since the reflected component shows evidence of absorption by a column of @xmath89 @xmath60 , the reflector should be thomson thick . the solid angle of the reflector , as viewed from the central x - ray source , should be about @xmath104 . this reflector would be a so - called dust torus , which is generally believed to exist around the central engine in seyfert galaxies . it has been shown that the continuum spectrum of ngc 4151 needs a partial covering by x - ray absorbers with a column density of ( 12)@xmath91 @xmath60 . the absorption column and the covering factor have been found to vary on a time scale of @xmath63@xmath56 s. as already discussed above , this leaky absorber should exist around or inside the line emitting region . if a dusty torus emits the line , a geometrical relation between a dusty torus and a broad line region seems to be consistent with the observed constraint on the geometrical relation between the line emitting region and the leaky absorber . the absorbers could be relatively cold clouds in the broad line region . if so , the typical time scale of a change of the leaky absorption should be roughly estimated by @xmath105 , where @xmath106 is the size of the x - ray emitting region and @xmath107 the velocity of the x - ray absorbing clouds . since the typical velocity of the broad line clouds is several times @xmath108 cm s@xmath13 and the typical time scale of the absorption change is a few times @xmath63 s , the size of the emission region may be about @xmath109 cm . this size corresponds to a few ten - times @xmath31 of a black hole with a mass of @xmath57 @xmath100 , and is consistent with a natural expectation that a region close to the central black hole should be emitting the continuum x - rays . arp , h. 1997 , , 319 , 33 burke , b. e. , mountain , r. w. , daniels , p. j. , & dolat , v. s. 1994 , ieee trans . 41 , 375 clavel , j. , altamore , a. , boksenberg , a. , bromage , g. e. , elvius , a. , pelat , d. , penston , m. v. , perola , g. c. , snijders , m. a. , & ulrich , m. h. 1987 , , 321 , 251 edelson , r , & nandra , k. 1999 , , 514 , 682 evans , i. n. , tsvetanov , z. , kriss , g. a. , ford , h. c. , caganoff , s. , & koratkar , a. p. 1993 , , 417 , 82 fabian , a. c. , rees , m. j. , stella , l. , & white , n. e. 1989 , , 238 , 729 fabian , a. c. , nandra , k. , reynolds , c. s. , brandt , w. n. , otani , c. , tanaka , y. , inoue , h. , & iwasawa , k. 1995 , , 277 , l11 fabian , a. c. , iwasawa , k. , reynolds , c. s. , & young , a. j. 2000 , , 112 , 1145 hayashida , k. , miyamoto , s. , kitamoto , s. , negoro , h. , & inoue , h. 1998 , , 500 , 642 inoue , h. 1985 , , 40 , 317 jahoda , k. , swank , j. h. , giles , a. b. , stark , m. j. , strohmayer , t. , zhang , w. , & morgan , e. h. 1996 , spie , 2808 , 59 makishima , k. , tashiro , m. , ebisawa , k. , ezawa , h. , fukazawa , y. , gunji , s. , hirayama , m. , idesawa , e. et al . 1996 , , 48 , 171 morrison , r. , & mccammon , d. 1983 , , 270 , 119 nandra , k. , george , i. m. , mushotzky , r. f. , turner , t. j. , & yaqoob , t. 1997 , , 477 , 602 nishiura , s. , murayama , t. , & taniguchi , y. 1998 , , 50 , 31 ogle , p. m. , marshall , h. m. , lee , j. c. , & canizares , c. r. 2000 , , 545 , l81 ohashi , t. , ebisawa , k. , fukazawa , y. , hiyoshi , k. , horii , m. , ikebe , y. , ikeda , h. , inoue , h. et al . 1996 , , 48 , 157 osterbrock , d. e. & koski , a. t. 1976 , , 176 , 61 piro , l. , nicastro , f. , feroci , m. , grandi , p. , parmar , a. , oosterbroek , t. , mineo , t. , piraino , s. et al . 1998 , b ( proc . suppl . ) , 69 , 481 serlemitsos , p. j. , jalota , l. , soong , y. , kunieda , h. , tawara , y. , tsusaka , y. , suzuki , h. , sakima , y. et al . 1995 , , 47 , 105 tanaka , y. , inoue , h. , & holt , s. s. 1994 , , 46 , l37 tanaka , y. , nandra , k. , fabian , a. c. , inoue , h. , otani , c. , dotani , t. , hayashida , k. , iwasawa , k. et al . 1995 , , 375 , 659 turner , t. j. , george , i. m. , nandra , k. , & mushotzky , r. f. 1998 , , 493 , 91 ulrich , m .- h . , & horne , k. 1996 , , 283 , 748 ulrich , m .- h . 2000 , , 10 , 135 wang , j .- x . , zhou , y .- y . , & wang , t .- 1999 , , 523 , l129 wang , j .- x . , wang , t .- , & zhou , y .- y . 2001 , , 549 , 891 weaver , k. a. , mushotzky , r. f. , arnaud , k. a. , serlemitos , p. j. , marshall , f. e. , petre , r. , jahoda , k. m. , smale , a. p. , & netzer , m. 1994 , , 423 , 621 weaver , k. a. , & reynolds , c. s. 1998 , , 503 , l39 yaqoob , t. , edelson , r. , weaver , k. a. , warwick , r. s. , mushotzky , r. f. , serlemitsos , p. j. , & holt , s. s. , 1995 , , 453 , l81 yang , y. , wilson , a. s. , & ferruit , p. 2001 , , 563 , 124 zdziarski , a. a. , johnson , w. n. , & magdziarz , p. 1996 , , 283 , 193 lccccccsequence number & 1 & 2 & 3 & 4 & 5 & 6 + @xmath64 [ @xmath110 atom @xmath60 ] & @xmath112 & @xmath113 & @xmath114 & @xmath115 & @xmath116 & @xmath116 + @xmath65 [ % ] & @xmath117 & @xmath118 & @xmath118 & @xmath119 & @xmath120 & @xmath121 + inclination angle [ deg ] & @xmath122 & @xmath123 & @xmath122 & @xmath124 & @xmath125 & @xmath126 + line flux [ @xmath127 photon @xmath60 s@xmath13 ] & @xmath129 & @xmath129 & @xmath129 & @xmath130 & @xmath131 & @xmath129 + @xmath132 & 1.10/140 & 1.02/140 & 0.99/140 & 1.03/140 & 1.02/140 & 1.03/140 + + lccccccsequence number & 1 & 2 & 3 & 4 & 5 & 6 + photon index ( @xmath38 ) & @xmath133 & @xmath134 & @xmath135 & @xmath136 & 1.55@xmath137 & 1.45@xmath138 + break energy [ kev ] & @xmath139 & @xmath140 & @xmath139 & @xmath140 & @xmath141 & @xmath142 + photon index ( @xmath39 ) & @xmath143 & @xmath144 & @xmath145 & @xmath146 & 1.89@xmath147 & 1.87@xmath148 + @xmath149 [ 10@xmath150 atom @xmath60 ] & @xmath151 & @xmath152 & @xmath153 & @xmath154 & @xmath155 & @xmath156 + @xmath65 [ % ] & @xmath157 & @xmath158 & @xmath159 & @xmath160 & @xmath161 & @xmath162 + line flux [ 10@xmath163 photon @xmath60 s@xmath13 ] & @xmath164 & @xmath165 & @xmath166 & @xmath167 & @xmath168 & @xmath169 + @xmath132 & 1.07/46 & 1.40/46 & 1.12/46 & 1.48/46 & 1.14/46 & 1.27/46 + sequence number & 7 & 8 & 9 & 10 & 11 + photon index ( @xmath38 ) & @xmath170 & @xmath171 & @xmath172 & @xmath134 & @xmath173 + break energy [ kev ] & @xmath139 & @xmath174 & @xmath175 & @xmath176 & @xmath174 + photon index ( @xmath39 ) & @xmath177 & @xmath178 & @xmath179 & @xmath180 & @xmath181 + @xmath149 [ 10@xmath150 atom @xmath60 ] & @xmath182 & @xmath183 & 9.7@xmath184 & @xmath185 & @xmath186 + @xmath65 [ % ] & @xmath187 & @xmath188 & @xmath189 & @xmath190 & @xmath191 + line flux [ 10@xmath163 photon @xmath60 s@xmath13 ] & @xmath192 & @xmath193 & @xmath194 & @xmath195 & @xmath196 + @xmath132 & 1.12/46 & 0.99/46 & 1.39/46 & 1.28/46 & 1.20/46 + + lccccccsequence number & 1 & 2 & 3 & 4 & 5 & 6 + photon index & @xmath197 & @xmath198 & @xmath199 & @xmath200 & @xmath201 & @xmath202 + @xmath64 [ 10@xmath150 atom @xmath60 ] & @xmath203 & @xmath112 & @xmath204 & @xmath205 & @xmath204 & @xmath203 + @xmath206 [ % ] & @xmath207 & @xmath208 & @xmath209 & @xmath210 & @xmath211 & @xmath212 + @xmath51 [ 10@xmath150 atom @xmath60 ] & @xmath213 & @xmath214 & @xmath215 & @xmath216 & @xmath217 & @xmath218 + @xmath48 [ % ] & @xmath219 & @xmath220 & @xmath221 & @xmath222 & @xmath223 & @xmath224 + line flux [ @xmath127 photon @xmath60 s@xmath13 ] & @xmath225 & @xmath226 & @xmath227 & @xmath228 & @xmath229 & @xmath230 + @xmath132 & 0.99/46 & 0.94/46 & 0.91/46 & 0.88/46 & 0.97/46 & 0.87/46 + sequence number & 7 & 8 & 9 & 10 & 11 + photon index & @xmath231 & @xmath232 & @xmath233 & @xmath234 & @xmath235 + @xmath64 [ 10@xmath150 atom @xmath60 ] & @xmath236 & @xmath205 & @xmath237 & @xmath140 & @xmath236 + @xmath206 [ % ] & @xmath238 & @xmath239 & @xmath240 & @xmath241 & @xmath242 + @xmath51 [ 10@xmath150 atom @xmath60 ] & @xmath243 & @xmath244 & @xmath245 & @xmath246 & @xmath247 + @xmath48 [ % ] & @xmath248 & @xmath249 & @xmath250 & @xmath251 & @xmath252 + line flux [ 10@xmath163 photon @xmath60 s@xmath13 ] & @xmath253 & @xmath254 & @xmath254 & @xmath255 & @xmath256 + @xmath132 & 0.98/46 & 0.91/46 & 0.99/46 & 0.91/46 & 0.90/46 + + lccccccsequence number & 1 & 2 & 3 & 4 & 5 & 6 + @xmath64 [ @xmath110 atom @xmath60 ] & @xmath139 & @xmath257 & @xmath141 & @xmath258 & @xmath259 & @xmath142 + @xmath65 [ % ] & @xmath260 & @xmath261 & @xmath262 & @xmath263 & @xmath264 & @xmath265 + @xmath30 [ @xmath110 atom @xmath60 ] & @xmath266 & @xmath267 & @xmath268 & @xmath225 & @xmath269 & @xmath270 + line flux [ @xmath127 photon @xmath60 s@xmath13 ] & @xmath271 & @xmath272 & @xmath273 & @xmath272 & @xmath273 & @xmath274 + @xmath132 & 1.04/140 & 0.97/140 & 0.94/140 & 0.99/140 & 0.99/140 & 1.02/140 +

we have studied the origin of the broad and skewed feature at 4.57.5 kev in the energy spectra of ngc 4151 using the asca and rxte data . the feature consists of a narrow peak at 6.4 kev and a broad wing extended between 4.57.5 kev . an analysis of the long - term variations revealed that the feature became variable only on a time scale longer than @xmath0 s. through a comparison with the continuum variabilities , we found that the emission region of the excess flux at 4.57.5 kev has an extent of @xmath1 cm . the broad and skewed feature at 4.57.5 kev may be explained by the so - called `` disk - line '' model . if so , the size of the line - emitting region , @xmath1 cm , should be equal to several or ten - times the schwarzschild radius of the central black hole . this results in a black hole mass of @xmath2 @xmath3 , which may be too large for ngc 4151 . we propose an alternative explanation for the broad and skewed feature , i.e. a `` reflection '' model , which can also reproduce the overall energy spectra very well . in this model , cold matter with a sufficiently large column density is irradiated by x - rays to produce a reflected continuum , which constitutes the broad wing of the feature , and narrow fluorescent lines . the equivalent width of the iron fluorescent line ( @xmath42 kev ) and the upper limit of its width ( @xmath5 ev ) are also consistent with this model . from these results and considerations , we conclude that the `` disk - line '' model has difficulty to explain the spectral variations of ngc 4151 , and the reflection model is more plausible .

transient lunar phenomena ( tlps or ltps ) are defined for the purposes of this investigation as localized ( smaller than a few hundred km across ) , transient ( up to a few hours duration , and probably longer than typical impact events - less than 1s to a few seconds ) , and presumably confined to processes near the lunar surface . how such events are manifest is summarized by cameron ( 1972 ) . in paper i ( crotts 2008 ; see also crotts 2009 ) we study the systematic behavior ( especially the spatial distribution ) of tlp observations - particularly their significant correlations with tracers of lunar surface outgassing , and we are thereby motivated to understand if this correlation is directly causal . numerous works have offered hypotheses for the physical cause of tlps ( mills 1970 , garlick et al . 1972a , b , geake & mills 1977 , cameron 1977 , middlehurst 1977 , hughes 1980 , robinson 1986 , zito 1989 , carbognani 2004 , davis 2009 ) , but we present a methodical examination of the influence of outgassing , exploring quantitatively how outgassing might produce tlps . furthermore , it seems likely that outgassing activity is concentrated in several areas , which leads one to ask how outgassing might interact with and alter the regolith presumably overlying the source of gas . reviews of similar processes exist but few integrate apollo - era data e.g. , stern ( 1999 ) , mukherjee ( 1975 ) , friesen ( 1975 ) . as the final version of this paper approached completion , several papers were published regarding the confirmed discovery of hydration of the lunar regolith . fortunately , we deal here with the special effects of water on lunar regolith and find that many of our predictions are borne out in the recently announced data . we will deal with this explicitly in 5 . several experiments from apollo indicate that gas is produced in the vicinity of the moon , even though these experiments disagree on the total rate : 1 ) lace ( lunar atmosphere composition experiment on _ apollo 17 _ ) , @xmath10.1 g s@xmath2 over the entire lunar surface ( hodges et al . 1973 , 1974 ) ; 2 ) side ( suprathermal ion detector experiment on _ apollo 12 , 14 , 15 _ ) , @xmath17 g s@xmath2 ( vondrak et al . 1974 ) ; 3 ) ccge ( cold cathode gauge experiment on _ apollo 12 , 14 , 15 _ ) , @xmath360 g s@xmath2 ( hodges et al . 1972 ) . these measurements not only vary by more than two orders of magnitude but also in assayed species and detection methods . lace results here applies only to neutral @xmath4ar , @xmath5ar and @xmath6ne . by mass @xmath4ar predominates . side results all relate to ions , and perhaps include a large contribution from molecular species ( vondrak et al . 1974 ) . ccge measures only neutral species , not easily distinguishing between them . the lace data indicate @xmath4ar episodic outgassing on timescales of a few months or less ( hodges & hoffman 1975 ) , but resolving this into faster timescales is more ambiguous . in this discussion we adopt the intermediate rate ( side ) , about 200 tonne y@xmath2 for the total production of gas , of all species , ionized or neutral . the lace is the only instrument to provide compositional ratios , which also include additional , rarer components in detail . we will use these ratios and in some cases normalize them against the side total . much of the following discussion is only marginally sensitive to the actual composition of the gas . for many components of molecular gas at the lunar surface , however , there is a significant possible contribution from cometary or meteoritic impacts , and a lesser amount from solar wind / regolith interactions . the influx of molecular gas from comets and meteorites are variously estimated , usually in the range of tonnes or tens of tonnes per year over the lunar surface ( see anders et al . 1973 , morgan & shemansky 1991 ) . cometary contributions may be sporadically greater ( thomas 1974 ) . except for h@xmath7 , solar wind interactions ( mukhergee 1975 ) provide only a small fraction of the molecular concentration seen at the surface ( which are only marginally detected : hoffman & hodges 1975 ) . there is still uncertainty as to what fraction of this gas is endogenous . current data do not succeed in resolving these questions , but we will return to consider them later in the context of gas seepage / regolith interactions . in this paper we consider various effects of outgassing through the regolith , and find the most interesting simple effect occurs when the flow is high enough to cause disruption of the regolith by an explosion to relieve pressure ( 2 ) , which we compare to fluidization . another interesting effect occurs when the gas undergoes a phase change while passing through the regolith ( 3 ) , which seems to apply only to water vapor . this leads primarily to the prediction of the likely production of subsurface ice , particularly in the vicinity of the lunar poles . these effects suggest a variety of observational / experimental approaches , which we summarize in 4 . in 5 we discuss the general implications of these findings , with specific suggestions as to how these might guide further exploration , particularly in respect to contamination by anthropogenic volatiles . we also discuss the relevance of the predictions in 4 to very recent discoveries regarding lunar regolith hydration . first , let us make a few basic points about outgassing and the regolith . one can easily picture several modes in which outgassing volatiles might interact with regolith on the way to the surface . these modes will come into play with increasing gas flow rate and/or decreasing regolith depth , and we simply list them with mneumonic labels along with descriptions : \1 ) choke : complete blockage below the regolith , meaning that any chemistry or phase changes occur within the bedrock / megaregolith ; \2 ) seep : gas is introduced slowly into the regolith , essentially molecule by molecule ; \3 ) bubble : gas is introduced in macroscopic packets which stir or otherwise rearrange the regolith ( such as `` fluidization '' e.g. , mills 1969 ) ; \4 ) gulp : gas is introduced in packets whose adiabatic expansion deposits kinetic energy into regolith and cools the gas , which therefore might even undergo a phase change ; \5 ) explode : gas is deposited in packets at base of the regolith leading to an explosion ; and \6 ) jet : gas simply flows into the vacuum at nearly sound speed with little entrained material . while the intermediate processes might prove interesting , the extreme cases are probably more likely to be in effect and will receive more of our attention . in fact choking behavior might lead to explosions or geysers , when the pressure blockage is released . since these latter two processes involve primarily simple hydrodynamics ( and eventually , newtonian ballistics ) , we will consider them first , and how they might relate to tlps . if outgassing occurs at a rate faster than simple percolation can sustain , and where regolith obstructs its path to the surface , the accumulation of the gas will disrupt and cause bulk motion of the intervening regolith . the outgassing can lift the regolith into a cloud in the temporary atmosphere caused by the event . the presence of such a cloud has the potential to increase the local albedo from the perspective of an outside observer due to increased reflectivity and possible mie scattering of underlying regolith . additionally , volatiles buried in the regolith layer could become entrained in this gas further changing the reflective properties of such a cloud . garlick et al . ( 1972b ) describe fluidization of lunar regolith , in which dust is displaced only temporarily and/or over small distances compared to ballistic trajectories , but we will assume that we are dealing with more rapid changes . let us construct a simple model of explosive outgassing through the lunar surface . for such an event to occur , we assume a pocket of pressurized gas builds at the base of the regolith , where it is delivered by transport through the crust / megaregolith presumably via channels or cracks , or at least faster diffusion from below . given a sufficient flow rate ( which we consider below ) , gas will accumulate at this depth until its internal pressure is sufficient to displace the overlying regolith mass , or some event releases downward pressure e.g. , impact , moonquake , incipient fluidization , puncturing a seal , etc . we can estimate the minimal amount of gas alone required to cause explosive outgassing by assuming that the internal energy of the buried gas is equal to the total energy necessary to raise the overlying cone of regolith to the surface . this `` minimal tlp '' is the smallest outgassing event likely to produce potentially observable disruption at a new site , although re - eruption through thinned regolith will require less gas . we consider the outgassing event occurring in two parts illustrated in figure 1 . initially , the gas bubble explodes upward propelling regolith with it until it reaches the level of the surface ; we assume that the plug consists of a cone of regolith within 45@xmath8 of the axis passing from the gas reservoir to the surface , normal to the surface . through this process the gas and regolith become mixed , and we assume they now populate a uniform hemispherical distribution of radius @xmath9 m on the surface . at this point , the gas expands into the vacuum and drags the entrained regolith outward until the dust cloud reaches a sufficiently small density to allow the gas to escape freely into the vacuum and the regolith to fall eventually to the surface . we consider this to be the `` minimal tlp '' for explosive outgassing , as there is no additional reservoir that is liberated by the event beyond the minimum to puncture the regolith . one could also imagine triggering the event by other means , many of which might release larger amounts of gas other than that poised at hydrodynamical instability . for the initial conditions of the first phase of our model , we assume the gas builds up at the base of the regolith layer at a depth of 15 m ( for more discussion of this depth , see 3 ) . we set the bulk density of the regolith at @xmath10 g @xmath11 ( mckay et al . 1991 ) , thereby setting the pressure at this depth at 0.45 atm . because of the violent nature of an explosive outgassing , we assume that the cone of dust displaced will be 45@xmath8 from vertical ( comparable to the angle of repose for a disturbed slope of this depth : carrier , olhoeft & mendell 1991 ) . the mass of overlying regolith defined by this cone is @xmath12 kg . in order to determine the mass of gas required to displace this regolith cone , we equate the internal energy of this gas bubble with the potential energy ( @xmath13 , where @xmath14 m s@xmath15 ) required to lift the cone of regolith a height @xmath16 m to the surface , requiring 47,000 moles of gas . much of the gas found in outgassing events consists of @xmath17he , @xmath5ar and @xmath4ar ( see 1 ) , so we assume a mean molar mass for the model gas of @xmath18 g mol@xmath2 , hence 940 kg of gas is necessary to create an explosive outgassing event . the temperature at this depth is @xmath19c ( see 3 ) , consequently implying an overall volume of gas of 2400 m@xmath20 or a sphere 8.3 m in radius . what flow rate is needed to support this ? using fick s diffusion law , @xmath21 , where the gas number density @xmath22 @xmath11 is taken from above , and drops to zero through 15 m of regolith in @xmath23 . the diffusivity @xmath24 is 7.7 and 2.3 cm@xmath0 s@xmath2 for he and ar , respectively , in the knudsen flow regime for basaltic lunar soil simulant ( martin et al . 1973 , where @xmath26 is the absolute temperature . the sticking time of the gas molecules , or heat of absorption , becomes significant if the gas is more reactive or the temperature is reduced . unfortunately we find no such numbers for real regolith , although we discuss realistic diffusivities for other gases below . ] ) , so we adopt @xmath27 cm@xmath0 s@xmath2 for our assumed he / ar mixture . ( for any other gas mixture of this molecular weight , @xmath24 would likely be smaller ; below we also show that @xmath24 tends to be lower for real regolith . ) over the area of the gas reservoir , this implies a mass leakage rate of 2.8 g s@xmath2 , or @xmath140% of the total side rate . with the particular approximations made about the regolith diffusivity , this is probably near the upper limit on the leakage rate . at the surface of the regolith , this flow is spread to a particle flux of only @xmath28 cm@xmath15 , which presumably causes no directly observable optical effects . the characteristic time to drain ( or presumably fill ) the reservoir is 4 d. the second phase of the simulation models the evolution of dust shells and expanding gas with a spherically symmetric 1d simulation centered on the explosion point . the steps of the model include : 1 ) the regolith is divided into over 600 bins of different mean particle size . these bins are logarithmically spaced over the range @xmath30 mm to @xmath31 m according to the regolith particle size distribution from sample 72141,1 ( from mckay et al . the published distribution for sample 72141,1 only goes to 2@xmath32 m , but other sources ( basu & molinaroli 2001 ) indicate a component extending below 2@xmath32 m , so we extend our size distribution linearly from 2@xmath32 m to 0@xmath32 m . furthermore , we assume the regolith particles are spherical in shape and do not change in shape or size during the explosion ; 2 ) to represent the volume of regolith uniformly entrained in the gas , we create a series of 1000 concentric hemispherical shells for each of the different particle size bins ( i.e. , roughly 600,000 shells ) . each of these shells is now independent of each other and totally dependent on the gas - pressure and gravity for motion ; 3 ) we further assume that each regolith shell remains hemispherical throughout the simulation . explicitly , we trace the dynamics of each shell with a point particle , located initially 45 degrees up the side of the shell ; 4 ) we calculate the outward pressure of the gas exerted on the dust shells . the force from this pressure is distributed among different shells of regolith particle size weighted by the total surface area of the grains in each shell . we calculate each shell s outward acceleration and consequently integrate their equations of motion using a timestep @xmath33 s ; 5 ) we calculate the diffusivity of each radial shell ( in terms of the ability of the gas to move through it ) by dividing the total surface area of all dust grains in a shell by the surface area of the shell itself ( assuming grains surfaces to be spherical ) ; 6 ) starting with the largest radius shell , we sum the opacities of each shell until we reach a gas diffusive opacity of unity . gas interior to this radius can not `` see '' out of the external regolith shells and therefore remains trapped . gas outside of this unit opacity shell is assumed to escape and is dropped from the force expansion calculation . dust shells outside the unit opacity radius are now assumed to be ballistic ; 7 ) we monitor the trajectory of each dust shell ( represented by its initially 45@xmath8 particle ) until it drops to an elevation angle of 30@xmath8 ( when most of the gas is expanding above the particle ) , at which time this particle shell is no longer supported by the gas , and is dropped from the gas - opacity calculation ; 8) an optical opacity calculation is made to determine the ability of an observer to see the lunar surface when looking down on the cloud . we calculate the downward optical opacity ( such as from earth ) by dividing the total surface area of the dust grains in a shell by the surface area of the shell as seen from above ( @xmath34 ) . starting with the outmost dust shell , we sum downward - view optical opacities until we reach optical depth @xmath35 and @xmath36 to keep track the evolution of this cloud s appearance as seen from a distance ; 9 ) we return to step # 4 above and iterate another timestep @xmath37 , integrating again the equations of motion . we continue this algorithm until all gas is lost and all regolith has fallen to the ground . finally , when all dust has fallen out , we calculate where the regolith ejecta have been deposited . because we re representing each shell as a single point for the purposes of the equations of motion calculations , we want to do more than simply plot the location of each shell - particle on the ground to determine the deposition profile of ejected regolith . thus we create a template function for the deposition of a ballistic explosion of a single spherical shell of material . by applying this template to each shell - particle s final resting location , we better approximate the total deposition of material from that shell . we then sum all of the material from the @xmath38 shells to determine the overall dust ejecta deposition profile . there are obvious caveats to this calculation . undoubtably the release mechanism is more complex than that adopted here , but this release mode is sufficiently simple to be modeled . secondly , the diffusion constant , and therefore the minimal flow rate , might be overestimated due to the significant ( but still largely unknown ) decrease in regolith porosity with increasing depth on the scale of meters ( carrier et al . 1991 ) , plus the liklihood that simulants used have larger particles and greater porosity than typical regolith . lastly , the regolith depth of 15 meters might be an overestimate for some of these regions , which are among the volcanically youngest and/or freshest impacts on the lunar surface . this exception does not apply to plato and its active highland vicinity , however , and aristarchus is thickly covered with apparent pyroclastic deposits which likely have different but unknown depths and diffusion characteristics . we find results for this `` minimal tlp '' numerical model of explosive outgassing through the lunar regolith interesting in terms of the reported properties of tlps . figure 2 shows the evolution of the model explosion with time , as might be seen from an observer above , in terms of the optical depth @xmath39 and @xmath40 profiles of the model event , where @xmath39 is a rough measure of order unity changes in the appearance of the surface features , whereas @xmath40 is close to the threshold of the human eye for changes in contrast , which is how many tlps are detected ( especially the many without noticable color change ) . in both cases the cloud at the particular @xmath41 threshold value expands rapidly to a nearly fixed physical extent , and maintains this size until sufficient dust has fallen out so as to prevent any part of the cloud from obscuring the surface to this degree . easily - seen effects on features ( @xmath39 ) lasts for 50 s and extends over a radius of 2 km , corresponding to 2 arcsec in diameter , resolvable by a typical optical telescope but often only marginally so . in contrast , the marginally detectable @xmath40 feature extends over 14 km diameter ( 7.5 arcsec ) , lasting for 90 s , but is easily resolved . this model `` minimal tlp '' is an interesting match to the reported behavior of non - instantaneous ( not @xmath42 s ) tlps : about 7% of duration 90 s or less , and half lasting under about 1300 s. certainly there should be selection biases suppressing reports of shorter events . most tlp reports land in an envelope between about two minutes and one hour duration , and this model event lands at the lower edge of this envelope . furthermore , most tlps , particularly shorter ones , are marginally resolved spatially , as would be the easily - detectable component of the model event . this correspondence also seems interesting , given the simplicity of our model and the state of ignorance regarding relevant parameters . how might this dust cloud actually affect the appearance of the lunar surface ? first , the cloud should cast a shadow that will be even more observable than simple surface obscuration , blocking the solar flux from an area comparable to the @xmath39 region and visible in many orientations . experiments with agitation of lunar regolith ( garlick et al . 1972b ) show that the reflectance of dust is nearly always increased under fluidization , typically by about 20% and often by about 50% depending on the particular orientation of the observer versus the light source and the cloud . similar results should be expected here for our simulated regolith cloud . these increases in lunar surface brightness would be easily observable spread over the many square kilometers indicated by our model . furthermore , because the sub - micron particle sizes dominate the outer regions of the cloud , it seems reasonable to expect mie - scattering effects in these regions with both blue and red clouds expected from different sun - earth - moon orientations . figure 3 shows the typical fall - out time of dust particles as a function of size . particles larger than @xmath130 @xmath32 m all fall out within the first few seconds , whereas after a few tens of seconds , particles are differentiated for radii capable of contributing to wavelength - dedendent scattering . later in the event we should expect significant color shifts ( albeit not order - unity changes in flux ratios ) . the larger dynamical effects in the explosion cloud change rapidly over the event . half of the initially entrained gas is lost from the cloud in the first 3 s , and 99% is lost in the first 15 s. throughout the observable event , the remaining gas stays in good thermal contact with the dust , which acts as an isothermal reservoir . gas escaping the outer portions of the dust cloud does so at nearly the sound speed ( @xmath43 m s@xmath2 ) , and the outer shells of dust also contain particles accelerated to similar velocities . gas escaping after about 3 s does so from the interior of the cloud in parcels of gas with velocities decreasing roughly inversely with time . one observable consequence of this is the expectation that much of the gas and significant dust will be launched to altitudes up to about 50 km , where it may be observed and might affect spacecraft in lunar orbit . the longterm effects of the explosion are largely contained in the initial explosion crater ( nominally 14 m in radius ) , although exactly how the ejecta ultimately settle in the crater is not handled by the model . at larger radii the model is likely to be more reliable ; figure 4 shows how much dust ejecta is deposited by the explosion as a function of radius . beyond the initial crater , the surface density of deposited material varies roughly as @xmath44 , so it converges rapidly with distance . inside a radius of @xmath45 m , the covering factor of ejecta is greater than unity ; beyond this one expects coverage to be patchy . this assumes that the crater explosion is symmetric and produces few `` rays . '' the explosion can change the reflectivity by excavating fresh material . this would be evidenced by a @xmath110% drop in reflectance at wavelength @xmath46 nm caused by surface fe@xmath47 states in pyroxene and similar minerals ( adams 1974 , charette et al . likewise there is an increase in reflectivity in bluer optical bands ( buratti et al . 2000 ) over hundreds of nm . even though these photometric effects are compositionally dependent , we are interested only in differential effects : gradients over small distances and rapid changes in time . the lifetime of even these effects at 300 m radius is short , however , due to impact `` gardening '' turnover . the half - life of the ejecta layer at 300 m radius is only of order 1000 y ( from gault et al . 1974 ) , and shorter at large radius ( unless multiple explosions accumulate multiple layers of ejecta ) . at 30 m radius the half - life is of order 10@xmath48 y. from maturation studies of the 950 nm feature ( lucey et al . 2000 , 1998 ) , even at 30 m , overturn predominates over optical maturation rates ( over hundreds of my ) . the scale of outgassing in this model event , both in terms of gas release ( @xmath49 1 tonne ) and timescale ( @xmath49 4 d ) , are consistent with the total gas output and temporal granularity of outgassing seen in @xmath4ar , a dominant lunar atmospheric component . the fact that this model also recovers the scale of many features actually reported for tlps lends credence to the idea that outgassing and tlps might be related to each other causally in this way , as well as circumstantially via the rn@xmath50 episodes and tlp geographical correlation ( paper i ) . how often such an explosive puncturing of the regolith layer by outgassing should occur is unknown , due to the uncertainty in the magnitude and distribution of endogenous gas flow to the surface , and to some degree how the regolith reacts in detail to large gas flows propagating to the surface . also , a new crater caused by explosive outgassing will change the regolith depth , its temperature structure , and eventually its diffusivity . we will not attempt here to follow the next steps in the evolution of an outgassing `` fumerole '' in this way , but are inspired to understand how regolith , its temperature profile , and gas interact , as in the next section . furthermore , such outgassing might happen on much larger scales , or might over time affect a larger area . indeed , such a hypothesis is offered for the scoured region of depleted regolith forming the ina d feature and may extend to other regions around imbrium ( schultz et al . 2006 ) . our results here and in paper i bear directly on the argument of vondrak ( 1977 ) that tlps as outgassing events are inconsistent with side episodic outgassing results . the detection limits from alsep sites _ apollo 12 , 14 _ and _ 15 _ correspond to 16 - 71 tonne of gas per event at common tlp sites , particularly aristarchus . ( vondrak states that given the uncertainties in gas transportation , these levels are uncertain at the level of an order of magnitude . ) our `` minimal tlp event '' described above is 20 - 80 times less massive than this , however , and still visible from earth . it seems implausible that a spectrum of such events would never exceed the side limit , but it is not so obvious such a large event would occur in the seven - year alsep operations interval . also , this side limit interpretation rests crucially on alphonsus ( and ross d ) as prime tlp sites , both features which are rejected by our robust geographical tlp sieve in paper i. dust elutriation or particle segregation in a cloud agitated by a low - density gas , occurring in this model , could potentially generate large electrostatic voltages , perhaps relating to tlps ( mills 1970 ) . luminous discharges are generated in terrestrial volcanic dust clouds ( anderson et al . 1965 , thomas et al . 2007 ) . above we see dust particles remain suspended in a gas of number density @xmath51 to @xmath52 @xmath11 on scales of several tenths of a km to several km , a plausible venue for large voltages . in the heterogeneous lunar regolith , several predominant minerals with differing particle size may segregate under gas flow suspension and acceleration . assuming a typical particle size of @xmath53 m , and typical work function differences for particles of even well - defined compositions is problematic due to surface effects such as solar - wind / micrometeoritic weathering and exposed surface fe@xmath47 states . the following analysis suffices for two particles of different conducting composition ; a similar result arises via triboelectric interaction of two different dielectrics although the details are less understood . disturbed dust is readily charged for long periods in the lunar surface environment ( stubbs , vondrak & farell 2005 ) . ] of @xmath54 ev , two particles exchange charge upon contact until the equivalent of @xmath550.25v is maintained , amounting to @xmath56 coul = 1700 e@xmath57 . when these particles separate to distance @xmath58 , their mutual capacitance becomes @xmath59 . for @xmath60 m , if particles retain @xmath61 , voltages increase by @xmath62 times ! such voltages can not be maintained . paschen s coronal discharge curve reaches minimum potential at 137v for ar , 156v for he , for column densities @xmath63 of @xmath64 and @xmath65 cm@xmath15 , respectively , and rises steeply for lesser column densities ( and roughly proportional to @xmath63 for larger @xmath63 ) . similar optimal @xmath63 are found for molecules , with minimum voltages a few times higher e.g. , 420v at @xmath66 cm@xmath15 for co@xmath7 , 414v for h@xmath7s , 410v for ch@xmath67 , and @xmath63 for other molecules @xmath68 cm@xmath15 . the visual appearance of atomic emission in high voltage discharge tubes is well know , with he glowing pink - orange ( primarily at 4471.5 and 5875.7 : reader & corliss 1980 , pearse & gaydon 1963 ) , and ar glowing violet ( from lines 4159 - 4880 ) . if this applies to tlps , the incidence of intense red emission in some tlp reports ( cameron 1978 ) argues for another gas . to 13 ( mag arcsec@xmath15 ) in v , compared to 3.4 at full moon , so visible sources could be faint . ] ne is not an endogenous gas . common candidate molecules appear white or violet - white ( co@xmath7 , so@xmath7 ) or red ( water vapor - primarily h@xmath69 , which is produced in many hydrogen compounds ; ch@xmath67 - balmer lines plus ch bands at 390 and 431 nm ) . the initial gas density at the surface from a minimal tlp is @xmath70 @xmath11 , so initially the optimal @xmath63 for coronal discharge is on cm scales ( versus the initial outburst over tens of meters ) . as the tlp expands to 1 km radius , @xmath71 drops to @xmath72 @xmath11 , so the optimal @xmath63 holds over the scale of the entire cloud , likely the most favorable condition for coronal discharge . if gas kinetic energy converts to luminescence with , for instance , 2% efficiency , at this density this amounts to @xmath73 j m@xmath74 , or 100 j m@xmath15 , compared to the reflected solar flux of 100 w m@xmath15 , capable of a visible color shift for several seconds . perhaps a minimal tlp could sustain a visible coronal discharge over much of its @xmath75 min lifetime . these should also be observable on the nightside surface , too , since solar photoionization is seemingly unimportant in initiating the discharge , and there are additional factors to consider . referring back to scenarios ( 2 ) and ( 3 ) in 1 , the onset of fluidization ( mills 1969 ) marks the division between these two regimes of seepage and `` bubbling '' and has been studied ( siegal & gold 1973 , schumm 1970 ) . although laboratory test are made with coarser sieve particulates and much thinner dust layers in 1 @xmath76 gravity , we can scale the gas pressure needed for incipient fluidization by @xmath77 and thickness @xmath78 to find the threshold @xmath79 atm ( siegal & gold 1973 ) . correcting for less diffusive regolith , this pressure estimate is likely a lower limit . below this pressure simple gas percolation likely predominates . what processes occur during `` simple '' percolation ? were it not for phase changes of venting gas within the regolith , the composition of the gas might be a weak consideration in this paper ( except for perhaps the molecular / atomic mass ) , and temperature would likely only affect seepage as @xmath80 in the diffusivity . water plays a special role in this study ( separate from concerns regarding resource exploitation or astrobiology ) , in that it is the only common substance encountering its triple point temperature in passing through the regolith , at least in many locations . in this case water might not contribute to overpressure underneath the regolith leading to explosive outgassing . this would also imply that even relatively small volatile flows containing water would tend to freeze in place and remain until after the flow stops . for water this occurs at 0.01@xmath8c , corresponding to 0.006 atm in pressure ( the pressure dropping by a factor of 10 every @xmath125@xmath8 . ) effectively , water is the only relevant substance to behave in this fashion . the next most common substances may be large hydrocarbons such as nonane or benzene , obviously not likely abundant endogenous effluents from the interior . also h@xmath7so@xmath67 reaches its triple point , but changes radically with even modest concentrations of water . a similar statement can be made about hno@xmath81 , not a likely outgassing constituent . these will not behave as their pure state , either ; this leaves only h@xmath7o . water ( and sulfur ) has been found in significant concentration in volcanic glasses from the deep lunar interior ( saal et al . 2008 , friedman et al . 2009 ) , and has been liberated in large quantities in past volcanic eruptions . the measured quantities of tens of ppm imply juvenile concentrations of hundreds of ppm . from the heat flow measurements at the _ apollo 15 _ and _ 17 _ lunar surface experiment ( alsep ) sites ( langseth & keihm 1977 ) , we know that just below the surface , the stable regolith temperature is in the range of 247 - 253k ( dependent on latitude , of course ) , with gradients ( below 1 - 2 m ) of 1.2 - 1.8 deg m@xmath2 , which extrapolates to @xmath82c at @xmath83 m depths subsurface . with the exception of the outermost few centimeters , the entire regolith is below the triple point temperature and is too deep to be affected significantly by variations in heating over monthly timescales . this is an interesting depth , since in many areas the regolith is not quite this deep , as small as under a few meters near lichtenberg ( schultz & spudis 1983 ) and at the surveyor 1 site near flamsteed ( shoemaker & morris 1970 ) to depths at apollo sites ( summarized by mckay et al . 1991 ) near the @xmath82c depths calculated above , up to probably 20 m or more in the highlands , and 40 m deep north of the south pole - aitken basin ( bart & melosh 2005 ) . presumably , the fractured megaregolith supporting the regolith likely does not contain as many small particles useful for retaining water ice , as we detail below , but it may accumulate ice temporarily . recent heat flow analyses ( saito et al . 2007 ) account for longer timescale fluctuations placing the @xmath82c depth twice as far subsurface , increasing the lifetime of retained volatiles against sublimation accordingly ; for now we proceed with a more conventional , shorter - lived analysis . the escape of water and other volatiles into the vacuum is regulated by the state of the regolith and is presumably largely diffusive . we assume the knudsen flow regime ( low - density , non - collisional gas ) . of special importance is the measured abundance of small dust grains in the upper levels of the regolith , which perhaps pertains to depths @xmath84 m ( where bulk density is probably higher : carrier et al . assuming that particle distributions are self - similar in size distribution ( constant porosity ) , for random - walk diffusion out of a volume element @xmath85 , the diffusion time step presumably scales with the particle size @xmath86 , so the diffusion time @xmath87 . for particles of the same density , therefore , one should compute the diffusion time by taking a @xmath88-weighted average of particle sizes counted by mass , @xmath89 . this same moment of the distribution is relevant in 2 . published size distributions measured to sufficiently small sizes include again mckay et al . ( 1974 ) with @xmath90 @xmath32 m , and supplemented on smaller sizes with _ apollo 11 _ sample 10018 ( basu & molinaroli 2001 ) , which reduces the average to about 20 @xmath32 m . this is an overestimate because a large fraction ( 34 - 63% ) are agglutinates , which are groupings of much smaller particles . many agglutinates have large effective areas e.g. , @xmath91 , with values of a few up to 8 . ( here @xmath92 is a mean radius from the center of mass to a surface element . ) to a gas particle , the sub - particle size is more relevant than the agglutinate size , so the effective particle size of the entire sample might be much smaller , conceivably by a factor of a few . we compare this to experimental simulations , a reasonably close analogy being the sublimation of a slab of ice buried up to 0.2 m below a medium of simulant jsc mars-1 ( allen et al . 1998 ) operating at @xmath93k and 7 mbar ( chevrier et al . 2007 ) , close to lunar regolith conditions . this corresponds to the lifetime of 800 y for a 1 m thick ice layer covered by 1 m of regolith . the porosity of jsc mars-1 is 44 - 54% , depending on compactification whereas lunar soil has @xmath149% at the surface , perhaps 40% at a depth of 60 cm , and slightly lower at large depths ( carrier et al . lunar soil is somewhat less diffusive by solely this measure . the mean size @xmath89 of jsc mars-1 is 93 @xmath32 m , @xmath4910 times larger than that for _ apollo 17 _ and _ 11 _ regolith , accounting for agglutinates , so the sublimation timescale for regolith material is , very approximately , @xmath4910 ky ( perhaps up to @xmath130 ky ) . other simulants are more analogous to lunar regolith , so future experiments might be more closely relevant . converting a loss rate for 1 m below the surface to 15 m involves the depth ratio @xmath94 . farmer ( 1976 ) predicts an evaporation rate scaling as @xmath95 ( as opposed to the no - overburden analysis : ingersoll 1970 ) . experiments with varying depths of simulated regolith ( chevrier et al . 2007 ) show that the variation in lifetime indeed goes roughly as @xmath96 , implying a 1 m ice slab lifetime at 15 m on the order of @xmath97 to @xmath98 y. the vapor pressure for water ice drops a factor of 10 in passing from @xmath82c to current temperatures of about @xmath99c just below the surface ( also the naked - ice sublimation rate : andreas 2007 ) , which would indicate that @xmath190% of water vapor tends to stick in overlying layers ( without affecting the lifetime of the original layer , coincidentally ) . this begs the question of the preferred depth for an ice layer to form . the regolith porosity decreases significantly between zero and 1 m depth ( carrier et al . 1991 ) which argues weakly for preferred formation at greater depth . at 30 m depth or more , the force of overburden tends to close off porosity . the current best limit on water abundance is from the sunrise terminator abundances from lace , which produces a number ratio of h@xmath7o/@xmath4ar with a central value of 0.014 ( with @xmath100 limits of 0 - 0.04 ) . this potentially indicates an actual h@xmath7o/@xmath4ar outgassing rate ratio up to 5 times higher ( hoffman & hodges 1975 ) . adopting the side rate of 7 g s@xmath2 in the @xmath120 - 44 amu mass range , and assuming most of this is @xmath4ar ( vondrak , freeman & lindeman 1974 : given the much lower solar wind contributions of other species in this range ) , this translates to 0.1 g s@xmath2 of water ( perhaps up to 0.5 g s@xmath2 or 15 tonne y@xmath2 ) , in which case most of the gas must be ionized . the disagreement between side and lace is a major source of uncertainty ( perhaps due to the neutral / ionized component ambiguity ) . we discuss below that at earlier times the subsurface temperature was likely lower , but let us consider now the situation in which a source arises into pre - established regolith in recent times . we assume a planar diffusion geometry , again . in this case , we take spatial gradients over 15 m and scale the jsc mars-1 diffusivity of 1.7 cm@xmath0 s@xmath2 to 0.17 cm@xmath0 s@xmath2 for lunar regolith . since the triple - point pressure corresponds to number density @xmath101 @xmath11 , the areal particle flux density is @xmath102 s@xmath2 cm@xmath15 . for a large outgassing site , with the same water fraction of water indicated by lace e.g. , total outgassing of 7 g s@xmath2 including 0.1 g s@xmath2 of water , this rate can maintain a total area of 0.012 km@xmath0 at the triple - point pressure i.e. , a 125 m diameter patch . this is much larger than the 15 m regolith depth , bearing out our assumed geometry . if this ice patch were 1 m thick , for example , the ice would need to be replenished every 4000 y. of course , this is a simple model and many complications could enter . we consider briefly the effects of latitude , change in lunar surface temperature over geological time , and the effects of aqueous chemistry on the regolith . the temperature just below the surface is legislated by the time - averaged energy flux in sunlight , so it scales according to the stefan - boltzmann law from the temperature at the equator ( @xmath103 ) according to @xmath104 . this predicts a 6k temperature drop from the equator ( at about 252k ) to the latitude of the aristarchus plateau ( @xmath105 ) or the most polar subsurface temperature measurement by _ apollo 15 _ , a drop to 224k at plato ( @xmath106 ) , @xmath107200k for the coldest 10% of the lunar surface ( @xmath108 ) and @xmath107150k for the coldest 1% ( @xmath109 ) . , but there are flooded craters much higher . ] these translate into a regolith depth at the water triple - point of @xmath14 , 18 , 33 or 65 m deeper than at the equator , respectively , probably deeper in the latter cases than the actual regolith layer . permanently shadowed cold traps , covering perhaps 0.1% of the surface , have temperatures @xmath10760k ( e.g. , adorjan 1970 , hodges 1980 ) . ( note that the lunar south pole is a minor tlp site responsible for about 1% of robust report counts . ) since even at the equator the h@xmath7o triple point temperature occurs @xmath113 m below the surface , at increasing latitude this zone quickly moves into the megaregolith where the diffusivity is largely unknown but presumably higher ( neglecting the decrease in porosity due to compression by overburden ) . to study this , we assume a low diffusivity regolith layer 15 m deep overlying a high diffusivity layer which may contain channels directing gas quickly upward ( although perhaps not so easily horizontally ) . the diffusivity of the regolith near 0@xmath8c is dominated by elastic reflection from mineral surfaces , without sticking , whereas at lower temperatures h@xmath7o molecules stick during most collisions ( haynes , tro & george 1992 ) . this is especially the case if the surfaces are coated with at least a few molecular layers of h@xmath7o molecules , of negligible mass . the sticking behavior of h@xmath7o molecules on water ice has been studied over most of the temperature range relevant here ( washburn et al . 2003 ) ; but does depend somewhat on whether the ice is crystalline or amorphous ( speedy et al . in contrast the sticking behavior of h@xmath7o molecules on lunar minerals is much less well known . the lunar simulant diffusivity value above corresponds to a mean free path time of @xmath11 @xmath32s for h@xmath7o molecules near 0@xmath8c . in contrast the timescale for h@xmath7o molecules sticking on ice is ( from schorghofer & taylor [ 2007 ] and references therein ) : @xmath110 where @xmath111 is the areal density of h@xmath7o molecules on ice @xmath112 cm@xmath15 for density @xmath113 , and molecular mass @xmath32 . the sticking fraction @xmath69 varies from about 70% to 100% for @xmath114k to 120k . the equilibrium vapor pressure is given by @xmath115 $ ] where @xmath116 and @xmath117 are the triple point pressure and temperature , respectively , and sublimation enthalpy @xmath118 kj / mole . this expression and laboratory measurements imply a sticking timescale @xmath119s at @xmath120k , 1 ms at 200k , 1 s at 165k , 1 hr at 134k , and 1 yr at 113k . the sticking timescale quickly and drastically overwhelms the kinetic timescale at lower temperatures . this molecular behavior has a strong effect on the size of the ice patch maintained by the example source considered above . simply scaling by the time between molecular collisions , corresponding to a 125 m diameter ice patch at @xmath103 , we find at the base of the regolith a 160 m patch at @xmath121 ( aristarchus plateau ) , 580 m at @xmath122 ( plato ) , 2.3 km at @xmath123 ( 10% polar cap ) , and an essentially divergent value , 522 km at @xmath124 ( 1% polar cap ) . if in fact the regolith layer is much deeper than suspected , the added depth of low diffusivity dust significantly increases the patch area : 170 m at @xmath125 , 830 m at @xmath106 , and 4 km at @xmath123 . figure 5 presents graphically how the growth of the ice patch varies with latitude , plus also the effects of flow rate and the assumed regolith depth . most portions of the lunar surface have been been largely geologically inactive during the past 3 gy or more ( with some of the notable exceptions listed above ) . during this time several important modifications of the scenario above are relevant . the current heat flow from the lunar interior , @xmath126 w m@xmath15 ( langseth et al . 1972 , 1973 ) , is only a @xmath127 part of the solar constant , so it affects the temperature near the lunar surface at the level of only @xmath128 millidegree . there were times in the past , however , when interior heating likely pushed the temperature near the surface over @xmath82c . a zero - degree zone near the maria presumably could not form until @xmath13 gy ago , probably sufficient for the moon globally ( see spohn et al . after this the @xmath82c depth receded into the regolith , while the regolith layer was also growing . simultaneously , the average surface temperature was cooler by @xmath129 degree due to standard solar evolution ( gough 1981 perhaps @xmath130 lower in the highlands at 4 gy ago ) . since the the thickness of regolith after 3 gy ago grows at only about 1 m per gy ( quaide & oberbeck 1975 ) , within the maria the @xmath82c depth sinks into bedrock / fractured zone . whatever interaction and modification might be involved between the regolith and volatiles will proceed inwards , leaving previous epochs effects between the surface vacuum and the @xmath82c layer now at @xmath131 m. another issue to consider is possible regolithic chemical reactions with outgassing volatiles , especially over prolonged geological timescales . the key issue is the possible presence of water vapor , and perhaps so@xmath7 . there is little experimental work on the aqueous chemistry of lunar regolith ( which will vary due to spatial inhomogeneity ) . dissolution of lunar fines by water vapor is greatly accelerated in the absence of other gases such as o@xmath7 and n@xmath7 ( gammage & holmes 1975 ) and appears to proceed by etching the numerous damage tracks from solar - wind particles . this process acts in a way to spread material from existing grains without reducing their size ( which would otherwise tend to increase porosity ) . liquid water is more effective than vapor , not surprisingly , and ice tends to establish a pseudo - liquid layer on its surface . this is separate from any discussion of water retention on hydrated minerals surfaces robust to temperatures above 500@xmath8c ( cocks et al . 2002 & op cit . ) . this is a complex chemical system that will probably not be understood without simulation experiments . the major constituents are presumably silicates , which will migrate in solution only over geologic time . ( on earth , consider relative timescales of order 30 my typical migration times for quartz , 700 ky for orthoclase feldspar , kalsi@xmath81o@xmath132 and 80 ky for anorthite , caal@xmath7si@xmath7o@xmath132 : brantley 2004 . ) one might also expect the production of ca(oh)@xmath7 , plus perhaps mg(oh)@xmath7 and fe(oh)@xmath7 . it is not clear that fe(oh)@xmath7 would oxidize to more insoluble feo(oh ) , but any free electrons would tend to encourage this . it seems that the result would be generally alkaline . since feldspar appears to be a major component in some outgassing regions e.g. , aristarchus ( mcewen et al . 1994 ) , one should also anticipate the production of clays . this is not accounting for water reactions with other volatiles e.g. , ammonia , which has been observed as a trace gas ( hoffman & hodges 1975 ) perhaps in part endogenous to the moon , and which near @xmath82c can dissolve in water at nearly unit mass ratio ( also to make an alkaline solution ) . carbon dioxide is a possible volatile constituent , and along with water can metamorphose olivine / pyroxene into mg@xmath81si@xmath67o@xmath133(oh)@xmath7 i.e. talc , albeit slowly under these conditions ; in general the presence of co@xmath7 and thereby h@xmath7co@xmath81 opens a wide range of possible reactions into carbonates . likewise the presence of sulfur ( or so@xmath7 ) opens many possibilities e.g. , caso@xmath1342h@xmath7o ( gypsum ) , etc . since we do not know the composition of outgassing volatiles in detail , we will probably need to inform simulation experiments with further remote sensing or in situ measurements . the mechanical properties of this processed regolith are difficult to predict . some possible products have very low hardness and not high ductility . some of these products expand but will likely fill the interstitial volume with material , which will raise its density and make it more homogeneous . regolith is already ideal in having a nearly power - law particle distribution with many small particles . it seems likely that any such void - filling will sharply reduce diffusivity . the volatiles actually discovered in volcanic glasses from the deep interior ( saal et al . 2008 , friedman et al . 2009 ) include primarily h@xmath7o and so@xmath7 but not co@xmath7 or co. with the addition of water , regolithic mineral combinations tend to be cement - like , and experiments with anorthositic lunar chemical simulants have produced high quality cement without addition of other substances , except sio@xmath7 ( horiguchi et al . 1996 , 1998 ) . whether this happens @xmath135 @xmath136 depends upon whether over geological time ( cao)@xmath81sio@xmath7 or other ca can act as a binder without heating to sintering temperatures . the possible production of gypsum due to the high concentrations of sulfur would add to this cement - like quality . the extent to which ordinary mixes such as portland cement lose water into the vacuum depend on their content of expansive admixture ( kanamori 1995 ) . portland cement mixes show little evidence of loss of compressional strength in a vacuum ( cullingford & keller 1992 ) . we need to think in terms of possibly cemented slabs in some vicinities , and need to consider the effects of cracks or impacts into this concrete medium . this is probably not a dominant process , since the overturn timescale to depths even as shallow as 1 m is more than 1 gy ( gault et al . 1974 , quaide & oberbeck 1975 ) , whereas we discuss processes at @xmath115 m or more . craters 75 m in diameter will permanently excavate to a 15 m depth ( e.g. , collins 2001 , and ignoring the effects of fractures and breccia formation ) , and are formed at a rate of about 1 gy@xmath2 km@xmath15 ( extending neukum et al . [ 2001 ] with a shoemaker number / size power - law index 2.9 ) . this will affect some of the areal scales discussed above , but not all . we speculate that vapor or solution flow might tend to deliver ice and/or solute to these areas and eventually act to isolate the system from the vacuum . finally , we note above that over geological timescales this ice layer will tend to sink slowly into the regolith , at a rate of order 1 m gy@xmath2 , setting up a situation where any relatively impermeable concrete zone will tend to isolate volatiles from the vacuum . in this case volatile leakage will tend to be reduced to a peripheral region around the ice patch . assuming that volatiles leak out through the entire 15 m thickness of regolith at the patch boundary , the 125 m diameter patch area for @xmath137 from above corresponds to a peripheral zone expected from a 520 m diameter patch . thus any such concrete overburden will encourage growth of small patches , and will do so even more for larger ones ( assuming @xmath10715 m regolith depth ) . how much water might reasonably be expected to outgas at these sites ? the earliest analyses of apollo samples argued for extreme scarcity of water and other volatiles ( anders 1970 , charles , hewitt & wones 1971 , epstein & taylor 1972 ) . on the earth , water is the predominant juvenile outgassing component ( gerlach & graeber 1985 , rubey 1964 ) , whereas even the highest water concentrations discussed below ( saal et al . 2008 ) imply values an order of magnitude smaller . on the moon , water content is drastically smaller , with a current atmospheric water content much less than what would affect hydration in lunar minerals ( mukherkjee & siscoe 1973 ) , although some lunar minerals seem to involve water in their formation environments ( agrell et al . 1972 , williams & gibson 1972 , gibson & moore 1973 , and perhaps akhmanova et al . 1978 ) . the origin of the water in volcanic glasses ( saal et al . 2008 , friedman et al . 2009 ) is still poorly understood but implies internal concentrations that at first look seems in contradiction with earlier limits e.g. , anders 1970 . for much different lunar minerals , high concentration is implied for water ( mccubbin et al . 2007 ) as well as other volatiles ( krhenbhl et al . 1973 ) . it is not a goal of this paper to explain detected water in lunar samples , but its origin at great depth is salient here . as a point of reference , hodges and hoffman ( 1975 ) show that the @xmath4ar in the lunar atmosphere derives from deep in the interior , of order 100 km or more . they hypothesize that the gas could just as easily derive from the asthenosphere , 1000 km deep or more ( see also hodges 1977 ) . the picritic glasses analyzed by saal , friedman , et al . derive from depths of @xmath1300 - 400 km or greater ( elkins - tanton et al . 2003 , shearer , layne & papike 1994 ) . o@xmath7 fugacity measurements e.g. , sato 1979 , are based on glasses from equal or lesser depths . water originating from below the magma ocean might provide one explanation ( saal et al . 2008 ) , as might inhomogeneity over the lunar surface . differentiation might not have cleared volatiles from the deep interior despite its depletion partially into the mantle . one might also consider that geographical variation between terranes e.g. , kreep ( k - rare earth element - p ) or not , might be important . in the moon s formation temperatures of proto - earth and progenitor impactor material in simulations grow to thousands of kelvins , sufficient to drive off the great majority of all volatiles , but these are not necessarily the only masses in the system . either body might have been orbited by satellites containing appreciable volatiles , which would likely not be heated to a great degree and which would have had a significant probability of being incorporated into the final moon . furthermore , there is recent discussion of significant water being delivered to earth / moon distances from the sun in the minerals themselves ( lunine et al . 2007 , drake & stimpfl 2007 ) , and these remaining mineral - bound even at high temperatures up to 1000k ( stimpfl et al . the volume of surface water on earth is at least @xmath138 km@xmath20 , so even if the specific abundance of lunar water is depleted to @xmath139 terrestrial , one should still expect over @xmath140 tonnes endogenous to the moon , and it is unclear that later differentiation would eliminate this . this residual quantity of water would be more than sufficient to concern us with the regolith seepage processes outlined above . for carbon compounds , models of the gas filling basaltic vesicles ( sato 1976 ; also ohara 2000 , wilson & head 2003 , taylor 1975 ) predict co , cos , and perhaps co@xmath7 as major components . negligible co@xmath7 is found in fire - fountain glasses originating from the deep interior ( saal et al . 2008 ) ; this should be considered in light of co on the moon ( and co@xmath7 on earth ) forming the likely predominant gas driving the eruption ( rutherford & papale 2009 ) . we suspect that water outgassing was likely higher in the past than it is now . furthermore , no site of activity traced by @xmath50rn or by robust tlp counts ( paper i ) has been sampled . ( the sample return closest to aristarchus , _ apollo 12 _ , is 1100 km away . ) from our discussion above the behavior of outgassing sites near the poles versus near the equator might differ greatly , with volatile retention near the poles being long - term and perhaps making the processing of volatiles much more subterranean and covert . ( note that the lunar south pole is a minor tlp site responsible for about 1% of robust report counts , as per paper i. ) with these uncertainties we feel unable to predict exactly how or where particular evidence of lunar surface outgassing might be found , although the results from above offer specific and varied signals that might be targeted at the lunar surface . for this reason we turn attention to how such effects might be detected realistically from the earth , lunar orbit , and near the moon s surface , and we suggest strategies not only for how these might be tested but also how targeted observations might economically provide vital information about the nature of lunar outgassing . we consider the impact of recent hydration detections in 5 . we appreciate the controversial nature of suggesting small but significant patches of subsurface water ice , given the history of the topic . we take care to avoid `` cargo cult science '' - selection of data and interpretion to produce dramatic but subjectively biased conclusions that do not withstand further objective scrutiny ( feynman 1974 ) . despite the advances made primarily by apollo - era research , we are still skirting the frontiers of ignorance . we are operating in many cases in a regime where interesting observations have been made but the parameters e.g. , the endogenous lunar molecular production ( water vapor or otherwise ) , required to evaluate alternative models and interpretations are sufficiently uncertain to frustrate immediate progress . below we offer several straightforward and prompt tests of our conclusions and hypotheses which offer prospects of settling many of these issues . tlps are rare and short - lived , which hampers their study . we advance supplanting the current anecdotal catalog with data with _ a priori _ explicit , calculable selection effects . this might seem daunting ; paper i used in essence all known reports from lunar visual observers since the telescope s invention ! with modern imaging and computing , it is tractable . another problem clear above is the variety of ways in which outgassing can interact with the regolith . in cases of slow seepage , gases may long delay their escape from the regolith . if the gases are volcanic , they might interact along the way , and water vapor might trap it and other gases in the regolith . these factors bear on designing future investigations . we can make significant headway exploiting more modern technology . table 1 lists the many methods detailed in this section . there has been no areal - encompassing , digital image monitoring of the near side with appreciable time coverage using modern software techniques to isolate transients . numerous particle detection methods are promising . the relevant experiments on apollo were limited in duration , a week or less , or 5 - 8 years in the case of alsep . furthermore _ clementine _ and the relevant portion of _ lunar prospector _ were also relatively short . these limitations serve as background to the following discussions . in this section we provide a potential roadmap to detailed study of outgassing . -0.5 in optical imaging advances several goals . transient monitoring recreates how tlps were originally reported . not yet knowing tlp emission spectra , our bandpass should span the visual , 400 - 700 nm . after an event , surface morphology / photometry changes might persist , betrayed by 0.95 and 1.9 @xmath32 m surface fe@xmath47 bands and increased blue reflectivity ( 2.2 ) . hydration is manifest in the infrared . asteroidal regolith 3 @xmath32 m hydration signals are common ( lebofsky et al . 1981 , rivkin et al . 1995 , 2002 , volquardsen et al . 2004 ) , and stronger than those at 700 nm ( vilas et al . 1999 ) seen in lunar polar regions . absorption near 3 @xmath32 m appears in lunar samples exposed to terrestrial atmosphere for a few years ( markov et al . 1980 , pieters et al . 2005 ) but not immediately ( akhmanova et al . 1972 ) , disappearing within a few days in a dry environment . further sample experiments are needed . -0.5 in earth - based monitoring favors the near side , as do tlp - correlated effects : @xmath50rn outgassing ( all four events on nearside , plus most @xmath141po residual ) and mare edges . the best , consistent resolution comes from the @xmath142 @xmath143 @xmath144 with @xmath145 arcsec fwhm ( @xmath1150 m ) but with large overhead times . competing high - resolution imaging from `` lucky exposures '' ( le , also `` lucky imaging '' ) exploits occasionally superlative imaging within a series of rapid exposures ( fried 1978 , tubbs 2003 ) . amateur setups achieve excellent le results , and the cambridge group ( law , mackay & baldwin 2006 ) attains diffraction - limited imaging on a 2.5-meter telescope , @xmath1200 - 300 m fwhm . only @xmath1461% of observing time survives image selection , but for the moon this requires little time . le resolution is limited to a seeing isoplanatic patch , @xmath11000 arcsec@xmath0 , 3000 times smaller than the moon . likewise , @xmath147 s wide field camera 3 , covering 3000 arcsec@xmath0 , can not practically survey the near side . -0.1 in high resolution imaging can monitor small areas over time or in one - shot applications compared with other sources i.e. , lunar imaging missions . le or @xmath147 match the resolution of global maps from _ lunar reconnaissance orbiter _ camera s ( lroc ) wide - angle camera ( robinson et al . 2005 ) , and @xmath148/uvis , over 0.3 - 1 @xmath32 m . @xmath149 s multiband imager ( ohtake et al . 2007 ) has 40-m 2-pixel resolution . lroc s narrow angle camera has 2 m resolution in one band , targeted . _ change-1_/ccd ( yue et al . 2007 ) might also aid `` before / after '' sequences . lunar orbiter images , resolving to @xmath11 m , form excellent `` before '' data for many sites , for morphological changes e.g. , cores of explosive events over 40 years . the prime technique for detecting changes between epochs of similar images is image subtraction , standard in studying supernovae , microlensing and variable stars . this produces photon poisson noise - limited performance ( tomaney & crotts 1996 ) and is well - matched to ccd or cmos imagers , which at 1 - 2 arcsec fwhm resolution cover the moon with 10 - 20 mpixels , readily available . one needs @xmath492 pixels fwhm , otherwise non - poisson residuals dominate . our group has automated tlp monitors on the summit of cerro tololo , chile and at rutherfurd observatory in new york that produce regular lunar imaging ( crotts et al . 2009 ) , often simultaneously . each cover the near side at 0.6 arcsec / pixel with images processed in 10s . this is sufficient to time - sample nearly all reported tlps ( see paper i ) and produce residual images free of systematic errors at poisson levels ( figure 6 ) . imaging monitors open several possibilities for tlp studies , with extensive , objective records of changes in lunar appearance , at sensitivity levels @xmath110 times better than the human eye . an automated system can distinguish contrast changes of 1% or better , whereas the human eye is limited to @xmath4910% . we will measure the frequency of tlps soon enough ; paper i indicates perhaps one tlp per month visible to a human observing at full duty cycle . tlp monitors open new potential to alert other observers , triggering le imaging of an active area , or spectroscopy of non - thermal processes and the gas associated with tlps . spatially resolved spectroscopy can 1 ) elucidate tlp physics , including identification of gas released , or 2 ) probe quasi - permanent changes in tlp sites . we must find changes in a four - dimensional dataset : two spatial dimensions , wavelength , and time , too much to monitor for transients . fortunately , tlp monitoring can alert to an event in under 1000s , and a larger telescope with a spectrograph can observe the target ( within @xmath1300s ) . whereas `` hyperspectral '' imaging usually refers to resolving power @xmath150 , where @xmath151 is the fwhm resolution , tlp emission might be much narrower , thereby diluted at low resolution . for line emission , rejecting photons beyond the line profile yields contrasts up to 10@xmath17 times better than the human eye using a telescope . ir hydration band near 3.4 @xmath32 m have substructure over @xmath120 nm , requiring @xmath152 , compared to the ir spex on the nasa infrared telescope facility with @xmath153 . the 950 nm and 1.9 @xmath32 m pyroxene bands show compositional shifts ( hazen , bell & mao 1978 ) seen at @xmath154 . differentiating pyroxenes from fe - bearing glass ( farr et al . 1980 ) requires @xmath155 . observations involve scanning across the lunar face with a long slit spectrograph ( figure 7a ) . since lunar surface spectral reflectance is homogenized by impact mixing , @xmath15699% of the light in such a spectrum is `` subtracted away '' by imposing this average spectrum and looking for deviations ( figure 7b ) . the data cube can be sliced in any wavelength to construct maps of lunar features in various bands . figure 8 shows that surface features are reconstructed in detail and fidelity . what narrow lines might we search for ? the emission measure of gas in our model excited by solar radiation is undetectable except for the first few seconds . coronal discharge offers a caveat . reddish discharge may indicate h@xmath69 from dissociation of many possible molecules . emission from aristarchus ( absent balmer lines ) , and transient c@xmath7 swan bands ( kozyrev 1958 ) . we do not advance a model to explain these observations . ] rather than relying on h@xmath69 plus faint optical lines / bands to distinguish molecules , note that near - ir vibrational / rotational bands are brighter and more discriminatory . as in 3 , internal water vapor might have produced ice in the regolith @xmath10715 m subsurface , a venue for ground - penetrating radar , from lunar orbit . while epithermal neutrons and gamma radiation can detect hydrogen , they can not penetrate @xmath491 m. near the poles or subject to chemical modification ( 3.3 ) , ice might range closer to the surface . past and current lunar radar include _ apollo 17 _ s lunar sounder experiment ( lse ) ( brown 1972 , porcello 1974 ) at 5 , 16 and 260 mhz , @xmath149 s lunar radar sounder ( ono & oya 2000 ) at 5 mhz ( optionally , 1 mhz or 15 mhz ) , @xmath157 s mini - rf ( mini radio - frequency technology demonstration ) at 3 and @xmath110 ghz , and mini - sar on _ chandrayaan-1 _ at 3 ghz ( bussey et al . shorter wavelength radar could map possible changes in surface features in explosive outgassing , over tens of meters , in before / after radar sequences meshed with optical monitoring e.g. , with _ mini - rf ( chin et al . 2007 ) . for regolith and shallow bedrock , we need @xmath1100 - 300 mhz ; lse operated only a few orbits and near the equator . near side maps at @xmath11 km resolution at 430 ghz ( campbell et al . 2007 ) could improve with intensive ground - based programs , or from lunar orbit , penetrating @xmath158 m. orbital missions can combine different frequencies and/or reception angles to improve spatial resolution and ground clutter , and reduce interference speckle noise . earth - based radar maps exist at 40 , 430 and 800 mhz ( thompson & campbell 2005 ) , also 2.3 ghz ( stacy 1993 , campbell et al . 2006a , b ) . angles of incidence from earth are large e.g. , @xmath159 , with echoes dominated by diffuse scattering not easily modulated . circular polarization return can probe for surface water ice ( nozette 1996 , 2001 ) but is questioned ( simpson 1998 , campbell et al . 2006a ) . applying these to subsurface ice is at least as problematic , especially at @xmath1300 mhz to penetrate @xmath115 m. finding subsurface ice is challenging . the dielectric constant is @xmath160 for regolith , water ice ( slightly higher ) , and many relevant powders of comparable specific gravity e.g. , anorthosite and various basalts . these have attenuation lengths similar to ice , as well . using net radar return alone , it will be difficult to distinguish ice from regolith . in terrestrial situations massive ice bodies reflect little internally ( moorman , robinson & burgess 2003 ) . ice - bearing regions should be relatively dark in radar images , if lunar ice - infused volumes homogenize or `` anneal , '' either forming a uniform slab or by binding together regolith with ice in a uniform @xmath24 bulk . on the other hand , hydrated regolith has @xmath24 much higher than unhydrated ( up to 10 times ) , and attenuation lengths over 10 times shorter ( chung 1972 ) . hydration effects are largest at lower frequencies , even below 100 mhz . if water ice perturbs regolith chemistry , increasing charge mobility as in a solution , @xmath24 and conductivity increase , raising the loss tangent ( conductivity divided by @xmath24 and frequency ) . this high-@xmath24 zone should cause reflections , depending strongly on the suddenness of the transition interface . the 430 mhz radar map ( ghent et al . 2004 ) of aristarchus and vicinity , site of @xmath150% of tlp and radon reports shows the 43-km diameter crater surrounded by low radar - reflectivity some 150 km across , especially downhill from the aristarchus plateau , which is dark to radar , except bright craters and vallis schrteri . the darker radar halo centered on aristarchus itself is uniquely smooth , indicating that it was probably formed or modified by the impact , a few hundred my ago . this darkness might be interpreted as high loss tangent , as above , or simply fewer scatterers ( ghent et al . 2004 ) i.e. , rocks of @xmath11 m size ; it is undemonstrated why the latter applies in the ejecta blanket within the bright radar halo within 70 km of the aristarchus center . other craters , some as large as aristarchus , have dark radar haloes , but none so extended ( ghent et al . the aristarchus region matches subsurface ice redistributed by impact melt : dark , smooth radar - return centered on the impact ( although tending downslope ) . one should search for dark radar areas around likely outgassing sites . _ alpha - particle spectrometry : _ a @xmath50rn atom random walks only @xmath1200 km before decaying ( or sticking to a cold surface ) . in under a day , @xmath50rn dispersal makes superfluous placing detectors @xmath146100 km above the surface ( excepting @xmath161 sensitivity considerations ) . alpha - particle spectrometers observed the moon successfully for short times . the latitude coverage was limited on _ apollo 15 _ ( @xmath162 for 145 hours ) and _ apollo 16 _ ( @xmath163 , 128 h ) . _ lunar prospector s _ alpha particle spectrometer ( covering the entire moon over 229 days spanning 16 months ) was partially damaged and suffered sensitivity drops due to solar activity ( binder 1998 ) . @xmath149 s alpha ray detector ( ard ) promised 25 times more sensitivity than apollo ( nishimura et al . 2006 ) , but sharing a failed power supply it has yet to produce results . _ apollo 15 _ observed outgassing events from aristarchus and grimaldi , _ apollo 16 _ none , and _ lunar prospector _ aristarchus and kepler integrated over the mission . apollo and _ lunar prospector _ detected decay product @xmath141po at mare / highlands boundaries from @xmath50rn leakage over the past @xmath1100 y. an expected detection rate might be grossly estimated , consistent with an event 1 - 2 times per month detectable by _ apollo 15 _ , and by _ lunar prospector _ over the mission , with aristarchus responsible for @xmath150% . a polar orbiting alpha - particle spectrometer with a lifetime of a year or more and instantaneous sensitivity equal to apollo s could produce a detailed map of outgassing on the lunar surface separate from optical manifestation . two in polar orbit could cover the lunar surface every 1.8 half - lives of @xmath50rn , nearly doubling sensitivity . sensitivity can be increased if detectors incorporate solar wind vetos , or operate during solar minimum , and if detectors orient towards the lunar surface . _ on - orbit mass spectrometry : _ unlike @xmath50rn and its long surface residence , other outgassing events call for several instruments for efficient localization e.g. , by mass spectrometry . with outgassing of hundreds of tons and tens of events per year , particle mass fluence from one outburst seen 1000 km away approaches @xmath164 amu cm@xmath15 . a burst that is seen by a few detectors could be well constrained . gas scale heights @xmath1100 km imply detectors near the ground . conversely , an instantaneous outburst seen 100 km away will disperse less than one minute in arrival ; detectors must operate rapidly . this was a problem e.g. , the _ apollo 15 _ orbital mass spectrometer experiment ( hoffman & hodges 1972 ) requiring 62s to scan through a factor of 2.3 in mass . clearly there are two separate modes of gas propagation above the surface , neutral and ionized ( vondrak , freeman & lindeman 1974 , hodges et al . 1972 ) , at rates of one to hundreds of tonne y@xmath2 for each . operational strategies of these detectors are paramount . consider an event 1000 km away , which will spread @xmath165s in arrival time . a simple gas pressure gauge is too insensitive ; with an ambient atmosphere not atypical e.g. , number density @xmath166 @xmath11 ( varying day / night e.g. , hodges , hoffman & johnson 2000 ) , the collisonal background rate in 500 s amounts to 10@xmath167 or more than the fluence for a typical outburst ( assuming @xmath158 amu particles ) . since interplanetary solar proton densities can vary by order unity in an hour or less ( e.g. , mcguire 2006 ) , pressure alone is insufficient . mass spectrometry subdivides incoming flux in mass , but also in direction , decreasing effective background rates . one satellite particle detector can not distinguish episodic behavior of outgassing versus spacecraft motion at @xmath168 km s@xmath2 . localizing such signals between two platforms is ideal , at least for neutral species , if they constrain temporal / spatial location of specific outbursts using timing and signal strength differences . a timing difference indicates the distance difference to the source , with the source confined to a hyperboloid locus . location on this hyperboloid is fixed by signal strengths , plus left / right ambiguity from detector directionality . a mean nearest satellite distance of 1000 km from arbitrary sources requires @xmath4910 low orbital platforms . mass spectrometers on the surface can maintain such density over smaller areas efficiently once we know roughly where sources may be . a mass spectrometer planned for _ lunar atmospheric and dust environment explorer _ ( @xmath169 ) sits on one platform in equatorial orbit ; geographical resolution of outgassing events will be poor.27 , 50 , 76 or @xmath170 ( e.g. , ramanan & adimurthy 2005 ) . if we want to maintain a position over the terminator ( sun - synchronous orbit ) , we requires a precession rate @xmath171 rad s@xmath2 . alternatively , ladee achieves this by a precessing , highly eccentric orbit , but spends a small fracton of its time near the lunar surface . precession is fixed by coefficient @xmath172 ( konopliv et al . 1998 ) according to @xmath173 , where @xmath86 is lunar radius , @xmath174 orbital angular speed , @xmath175 lunar mass and @xmath92 orbital radius . precession due to the sun and earth are much smaller . one can not effectively institute both sun synchronicity in a polar orbit , however , since the maximum inclination orbit with @xmath176 s@xmath2 occurs at @xmath177 ( or else below the surface ) . to force sun - synchronicity at @xmath178 , @xmath179 s@xmath2 , requires only @xmath180 mm s@xmath15 which could even be accomplished by a hall - effect ion engine or even a solar sail ( with 330 cm@xmath0 g@xmath2 ) . ] surveying _ in situ _ approaches to studying volatiles is beyond the scope of this paper ; we emphasize a few key points . the key effort is to focus from wide - ranging reconnaissance down to scales where lunar volatiles can be sampled near their source . primary global strategies are optical transient monitoring ( near side , resolution @xmath11 km ) and orbital @xmath50rn and 3@xmath32 m detection ( both hemispheres , @xmath1100 km and @xmath1461 km , respectively ) . even trusting that tlps trace volatiles and centroiding tlps to 10% of a resolution element , localization error ( @xmath1100 m ) could preclude easy _ in situ _ followup . ( in appendix i we outline improving this to @xmath181 m. ) two simple _ in situ _ technologies could isolate outgassing sources below 100 km scales . first , three alpha particle detectors on the surface can triangulate nearby @xmath50rn outgassing sources , using strength and time delay in arrival of random walking @xmath50rn . secondly , a mass spectrometer that can reconstruct the ballistic trajectory of neutrals from the source ( austin et al . 2008 , daly , radebaugh & austin 2009 ) can construct an `` image '' of transient outgassing sources over regions up to 1000 km across . this spectrometer is not overwhelmed by pulsed sources while measuring masses over a wide range . further technologies could pinpoint subsurface structure at 10 m scales from information at 1 km . , or tuned to one of several species vibrational - rotational states . on smaller scales ( 1 - 100 m ) several varieties of mass spectroscopy might prove effective , including downward - sniffing spectrometers , triangulating outburst detectors arrays , and pyrolysis mass spectrometers ( ten kate et al . 2009 ) which heat regolith samples in search of absorbed species from previous outgassing . ] by lace s deployment with the final apollo landing , the outgassing environment was contaminated by anthropogenic gas ( freeman & hills 1991 ) especially near landing sites ; each mission of human exploration will deliver tens of tonnes gases to the surface , with species relevant to endogenous volcanic gas , approaching or exceeding the annual endogenous output of such gases.o@xmath67 ( nitrogen teroxide ) and ch@xmath81n@xmath7h@xmath81 ( mono - methyl hydrazine ) , with altair propelled by liquid oxygen and hydrogen . future missions might use liquid o@xmath7 and ch@xmath67 . earth departure stages might deliver residual @xmath182 and @xmath183 in lunar impact . altair ( and eds ) produce water , and orion exhausts h@xmath7o , co@xmath7 and n@xmath7 . n@xmath7 was the prime candidate constituent in an outburst seen by the _ apollo 15 _ over mare orientale : hoffman & hodges 1972 , perhaps anthropogenically - hodges 1991 . ] depending on spacecraft orientations and trajectories when thrusting , they may deliver @xmath120 tonnes of mostly water to the surface , which will remain up to about one lunation , making suspect measurements of these and other species for years . the origin of tlps has been mysterious , and their correlation to outgassing , while strong , was only circumstantial . the plausible generation of tlp - like events as simple consequence of outgassing from the interior lends credence to a possible causal link . we present a model tied to outgassing from deep below the regolith that reproduces the time and spatial scale of reported tlps , suggesting a causal link to outgassing . radiogenic gas evolved from the regolith can not provide the concentration to produce a noticeable explosive event . apollo and later data were insufficiently sensitive to establish the level of outgassing beyond @xmath50rn , and isotopes of ar , plus he , presumably , and did detect marginally molecular gas , but of uncertain origin , particularly ch@xmath67 , reviewing the evidence and available techniques , there are several gases that should be highlighted as crucial outgassing tracers . @xmath50rn ( and its products e.g. , @xmath141po ) can be detected remotely of course and are unique in terms of their mapping potential , while being a minor fraction of escaping gas , presumably . @xmath4ar is a major mass constituent of the atmosphere and unlike @xmath17he is not confused with the solar wind . both @xmath50rn and @xmath4ar will favor kreep terrane in the western maria , presumably . if outgassing arises in the deep interior , one can not neglect indications that at one time this was dominated by volcanic , molecular gas . particular among these is water vapor , passing its triple point temperature in rising through the regolith . given a high enough concentration , therefore , one should expect the production of water ice . the conditions under the regolith , particularly near the lunar poles , are favorable for such ice to persist even over geological time interval . it is possible that ice generated there when outgassing was more active still remains . we further point out that the plausible chemical interaction of such molecular gases with the regolith is the production of cement - like compounds that might radically alter the diffusivity of the regolith . given the temperature evolution of the regolith , this non - diffusive layer would isolate the volatile outflow from the vacuum . the question remains how we will detect such molecular outgassing effects , given their largely covert nature ; this is greatly complicated by possible anthropogenic contamination in the future . among molecular gases , sulfur e.g. , so@xmath7 is the predominate volatile detected in deep - interior fire fountain glasses , a key factor in deciding what to pursue as a volcanic tracer . furthermore , essential no liquid or hybrid rocket propellant candidate contain sulfur , and the only sulfuric solid propellants are fairly outdated e.g. , black powder and zn - s . nasa and hopefully other space agencies have no plans to use these on lunar missions . despite the hypotheses and methods outlined above , there is great doubt regarding the nature of lunar outgassing . water is of obvious and diverse interest , and co@xmath7 and co , while missing as apparent constituents , are interesting as drivers for fire fountain eruption . plausibly the only way to study these components reliably is before the new introduction of large spacecraft into the lunar environment . given uncertainty of how these gases and so@xmath7 might interact with the regolith , this early study appears paramount . significantly , many years to come monitoring for optical transients will be best done from earth s surface , even considering the important contributions that will be made by lunar spacecraft probes in the near future . however , these spacecraft will be very useful in evaluating the nature of transient events in synergy with ground - based monitoring . given the likely behavior of outgassing events , it is unclear that in - situ efforts alone will necessarily isolate their sources within significant winnowing of the field by remote sensing . early placement of capable mass spectrometers of the lunar surface , however , might prove very useful in refining our knowledge of outgassing composition , in particular a dominant component that could be used as a tracer to monitor outgassing activity with more simple detectors . this should take place before significant atmospheric pollution by large spacecraft , which will produce many candidate tracer gases in their exhaust . finally , as we edit this paper s final version , several works have become available indicating confirmed lunar regolith hydration signals ( pieters et al . 2009 , clark 2009 , sunshine et al . 2009 ) in the 3@xmath32 m band , and we comments about these here . these show a strong increase in hydration signal towards the poles , as predicted in 4 . to our knowledge our model is uniquely consistent with this and the general hydration signal strength , in places @xmath49700 ppm by mass ( also with vilas et al . 1999 , 2008 ) . unfortunately the moon mineralogy mapper ( m@xmath20 : pieters et al . 2006 ) , in finding this signal , but not completely mapping it , provided tentative indication of its large scale distribution varying over a lunation . this variation can be studied from earth with a simple near / mid ir camera ( insb or red - extended hgcdte ) with on- and off - band filters for ir hydration bands ( or 0.7@xmath32 m : vilas et al . 1999 ) . to complete this valuable work , along with other instruments needing a lunar polar orbiter ( alpha - particle spectrometer , ground - penetrating radar , mass spectrometers , etc . ) , an instrument similar to m@xmath20 should probably fly again before human lunar missions . note that the same type and level of signal was detected by _ luna 24 _ ( akhmanova et al . 1978 ) , and these authors believed it not due to terrestrial contamination . they detected increasing hydration with depth into the regolith , a likely circumstance in our model . this core sample reached 2 meters depth , several times deeper than epithermal neutrons e.g. , seen on _ lunar prospector _ or _ lro _ , and corresponding to an impact gardening over @xmath12 g.y . such a gradient arises naturally from seepage of water vapor , but water and/or hydroxyl from solar wind proton implantation may not explain the concentration of 3@xmath32 m signal to the poles and the _ luna 24 _ hydration depth profile . this offers a challenge for this model , or for water delivered by comets and/or meteoroids . in a separate paper we will review further evidence supporting endogenous origin of lunar hydration . we would much like to thank alan binder and james applegate , as well as daniel savin , daniel austin , ed spiegel and the other members of aeolus ( `` atmosphere as seen from , earth , orbit and lunar orbit '' ) for helpful discussion . this research was supported in part by nasa ( 07-past07 - 0028 and 07-laser07 - 0005 ) , the national geographic society ( cre grant 8304 - 07 ) , and columbia university . given constraints on imaging from earth , we consider imaging monitors closer to the moon . we propose no special - purpose missions , but detectors that could ride on other platforms e.g. , does lunar exploration require communications with line - of - sight access to all points on the moon s surface ( except within deep craters , etc . ) ? this might also serve for comprehensive imaging monitoring . a minimal full network has a tetrahedral geometry with points @xmath160000 km above the surface : a single platform at earth - moon lagrange point l1 , covering most of the near side , and three points in wide halo orbits around l2 for the far side plus limb seen from earth . proposals exist for an l1 orbital transfer facility ( lo 2004 , ross 2006 ) . no single satellite sees the entire far side , especially since farside radio astronomy might restrict low - frequency transmission i.e. , lasers only . one l2 satellite covers @xmath10797% of the far side ( subtending @xmath184 , selenocentrically ) ; full coverage ( plus some redundancy ) requires three satellites ( plus l1 ) . with this configuration , the farthest point from a satellite will be typically @xmath185 ( selenocentrically ) , foreshortened by @xmath186 times . such an imaging monitor might be ambitious ; to achieve 100 m fwhm at the lunar sub - satellite point requires @xmath14 gpixels , aperture @xmath187 m , and field - of - view 3@xmath188 . each such monitor on an existing platform will cost perhaps $ 100 m . in the meantime , we should accomplish what we can from the ground . martin , r.t . , winkler , j.l . , johnson s.w . & carrier , iii , w.d . 1973 , `` measurement of conductance of apollo 12 lunar simulant taken in the molecular flow range for helium , argon , and krypton gases . '' unpublished report quoted in carrier et al . ( 1991 ) . map of tlp activity & imaging monitor , entire nearside , & optical & comprehensive schedulability ; more & limited resolution + & @xmath12 km resolution . & & sensitive than human eye & + & & & & + polarimetric study of & compare reflectivity in two & optical & easy to schedule ; further constrains & requires use of two monitors + dust & monitors with perpendicular & & dust behavior & + & polarizers & & & + & & & & + changes in small , & adaptive optic imaging , @xmath1100 m&0.95@xmath32 m , etc.&``on demand '' given good conditions&undemonstrated , depends on + active areas & resolution & & & seeing ; covers @xmath150 km + & & & & diameter maximum + & & & & + & `` lucky imaging , '' @xmath1200 m & 0.95@xmath32 m , etc.&on demand given good conditions & low duty cycle , depends on + & resolution & & & seeing + & & & & + & _ hubble space telescope _ , @xmath1100 m&0.95@xmath32 m , etc.&on demand given advanced notice & limited availability ; low + & resolution & & & efficiency + & & & & + & _ clementine / lro / chandrayaan-1 _ & 0.95@xmath32 m , etc.&existing or planned survey & limited epochs ; low flexibility + & imaging , @xmath1100 m resolution & & & + & & & & + & _ lro / kaguya / chang-1 _ imaging , & 0.95@xmath32 m , etc.&existing or planned survey & limited epochs ; low flexibility + & higher resolution & & & + & & & & + tlp spectrum & scanning spectrometer map , plus & nir , & may be best method to find & requires alert from tlp image + & spectra taken during tlp event & optical & composition & tlp mechanism & monitor ; limited to long events + & & & & + & & & & + regolith hydration & nir hydration bands seen before vs.&2.9 , 3.4@xmath32m&directly probe regolith / water & requires alert from monitor + measurement & after tlp in nir imaging & & chemistry ; may detect water & and flexible scheduling + & & & & + & scanning spectrometer map , then & 2.9 , 3.4@xmath32m&directly probe regolith / water & requires alert from monitor + & spectra taken soon after tlp & & chemistry ; may detect water & and flexible scheduling + & & & & + relationship between & simultaneous monitoring : @xmath50rn @xmath69&@xmath50rn @xmath69 & & refute / confirm tlp / outgassing&optical monitor only covers + tlps & outgassing & particles by @xmath149 & optical tlps & optical & correlation ; find outgassing loci & nearside ; more monitors better + & & & & + subsurface water ice & penetrating radar from earth & @xmath1430 mhz&directly find subsurface ice with&ice signal is easily confused + & & & existing technique & with others + & & & & + & penetrating radar from lunar orbit&@xmath1300 mhz&better resolution ; deeper than&ice signal is easily confused ; + & & & neutron or gamma probes & more expensive + & & & & + & surface radar from lunar orbit & @xmath1561 ghz & better resolution ; study tlp site & redundant with high resolution + & & & surface changes & imaging ? + & & & & + high resolution tlp & imagers at / near l1 , l2 points & optical & map tlps with greater resolution & & expensive , but could piggyback + activity map & covering entire moon , at 100 m & & sensitivity , entire moon & on communications network + & resolution & & & + & & & & + comprehensive @xmath50rn @xmath69 & two @xmath50rn @xmath69 detectors in polar&@xmath50rn @xmath69&map outgassing events at full&expensive ; even better response + particle map & orbits 90@xmath8 apart in longitude & & sensitivity & with 4 detectors + & & & & + comprehensive map of & two mass spectrometers in adjacent & ions & & map outgassing events & find & expensive ; even better with more + outgas components & polar orbits & neutrals & composition & detectors +

we follow paper i with predictions of how gas leaking through the lunar surface could influence the regolith , as might be observed via optical transient lunar phenomena ( tlps ) and related effects . we touch on several processes , but concentrate on low and high flow rate extremes , perhaps the most likely . we model explosive outgassing for the smallest gas overpressure at the regolith base that releases the regolith plug above it . this disturbance s timescale and affected area are consistent with observed tlps ; we also discuss other effects . for slow flow , escape through the regolith is prolonged by low diffusivity . water , found recently in deep magma samples , is unique among candidate volatiles , capable of freezing between the regolith base and surface , especially near the lunar poles . for major outgassing sites , we consider the possible accumulation of water ice . over geological time ice accumulation can evolve downward through the regolith . depending on gases additional to water , regolith diffusivity might be suppressed chemically , blocking seepage and forcing the ice zone to expand to larger areas , up to km@xmath0 scales , again , particularly at high latitudes . we propose an empirical path forward , wherein current and forthcoming technologies provide controlled , sensitive probes of outgassing . the optical transient / outgassing connection , addressed via earth - based remote sensing , suggests imaging and/or spectroscopy , but aspects of lunar outgassing might be more covert , as indicated above . tlps betray some outgassing , but does outgassing necessarily produces tlps ? we also suggest more intrusive techniques from radar to in - situ probes . understanding lunar volatiles seems promising in terms of resource exploitation for human exploration of the moon and beyond , and offers interesting scientific goals in its own right . many of these approaches should be practiced in a pristine lunar atmosphere , before significant confusing signals likely to be produced upon humans returning to the moon . 6.5 in 8.5 in 0.0 in 0.0 in

the past few years have been a watershed in our ability to directly observe galaxy evolution . deep field surveys such as the canada - france redshift survey ( cfrs lilly 1995 ) and color - selected field samples such as that of steidel ( 1996 , 1999 ) have provided critical information on the evolution of field galaxies . madau ( 1996 ) integrated the results at @xmath8 into a coherent picture of the star formation history of the universe , suggesting that the global star formation rate peaked between @xmath9 and @xmath10 . since then , recognition of the importance of both dust and cosmic variance has changed the steep decline in the cosmic star formation rate inferred at @xmath11 into a flat plateau for @xmath12 ( steidel 1999 ) . cowie ( 1999 ) also show a more gradual rise at @xmath13 than initially inferred by the cfrs . there remain four substantial caveats regarding these findings . first , the number of spectroscopically measured redshifts between @xmath14 and @xmath10 is small . second , since the uv dropout technique used to identify the @xmath15 population requires them to be uv bright , it is possible that a substantial amount of star - forming activity in dusty systems has been overlooked . third , redshift surveys from which cosmic star formation rates are measured must be of sufficient depth and wavelength coverage that star formation indicators ( , @xmath16(2800 ) ) can be measured with limited extrapolation over wide redshift intervals . finally , small area surveys , such as the hdf , are vulnerable to perturbations from large scale structure . infrared - selected surveys provide a powerful tool for addressing these issues ( see dickinson , these proceedings ) . among the benefits , infrared @xmath17-corrections are small and relatively independent of galaxy type , age , and redshift . since the long - wavelength light of galaxies is dominated by lower mass stars rather than short - lived high - mass stars , infrared luminosities track galaxy mass , thereby providing a more direct comparison to theories of galaxy formation without relying on the poorly - understood physics of star formation . infrared light is also less vulnerable to dust absorption . on the negative side , since spectroscopy is primarily performed at optical wavelengths , infrared - selected samples are challenging to follow - up . also , since evolved stars become important contributors to the long - wavelength flux of a galaxy , poorly - understood phases of stellar evolution can make interpretation of broad - band colors ambiguous ( spinrad 1997 ) . finally , since infrared - surveys do not select for young stars , they are suboptimal for studying the cosmic star - formation history , though they provide a natural basis for studying the mass - aggregation history . we present the spices survey ( eisenhardt 2001 , in prep . ) , a deep @xmath2 imaging and spectroscopic survey covering over 100 arcmin@xmath1 spread over four fields . table [ tab1 ] lists the vega magnitude 3@xmath18 depths in 3 diameter apertures for the imaging . the relatively large area mitigates the effects of large - scale structure while the @xmath19-band depth is more than sufficient to detect @xmath20 galaxies to @xmath21 . the area and depth are a significant improvement over several recent surveys ( cowie 1996 ) , but are modest compared to several programs currently in production mode ( cimatti , these proceedings ; mccarthy , these proceedings ) . an important strength of spices is the spectroscopic program : we currently have 626 spectroscopic redshifts of @xmath22 sources selected from the sample , approximately one - third of the complete @xmath22 sample ( see figure 1 ) . these spectroscopic redshifts are being used to directly construct an eigenbasis of galaxy spectral energy distributions with which to determine photometric redshifts for the complete sample ( see budavari 2000 ) . wu ( these proceedings ) discusses _ hst _ imaging of one of the spices fields . here we discuss two initial results from the survey . .depth of spices imaging ( vega magnitudes ) [ cols="<,^,^,^,^,^,^ " , ] [ tab1 ] extremely red objects ( ero s ) are an intriguing class of extragalactic object , likely associated with @xmath23 galaxies ( cimatti , these proceedings ) . we find that the surface density of these sources is elevated in the spices fields relative to some of the surface densities reported previously in the literature . for @xmath24 , we find a surface density of 1.4 ero s arcmin@xmath25 , with a range in this value of 1.3 to 1.9 across the four fields . for the same magnitude range and color criterion , barger ( 2000 ) find a surface density of @xmath26 ero s arcmin@xmath25 over a field of view of 61.8 arcmin@xmath1 while mccracken ( 2000 ) find a surface density of ero s in the herschel deep field of @xmath27 arcmin@xmath25 over a 47.2 arcmin@xmath1 field . similarly , if we consider ero s defined as @xmath28 sources with @xmath29 , the spices fields have 0.13 ero s arcmin@xmath25 with a range of @xmath30 ero s arcmin@xmath25 across the four fields . using the same definition , the cadis survey finds @xmath31 ero s arcmin@xmath25 across a 154 arcmin@xmath1 field ( thompson 1999 ) while daddi ( 2000 ) find 0.07 ero s arcmin@xmath25 across a 447.5 arcmin@xmath1 field with strong clustering reported . what is the source of this discrepancy ? one possibility is that the depth and area of the spices imaging are significantly improved over many of the surveys mentioned above : @xmath32 is a 10@xmath18 detection in the spices survey . another possibility is large scale structure . though the spices fields cover @xmath33 arcmin@xmath1 , larger than several of the above surveys , fluctuations in the ero surface density on these scales have been reported by more recent larger area deep infrared surveys ( daddi 2000 ; cimatti , these proceedings ; mccarthy , these proceedings ) . indeed , one of the spices fields ( the lynx field : @xmath34 ) has a higher surface density of red objects than the other three fields . keck / lris spectroscopy and has subsequently identified many of these red sources with galaxies in two x - ray emitting clusters at @xmath35 ( stanford 1997 ; rosati 1998 ) . the @xmath19-band luminosity function ( klf ) at @xmath9 offers a powerful constraint on theories of galaxy formation . since the @xmath19-band light tracks mass better than ultraviolet / optical light , the klf is more directly comparable to theories of the collapse and merging of galaxies . kauffmann & charlot ( kc98 ; 1998 ) show that pure luminosity evolution ( ple ) models , models in which galaxies collapse monolithically at high redshift with little subsequent merging activity , predict that many massive galaxies exist at @xmath9 : @xmath36% of an infrared - selected field galaxy sample with @xmath37 should be at @xmath7 . alternatively , their hierarchical model predicts only @xmath38% of @xmath37 field galaxies should be at @xmath7 . ignoring the spices field with the @xmath35 clusters and another field with very limited spectroscopy , we conservatively find that @xmath39% of @xmath37 spices sources are at @xmath7 . this assumes that @xmath40% of @xmath28 , @xmath41 ( red ) sources are at @xmath7 , as our spectroscopic program shows thus far , and we only count those @xmath37 , @xmath42 ( blue ) sources already spectroscopically confirmed to be at @xmath7 . early photometric redshift analysis on these fields suggests a value @xmath3% of the @xmath37 being at @xmath7 . these numbers show that neither ple nor the kc98 hierarchical model correctly predicts the @xmath43 klf , implying that substantial merging occurs at @xmath7 . the identification of two clusters at @xmath35 and one cluster at @xmath45 in the lynx spices fields has led to a deep , 190 ksec _ chandra _ map of the field . analysis of the diffuse high- and low - redshift cluster x - ray emission are discussed in stanford ( 2001 , submitted ) and holden ( 2001 , in prep . ) , respectively . stern ( 2001 , in prep . ) discusses x - ray background ( xrb ) results from this data set . we confirm results of recently published _ chandra _ studies ( giacconi 2001 ) : most of the @xmath46 kev xrb is resolved into discrete sources ; the fainter soft - band sources have harder x - ray spectra , providing a coherent solution to the long - standing ` spectral paradox ' ; and @xmath47% of the sources have optical / near - infrared identifications in deep ground - based imaging . a preliminary spectroscopic program shows a mix of obvious agn , apparently normal galaxies , and , perhaps surprisingly , several x - ray emitting stars , some with hard x - ray spectra . we are also targeting the spices fields with very deep imaging in @xmath48 to identify high - redshift sources using the lyman break technique . this work has led to the discovery of a faint quasar at @xmath49 ( stern 2000 ) and several high - redshift galaxies out to @xmath50 . strong emission - line galaxies have also been identified serendipitously during the spices spectroscopic campaign , the highest redshift source being a likely @xmath51 ly@xmath52 emitter with @xmath53 erg cm@xmath25 s@xmath55 . we present first results from the spices survey , an infrared - selected photometric and spectroscopic survey . we find an elevated surface density of ero s compared to several recent deep , infrared surveys , likely due to fluctuations in that quantity from large scale structure at moderate redshifts . perhaps relatedly , we also find a large fraction of infrared - bright ( @xmath28 ) galaxies residing at @xmath7 . a good measure of this quantity provides a powerful constraint on models of galaxy formation . budavri , t. ( 2000 ) , , 120 , 1588 . barger , a. ( 1999 ) , , 117 , 102 . cowie , l. ( 1996 ) , , 112 , 839 . cowie , l. , songaila , a. , & barger , a. ( 1999 ) , , 118 , 603 . daddi , e. ( 2000 ) , , 361 , 535 . giacconi , r. ( 2001 ) , , in press . kauffmann , g. & charlot , s. ( 1998 ) , , 297 , 23 . lilly , s. ( 1995 ) , , 455 , 108 . madau , p. ( 1996 ) , , 283 , 1388 . mccracken , h. j. ( 2000 ) , , 311 , 707 . rosati , p. ( 1999 ) , , 118 , 76 . spinrad , h. ( 1997 ) , , 484 , 581 . stanford , s.a . ( 1997 ) , , 114 , 2332 . steidel , c.c . ( 1996 ) , , 112 , 352 . steidel , c.c . ( 1999 ) , , 519 , 1 . stern , d. et al . ( 2000 ) , , 533 , l75 thompson , d. et al . ( 2000 ) , , 523 , 100

we present first results from spices , the spectroscopic , photometric , infrared - chosen extragalactic survey . spices is comprised of four @xmath0 arcmin@xmath1 high galactic latitude fields with deep @xmath2 imaging reaching depths of @xmath3 mag ( ab ) in the optical and @xmath4 mag ( ab ) in the near - infrared . to date we have 626 spectroscopic redshifts for infrared - selected spices sources with @xmath5 ( vega ) . the project is poised to address galaxy formation and evolution to redshift @xmath6 . we discuss initial results from the survey , including the surface density of extremely red objects and the fraction of infrared sources at @xmath7 . one of the spices fields has been the target of a deep 190 ksec _ chandra _ exposure ; we discuss initial results from analysis of that data set . finally , we briefly discuss a successful campaign to identify high - redshift sources in the spices fields .

as one of the most compelling evidences for new physics beyond the standard model ( sm ) , the cosmic dark matter ( dm ) has been widely studied in particle physics @xcite . recently , the cdms - ii collaboration observed three events which can be explained by a light dm with mass about 8.6 gev and a spin - independent dm - nucleon scattering cross section of about @xmath1 pb @xcite . the existence of such a light dm seems to be corroborated by other direct detections such as the cogent @xcite , cresst @xcite and dama / libra @xcite . moreover , a light dm is also hinted by fermi - lat , a satellite - based dm indirect detection experiment @xcite . recent analysis of the fermi - lat data exhibits peaks in the gamma - ray spectrum at energies around 1 - 10 gev , which could be interpreted in terms of the annihilation of a dm with mass low than about 60 gev into leptons or bottom quarks @xcite . about these experimental results , it should be noted that they are not completely consistent with each other , and more seriously , they conflict with the xenon data @xcite and the latest lux result @xcite . so the issue of light dm leaves unresolved and will be a focal point both experimentally and theoretically . on the experimental side , many experiments like lux , xenon , cdms and cdex @xcite will continue their searches , while on the theoretical side we need to examine if such a light dark matter can naturally be predicted in popular new physics theories such as low energy supersymmetry ( susy ) . previous studies @xcite showed that , in the framework of the next - to - minimal supersymmetric standard model ( nmssm ) @xcite , a light neutralino dm around 10 gev is allowed by the collider constraints and dm relic density ( in contrast such a light dm is not easy to obtain in the mssm @xcite or cmssm @xcite ) . in nmssm , due to the presence of a singlet superfield @xmath2 , we have five neutralinos , three cp - even higgs bosons ( @xmath3 ) and two cp - odd higgs bosons ( @xmath4 ) @xcite . the mass eigenstates of neutralinos are the mixture of the neutral singlino ( @xmath5 ) , bino ( @xmath6 ) , wino ( @xmath7 ) and higgsinos ( @xmath8 , @xmath9 ) ; while the cp - even ( odd ) higgs mass eigenstates are the mixture of the real ( imaginary ) part of the singlet scalar @xmath10 and the cp - even ( odd ) mssm doublet higgs fields . an important feature of the nmssm is that the lightest cp - even ( odd ) higgs boson @xmath11 can be singlet - like and very light , and the lightest neutralino ( @xmath12 ) can be singlino - like and also very light . as a result , the spin - independent neutralino - nucleon scattering cross section can be enhanced to reach the cdms - ii value by the @xmath13channel mediation of a light @xmath14 @xcite . meanwhile , the dm relic density can be consistent with the measured value either through the @xmath15channel resonance effect of @xmath11 in dm annihilation or through the annihilation into a pair of light @xmath14 or @xmath16 @xcite . note that such a light dm in the nmssm should be re - examined because the latest lhc data may give severe constraints . due to the presence of a light dm and concurrently a light @xmath16 or @xmath14 , the sm - like higgs boson ( @xmath17 ) can have new decays @xmath18 and @xmath19 @xcite . as analyzed in @xcite , such decays may be subject to stringent constraints from the current lhc higgs data @xcite . besides , since a certain amount of higgsino component in @xmath12 is needed to strengthen the coupling of @xmath20 ( or @xmath21 ) which is necessary for the dm annihilation , the higgsino - dominated neutralinos and the chargino @xmath22 are generally not very heavy and will be constrained by the searches for events with three leptons and missing transverse momentum ( @xmath23 + /@xmath24 ) at 8 tev lhc @xcite . in this work , we consider these latest lhc data and examine the status of a light dm in the nmssm . we note that a recent study @xcite tried to explain the cdms - ii results in terms of a light dm in the nmssm . compared to @xcite which only studied three representative benchmark points , we perform a numerical scan under various experimental constraints and display the allowed parameter space in comparison with the the direct detection results of cogent , cdms - ii and lux . we also perform a global fit of the higgs data using the package @xcite , in which we further consider the latest lhc results of higgs invisible decay from the channel @xmath25 @xcite . moreover , we consider the constraints from the searches for events with @xmath23 + /@xmath24 signal at 8 tev lhc @xcite . the paper is organized as follows . in sec.ii we list the experimental constraints and describe our scan . in sect.iii we present our results and perform detailed analysis . finally , we draw our conclusions in sec.iv . in order to reduce the number of free parameters in our scan over the nmssm parameter space , we make some assumptions on the parameters that do not influence dm properties significantly . explicitly speaking , we fix gluino mass and all the soft mass parameters in squark sector at 2 tev , and those in slepton sector at 300 gev . we also assume the soft trilinear couplings @xmath26 and let them vary to tune the higgs mass . moreover , in order to predict a bino - like light dm and also to avoid the constraints from @xmath27 invisible decay @xcite , we abandon the gut relation between @xmath28 and @xmath29 . the free parameters are then @xmath30 in the higgs sector , the gaugino and higgsino mass parameters @xmath31 and @xmath32 , and the soft trilinear couplings of the third generation squarks @xmath33 . in this work , we define all these parameters at @xmath34 scale and adopt the markov chain monte carlo ( mcmc ) method to scan the following parameter ranges using @xmath35 @xcite : @xmath36 note here that the ranges of @xmath37 and @xmath38 are motivated to avoid landau pole , generally corresponding to the requirement of @xmath39 . this has been encoded in @xmath35 including the consideration of the interplay between @xmath37 and @xmath38 in the renormalization group running . a relatively small @xmath32 is chosen to avoid strong cancelation in getting the @xmath27 boson mass @xcite , and as we will see below , the upper bound of @xmath40 for @xmath32 here suffices our study and does not affect our main conclusions . also note that we artificially impose a lower bound of 320 gev for @xmath29 . this is motivated by the fact that @xmath29 in our study is not an important parameter , and that as required by the @xmath23 + /@xmath24 constraint @xmath29 should be larger than about 320 gev in the simplified model discussed in @xcite ( also see the constraint ( viii ) discussed below ) . the relevant @xmath41 function for the mcmc scan is build to guarantee the dm relic density and the sm - like higgs boson mass around their measured values . in our discussion , we consider the samples surviving the following constraints : * @xmath42 and @xmath43 . * _ the constraints from b - physics_. the light cp - even / odd higgs bosons can significantly affect the b - physics observables . especially , the precise measurements of radiative decays @xmath44 @xcite , @xmath45 @xcite and @xmath46 @xcite can give stringent constraints . so we require the samples to satisfy these b - physics bounds at 2@xmath47 level . * _ dm relic density_. as the sole dark matter candidate , the lightest neutralino @xmath12 is required to produce the correct thermal relic density . we require the neutralino relic density to be in the @xmath48 range of the planck and wmap 9-year data , @xmath49 , where a 10% theoretical uncertainty is included @xcite . * _ muon g-2_. we require nmssm to explain the muon anomalous magnetic moment data @xmath50 @xcite at @xmath48 level . * _ the absence of landau pole_. we impose this constraint using @xmath35 @xcite , where the interplay of @xmath37 and @xmath38 in the renormalization group running has been considered . * _ lep searches for susy_. for the lep experiments , the strongest constraints come from the chargino mass and the invisible @xmath27 decay . we require @xmath51 and the non - sm invisible decay width of @xmath52 to be smaller than 1.71 mev , which is consistent with the precision electroweak measurement result @xmath53 mev at @xmath54 confidence level @xcite . * _ higgs data_. firstly , we consider the exclusion limits of the lep , tevatron and lhc in higgs searches with the package @xcite . this package also takes into account the results of the lhc searches for non - sm higgs bosons , such as @xmath55 and @xmath56 @xcite . secondly , noticing that a light @xmath14 ( or @xmath16 ) may induce the distinguished signal @xmath57 , we consider the limitation of the @xmath58 signal on the parameter space using the latest cms results @xcite . finally , since a large invisible branching ratio of the higgs may be predicted in the light dm case , we perform a global fit of the higgs data using the package @xcite , where the systematics and correlations for the signal rate predictions , luminosity and higgs mass predictions are taken into account . in our fit , we further consider the latest lhc results of higgs invisible decay from the channel @xmath25 @xcite . we require our samples to be consistent with the higgs data at @xmath48 level , which corresponds to @xmath59 with @xmath41 obtained with the and @xmath60 denoting the minimum value of @xmath41 for the surviving samples in our scan . * _ lhc searches for susy_. based on the 20 fb@xmath61 data collected at the 8 tev run , the atlas and cms collaborations performed a search for the @xmath62 production with @xmath23 + /@xmath24 signal in a simplified model , where both @xmath63 and @xmath64 are assumed to be wino - like with @xmath65 , and a 95% c.l . upper limit on @xmath66 was obtained on the @xmath67 plane @xcite . + in this work , in order to implement this constraint we perform an analysis similar to @xcite with the code checkmate @xcite for each sample surviving the constraints ( i ) - ( vii ) . we consider the contributions from all @xmath68 ( @xmath69 and @xmath70 ) associated production processes to the signal , and calculate the production rates and the branching ratios with the code prospino2 @xcite and nmsdecay @xcite , respectively . our analysis indicates that this constraint can exclude effectively those samples with small values of @xmath32 below 115 gev , and also some samples with moderate @xmath32 in the range from 115 gev to 200 gev . nevertheless , compared to the results without considering this constraint , our conclusions do not change much such as the upper bounds on @xmath71 presented below . in order to study the implication of the dm direct detection experiments on the nmssm , we also calculate the dm spin - independent elastic scattering cross section off nucleon with the formulae used in our previous work @xcite . in getting the cross section , we set the parameter of the strange quark content in the nucleon as @xmath72 . in the rest of this work , we categorize the dm by its component , i.e. either bino - like or singlino - like , in presenting our results . since the interactions of the neutralinos with the higgs bosons come from the following lagrangian @xmath73 where the fields @xmath74 , @xmath75 and @xmath76 denote the neutral scalar parts of the higgs superfields @xmath2 , @xmath77 and @xmath78 , respectively , one can infer that if the dm is bino - like , the coupling strength of the @xmath79 interaction is mainly determined by the higgsino - component in @xmath80 , or more basically by the value of @xmath32 . to be more specific , if @xmath81 is sm - like , the coupling strength is mainly determined by the first two terms in the second row of eq.[interaction ] , while if @xmath81 is singlet - dominated , the coupling of @xmath79 is mainly determined by the first term of eq.[interaction ] . however , if the dm is singlino - like , the coupling strength is fundamentally determined by the parameters @xmath37 and @xmath38 and a low @xmath32 value may be helpful to enhance the coupling . in this work , we are also interested in the couplings of the sm - like higgs to light singlet - like scalars @xmath14 and @xmath16 . these couplings are mainly determined by the following terms in the higgs potential @xcite @xmath82 this equation indicates that , if @xmath37 and @xmath38 approach zero , the couplings @xmath83 can not be very large ; while if both of them have a moderate value , accidental cancelation is very essential to suppress the couplings . in fig.[fig1 ] we project the samples surviving the above constraints on the plane of neutralino dark matter mass versus spin - independent neutralino - nucleon scattering cross section . about this figure , we want to emphasize two points . the first one is that some of the experimental constraints , such as the dm relic density and the higgs data , play an important role in limiting the parameter space of the nmssm . so in the following , we pay special attention to investigate how the samples in fig.[fig1 ] survive these constraints . the other one is that the various experimental constraints will cut into the parameter space and the interplay among them is very complicated . as a result , the sample distributions on the @xmath84 plane might be very wired . the strategy of analyzing this figure is to get a general picture of the current status of light dm confronting the direct detection results and then focus on some interesting regions . as we will discuss later , we will mainly focus on those samples that either can explain the cdms - ii results or can survive the first lux exclusion . we will not consider the up - right region ( @xmath85 ) in fig.[fig1 ] since it is not experimentally hinted . after carefully analyzing our results , we have the following observations from fig.[fig1 ] : 1 . in the nmssm , dm as light as @xmath86 is still allowed by the current higgs data . both the bino and singlino - like dm are capable of explaining the results of cdms - ii and cogent , or surviving the current lux results and future lux exclusion limits . 2 . as pointed out in @xcite , light dm in the nmssm may annihilate in the early universe through @xmath74-channel resonance effect of some mediators or into light higgs scalar pair to get a correct relic density . we checked that , for @xmath87 , singlino - like dm annihilated in the early universe mainly through the @xmath74-channel resonance effect of @xmath88 for the most case ; while bino - like dm might annihilate either through the resonance effect or into @xmath14 ( @xmath16 ) pair . we will discuss this issue in more detail later . + in fact , the long thick band of grey samples ( for bino - like dm ) around @xmath89 exactly corresponds to the resonance case , and samples along this band are characterized by @xmath90 with @xmath91 denoting the mediator mass . for @xmath92 and @xmath93 , the mediator is @xmath27 boson and the sm - like higgs boson , respectively , while in other cases the mediator is either @xmath14 or @xmath16 . these conclusions can also apply to the singlino - like dm ( see fig.[fig2 ] ) . 3 . for samples with @xmath94 , generally @xmath14 needs to be lighter than about @xmath95 to push up the scattering rate . for the bino - like dm with mass varying from @xmath96 to @xmath97 , such a light @xmath14 is difficult to obtain after considering the constraint from the relic density ( see fig.[fig2 ] ) . while in the @xmath27 ( @xmath17 ) resonance region , the relic density has rather weak limitation on @xmath14 properties . in this case , @xmath14 may be as light as several gev so that the scattering rate is rather large , or the coupling @xmath98 may be greatly reduced to result in a relatively small @xmath99 . 4 . for bino - like dm , generally it is not easy to obtain samples with @xmath100 . this is because the @xmath101 interaction is still sizable even after considering the various constraints ( see discussions on fig.[fig3 ] ) , and in this case , the @xmath17-mediated contribution to the dm - nucleon scattering is important . however , in the extreme case when the bino - like dm is close to about @xmath102 , due to the lower bound of @xmath32 , the higgsino component in the dm will get further reduced and result in an even smaller @xmath103 . 5 . when focusing on the xenon and lux experiments , the bino - like and singlino - like dm exhibit quite different behaviors . the first lux-300 kg result can exclude a large part of the allowed parameter space , but still leaves both the bino - like and singlino - like light dm viable . the future xenon-1 t and lux-7.2ton results can cut further deeply into the parameter space . especially , they limit tightly the bino - like dm case and constrain most of the bino - like dm mass to be lower than about 17 gev and 12 gev , respectively , while the singlino - like dm can still survive leisurely . 6 . for bino - like dm samples there is a gap in the right half part of the cdms - ii @xmath48 region . this is due to the tension between the lhc higgs data and the constraint from @xmath104 . as discussed in @xcite ( and see table i ) , the cdms - ii favored samples in bino - like dm scenario usually require a moderate @xmath37 along with a moderate @xmath38 to achieve the accidental cancelation in @xmath105 so that the sm - like higgs decay to @xmath14 or @xmath16 pair is suppressed . while on the other hand , this may increase the effective coupling of @xmath14 to down - type fermions which is proportional to @xmath106 with @xmath107 $ ] @xcite , and receive constraint from the measurement of @xmath104 . we checked that most of the excluded bino - like dm samples in the gap have a relatively large @xmath37 , while the singlino - like dm samples usually correspond to a small @xmath37 ( see following discussion on table i ) and thus receive less constraint . compared to bino - like dm which is restricted in certain areas on the @xmath84 plane , singlino - like dm can spread nearly to the whole region of the plane . this reflects the fact that singlino - like dm is more adaptable in light dm physics . .the ranges of relevant nmssm input parameters corresponding to part of the samples in fig.1 , which predict a dm lighter than 35 gev and meanwhile can explain the cdms - ii at 2@xmath47 level or survive the lux-300 kg exclusion limit . parameters with the mass dimension are in the unit of gev . [ cols="^ , < , < , < , < " , ] in the following , we concentrate on the samples in fig.1 that can either explain the cdms - ii experiment at @xmath48 level or survive the lux-300 kg exclusion limit . since the results of the cdms - ii and lux experiments are so incompatible , it would be interesting to investigate the difference of these two types of samples . to simply our analysis , we mainly consider the samples predicting a dm lighter than about 35 gev . these samples are not easy to obtain with traditional random scan method when exploring the susy parameter space due to its rather specific particle spectrum , but as we will see below , the underlying physics of these samples are clear and easy to understood . in table [ table1 ] , we list the ranges of relevant nmssm input parameters corresponding to these samples , which are classified by the component of the dm ( i.e. bino - like or singlino - like ) and meanwhile by its scattering cross section off the nucleon ( i.e. can explain the cdms - ii results at @xmath48 level or survive the lux-300 kg exclusion limit ) . from table i , one can learn the following facts : * the survived parameter ranges for lux - safe samples are generally wider than those of cdms - ii preferred samples . this is totally expectable from the experimental data of lux and cdms - ii . on the @xmath84 plane , cdms - ii @xmath48 region is constrained in a relatively narrow range @xmath108 and @xmath109 . to survive the first lux exclusion , however , a properly large @xmath110 for a certain @xmath111 will be enough . @xmath111 can cover the whole range @xmath112 and @xmath99 can vary from @xmath113 to @xmath114 . therefore , compared to cdms - ii region , there will be more freedom for the parameter space to satisfy the lux exclusion . * to obtain a dm lighter than 35 gev , one needs to have @xmath115 for the bino - like dm and @xmath116 for the singlino - like dm . this can be easily understood from the neutralino mass matrix @xcite . @xmath117 where @xmath118 and @xmath119 are gauge couplings , and @xmath120 and @xmath121 are higgs vacuum expectation values . in fact , a simple estimation can be made for singlino - like dm mass . table [ table1 ] shows that @xmath122 is usually at least one order smaller than @xmath37 . assuming @xmath123 and @xmath124 , we will have @xmath125 . * the cdms - ii samples usually have @xmath126 for both bino - like and singlino - like dm . the underlying reason is that a small value of @xmath32 and consequently a sufficient amount of higgsino component in the dm is helpful to increase the coupling strength of the dm with the light higgs bosons . this will in return push up the rate of the dm - nucleon scattering which is required by the cdms - ii results . * more interestingly , we find that for samples in the whole range of @xmath127 , the value of @xmath32 is upper bounded by about @xmath128 and @xmath129 for bino - like and singlino - like dm , respectively . two reasons can account for this . the first one is that in our scan , we required the nmssm to explain the muon anomalous magnetic moment . the parameter @xmath32 influences the contribution of the nmssm to the moment through chargino and neutralino mass , and a large value of @xmath32 will reduce the contribution significantly . another important reason is that , as mentioned above and also discussed below , in order to get a correct dm relic density , a light @xmath14 or @xmath16 must be present . noting that @xmath32 enters explicitly the squared mass of the singlet scalar @xcite , one can infer that too large values of @xmath32 can not be favored to get the desired light scalar masses . + we also want to emphasize that , for the bino - like dm , an upper bound of @xmath32 will result in a lower limit of the higgsino component in the dm and thus a lower bound of the invisible branching ratio for @xmath130 . this can be explicitly seen in the left panel of fig.[fig3 ] below . * for singlino - like dm case , both @xmath37 and @xmath38 are small and especially , @xmath122 is very close to 0 . as indicated by eqs.([interaction],[higgs_potential],[neutralino_mass_matrix ] ) , the couplings of sm - like higgs boson to dm and also to the light higgs scalars @xmath131 will usually be suppressed . this can result in a @xmath99 as low as @xmath132 ( see fig.[fig1 ] ) and also a relatively small rate for the decays @xmath133 ( see fig.[fig3 ] and fig.[fig4 ] ) . while for the bino - like dm case with a moderate value of @xmath37 and @xmath38 , accidental cancelation is very essential to suppress the couplings of @xmath17 to @xmath134 and obtain an allowed higgs signal . as discussed in fig.[fig1 ] , given @xmath135 , at least one light scalar is needed to accelerate the annihilation . in order to illustrate this feature , in fig.[fig2 ] we project the @xmath135 samples of fig.1 which can explain the cdms - ii results at 2@xmath47 level or survive the lux-300 kg exclusion limits on the plane of dm mass versus @xmath136 . red codes represent samples suggested by the cdms - ii experiment and meanwhile satisfying @xmath137 , while cyan ( blue ) codes correspond to samples surviving the lux-300 kg exclusion limits and also satisfying @xmath137 ( @xmath138 ) . note that due to the large scattering cross section favored by the cdms - ii results , a light @xmath14 is needed ( as the t - channel propagator ) and the case @xmath139 is absent . from fig.[fig2 ] we have the following observations : samples in fig.[fig1 ] which can explain the cdms - ii results at 2@xmath47 level or survive the lux-300 kg exclusion limits , projected on the plane of dm mass versus @xmath136 . red codes represent samples suggested by the cdms - ii experiment and meanwhile satisfying @xmath137 , while cyan(blue ) codes correspond to samples surviving the lux-300 kg exclusion limits and also satisfying @xmath137 ( @xmath138 ) . note that due to the large scattering cross section favored by the cdms - ii results , a light @xmath14 is needed ( as the t - channel propagator ) and the case @xmath139 is absent.,width=604 ] 1 . in both bino - like and singlino - like dm scenario , the straight line @xmath140 is very obvious , which corresponds to the s - channel resonance effect of @xmath14 or @xmath16 . however , in the singlino - like scenario with @xmath141 , there are some small regions where the line seems to be not continuous . in fact , this is not the case . we checked that there still exits a scalar ( either @xmath14 or @xmath16 ) with mass around @xmath142 . it is just that this scalar does not correspond to the lightest higgs boson . moreover , for the scalars shown in fig.[fig2 ] , we checked that they are highly singlet - dominated , which agree with previous study in @xcite . 2 . since @xmath14 contributes to the spin - independent dm - nucleon scattering as the t - channel propagator @xcite , a very light @xmath14 is needed to explain the cdms - ii result . for the bino - like dm , the cdms - ii samples are mainly distributed in low @xmath143 region with @xmath110 upper bounded by about 4 gev , while for the singlino - like dm , the corresponding samples spread a larger region in @xmath144 plane . moreover , when focusing on the cdms - ii samples , we checked that if the dm is bino - like , the channel @xmath145 plays the dominant role in contributing to the dm annihilation , while if the dm is singlino - like , the s - channel resonance effect is the main contribution . since the constraint from the lux-300 kg data on the scattering rate is rather weak in the very light dm region , @xmath14 as light as 1 gev is still allowed for @xmath146 . with the increase of dm mass , the constraint becomes much stronger and @xmath14 generally needs to be heavier than about 10 gev for @xmath147 in both scenarios . , but projected on the plane of the invisible branching fractions of the sm - like higgs boson versus dm mass , and extended the dm mass to about 60 gev.,width=604 ] for the sm - like higgs boson , since the decay channel @xmath148 is opened when @xmath149 , one can expect that the higgs data will impose rather tight constraints on this decay rate . in fig.[fig3 ] , we show the samples of fig.[fig2 ] on the plane of @xmath150 versus dm mass and extend the dm mass to about 60 gev . we have the following observations : 1 . the current higgs data still allow for an invisible decay branching ratio as large as @xmath0 at @xmath48 level . the tolerance of such a large invisible branching ratio is owe to the large uncertainties of the current data , especially the fact that atlas and cms data point to two opposite directions in the di - photon rate . obviously , an invisible decay branching ratio reaching @xmath0 may be easily tested at the 14 tev lhc with @xmath151 fb@xmath61 , where a 95% c.l . upper limit on the invisible decay , i.e. @xmath152 , can be imposed @xcite . 2 . in the bino - like dm scenario , due to the necessary higgsino component in the dm required by an efficient dm annihilation rate , the interaction between dm and @xmath17 can be relatively large . as a result , @xmath150 as large as @xmath0 is possible . note that for the cdms - ii samples , @xmath150 is always larger than about @xmath153 . the underlying reason is that , as we mentioned earlier , the channel @xmath145 plays an important role in contributing to the dm annihilation . this requires the strength of the @xmath154 interaction to be sufficiently large , and so is the @xmath155 interaction . also note that since @xmath32 is upper bounded for @xmath156 ( see table [ table1 ] ) , generally there is a lower bound of @xmath157 . 3 . in the singlino - like scenario , since the @xmath158 coupling is determined by @xmath37 and @xmath38 and table [ table1 ] indicates that these two parameters are generally small , @xmath150 is usually suppressed and can reach about @xmath159 in the optimal case . , but showing the branching fraction of decays @xmath160 versus dm mass.,width=604 ] due to the existence of light scalars in light dm scenario , the sm - like higgs may also decay into the lighter scalars , @xmath161 . unlike the @xmath162 coupling , the coupling strengthes of @xmath17 to these scalars are mainly determined by @xmath37 and @xmath38 ( see eq.([higgs_potential ] ) and also note that both @xmath14 and @xmath16 are highly singlet - dominated @xcite ) . consequently , according to table [ table1 ] , the maximum decay rate in the bino - like dm scenario should in principle be larger than that in the singlino - like case . similar to fig.[fig3 ] , we show the total branching fractions of these two decays versus dm mass in fig.[fig4 ] . one can learn that this branching ratio can reach @xmath0 in the bino - like dm scenario , while in the singlino - like case the maximum can only reach about @xmath159 . under current experimental constraints including the latest lhc higgs data and the dark matter relic density , we examined the status of a light nmssm dark matter and confronted it with the direct detection results of cogent , cdms - ii and lux . we have the following observations : ( i ) a dark matter as light as 8 gev is still allowed and its scattering cross section off the nucleon can be large enough to explain the cogent / cdms - ii favored region ; ( ii ) the lux data can exclude a sizable part of the allowed parameter space , but still leaves a light dark matter viable ; ( iii ) the sm - like higgs boson can decay into the light dark matter pair and its branching ratio can reach @xmath0 at @xmath48 level under the current lhc higgs data , which may be covered largely at the 14 tev lhc experiment . we thank nima arkani - hamed , archil kobakhidze , yang zhang and jie ren for helpful discussions . this work was supported by the arc center of excellence for particle physics at the tera - scale , by the national natural science foundation of china ( nnsfc ) under grant no . 10821504 , 11222548 , 11305049 and 11135003 , and also by program for new century excellent talents in university . s. schael _ et al . _ [ aleph and delphi and l3 and opal and sld and lep electroweak working group and sld electroweak group and sld heavy flavour group collaborations ] , phys . rept . * 427 * , 257 ( 2006 ) [ hep - ex/0509008 ] .

in susy , a light dark matter is usually accompanied by light scalars to achieve the correct relic density , which opens new decay channels of the sm - like higgs boson . under current experimental constraints including the latest lhc higgs data and the dark matter relic density , we examine the status of a light neutralino dark matter in the framework of nmssm and confront it with the direct detection results of cogent , cdms - ii and lux . we have the following observations : ( i ) a dark matter as light as 8 gev is still allowed and its scattering cross section off the nucleon can be large enough to explain the cogent / cdms - ii favored region ; ( ii ) the lux data can exclude a sizable part of the allowed parameter space , but still leaves a light dark matter viable ; ( iii ) the sm - like higgs boson can decay into the light dark matter pair with an invisible branching ratio reaching @xmath0 under the current lhc higgs data , which may be tested at the 14 tev lhc experiment .

understanding the formation mechanisms and evolution with cosmic time of galaxies is one of the major goals of observational cosmology . in the current picture of structure formation , dark matter halos build up in a hierarchical fashion controlled by the nature of the dark matter , the power spectrum of density fluctuations , and the parameters of the cosmological model . the assembly of the stellar content of galaxies is governed by much more complicated physics , such as the mechanisms of star formation , gaseous dissipation , the feedback of stellar and central supermassive black hole energetic output on the baryonic material of the galaxies , and mergers . the mean space density of galaxies per unit luminosity , or luminosity function ( lf ) , is one of the most fundamental of all cosmological observables , and it is one of the most basic descriptors of a galaxy population . the shape of the lf retains the imprint of galaxy formation and evolution processes ; the evolution of the lf as a function of cosmic time , galaxy type and environment provides insights into the physical processes that govern the assembly and the following evolution of galaxies . therefore , the lf represents one of the fundamental observational tools to constrain the free parameters of theoretical models . the local ( @xmath6 ) lf has been very well determined from several wide - area , multi - wave band surveys with follow - up spectroscopy ( @xcite ; @xcite ; @xcite ; @xcite ) . at intermediate redshifts ( @xmath16 ) , spectroscopic surveys found a steepening of the faint - end lf with increasing redshift in the global lf , mainly due to the contribution by later type galaxies ( @xcite ; @xcite ) . from the combo-17 survey , @xcite measured the rest - frame optical lf up to @xmath17 , finding that early - type galaxies show a decrease of a factor of @xmath18 in the characteristic density @xmath8 of the lf . the latest type galaxies show a brightening of @xmath19 mag in @xmath20 ( the characteristic magnitude ) and an increase of @xmath21 in @xmath8 in their highest redshift bin in the blue band . further progress in the measurement of the lf at @xmath22 was obtained with the vimos vlt deep survey ( vvds ; @xcite ) and the deep-2 galaxy redshift survey @xcite . from the vvds data , @xcite measured the rest - frame optical lf from @xmath23 to @xmath24 . from the same data set , @xcite performed a similar analysis for different spectral galaxy types , finding a significant steepening of the lf going from early to late types . their results indicate a strong type - dependent evolution of the lf , and identify the latest spectral types as responsible for most of the evolution of the uv - optical lf out to @xmath25 . contrary to low - redshift studies , the selection of high - redshift ( @xmath5 ) galaxies still largely relies on their colors . one of the most efficient ways to select high - redshift galaxies is the lyman drop - out technique , which enabled steidel and collaborators to build large samples of @xmath26 star - forming galaxies ( @xcite , 1999 ) . extensive studies of these optically ( rest - frame ultraviolet ) selected galaxies at @xmath26 ( lyman break galaxies [ lbgs ] ) and at @xmath27 ( bm / bx galaxies ; @xcite ; @xcite ) have shown that they are typically characterized by low extinction , modest ages , stellar masses @xmath28 m@xmath29 , and star formation rates of 10100 m@xmath29yr@xmath30 ( @xcite ; @xcite ; @xcite ) . @xcite recovered the rest - frame @xmath1-band lf of lbgs at @xmath26 from the rest - frame uv lf ( @xcite ; but see also @xcite ) , finding that the lbg lf is characterized by a very steep faint end . lbgs dominate the uv luminosity density at @xmath31 , as well as possibly the global star formation rate density at these redshifts @xcite . however , since the lyman break selection technique requires galaxies to be very bright in the rest - frame uv in order to be selected , it might miss galaxies that are heavily obscured by dust or whose light is dominated by evolved stellar populations . these objects can be selected in the near - infrared ( nir ) , which corresponds to the rest - frame optical out to @xmath26 . using the nir selection criterion @xmath32 ( also suggested by @xcite ) , @xcite and @xcite discovered a new population of high - redshift galaxies ( distant red galaxies [ drgs ] ) that would be largely missed by optically selected surveys . follow - up studies have shown that drgs constitute a heterogeneous population . they are mostly actively forming stars at @xmath33 ( @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ) . however , some show no signs of active star formation and appear to be passively evolving ( @xcite ; @xcite ; @xcite ) , while others seem to host powerful active galactic nuclei ( @xcite ; @xcite ) . compared to lbgs , drgs have systematically older ages and larger masses @xcite , although some overlap between the two exists ( @xcite ; @xcite ) . recently , @xcite have demonstrated that in a mass - selected sample ( @xmath34 @xmath35 ) at @xmath36 , drgs make up 77% in mass , compared to only 17% from lbgs ( see also @xcite ) , implying that the rest - frame optical lf determined by @xcite is incomplete . the global ( i.e. , including all galaxy types ) rest - frame optical lf at @xmath5 can be studied by combining multiwavelength catalogs with photometric redshift information . @xcite studied the @xmath0-band lfs of red and blue galaxies . they find that the @xmath0-band number densities of red and blue galaxies have different evolution , with a strong decrease of the red population at @xmath37 compared to @xmath38 and a corresponding increase of the blue population , in broad agreement with the predictions from their hierarchical cold dark matter models . as all previous works at @xmath5 are based on either very deep photometry but small total survey area ( @xcite ; @xcite ) or larger but still single field surveys ( @xcite ) , their results are strongly affected by field - to - field variations and by low number statistics , especially at the bright end . moreover , @xcite used an i - band selected data set from the fors deep field . the @xmath39 band corresponds to the rest - frame uv at @xmath27 , which means that significant extrapolation is required . in this paper we take advantage of the deep nir musyc survey to measure the rest - frame optical ( @xmath0 , @xmath1 , and @xmath2 band ) lfs of galaxies at @xmath3 . its unique combination of surveyed area and depth allows us to ( 1 ) minimize the effects of field - to - field variations , ( 2 ) better probe the bright end of the lf with good statistics , and ( 3 ) sample the lf down to luminosities @xmath40 mag fainter than the characteristic magnitude . to constrain the faint - end slope of the lf and to increase the statistics , we also made use of the fires and the goods - cdfs surveys , by constructing a composite sample . the large number of galaxies in our composite sample also allows us to measure the lfs of several subsamples of galaxies , such as drgs and non - drgs ( defined based on their observed @xmath15 color ) , and of intrinsically red and blue galaxies ( defined based on their rest - frame @xmath12 color ) . this paper is structured as follows . in [ sec - cs ] we present the composite sample used to measure the lf of galaxies at @xmath3 ; in [ sec - lf ] we describe the methods applied to measure the lf and discuss the uncertainties in the measured lf due to field - to - field variations and errors in the photometric redshift estimates ; the results ( of all galaxies and of the individual subsamples considered in this work ) are presented in [ sec - results ] , while the estimates of the number and luminosity densities and the contribution of drgs ( red galaxies ) to the global stellar mass density are given in [ sec - densities ] . our results are summarized in [ sec - concl ] . we assume @xmath41 , @xmath42 , and @xmath43 km s@xmath30 mpc@xmath30 throughout the paper . all magnitudes and colors are on the vega system , unless identified as `` ab '' . throughout the paper , the @xmath15 color is in the observed frame , while the @xmath12 color refers to the rest frame . the data set we have used to estimate the lf consists of a composite sample of galaxies built from three deep multiwavelength surveys , all having high - quality optical to nir photometry : the `` ultradeep '' faint infrared extragalactic survey ( fires ; @xcite ) , the great observatories origins deep survey ( goods ; @xcite ; chandra deep field south [ cdf - s ] ) , and the multi - wavelength survey by yale - chile ( musyc ; @xcite ; @xcite ) . photometric catalogs were created for all fields in the same way , following the procedures of @xcite . fires consists of two fields , namely , the hubble deep field south proper ( hdf - s ) and the field around ms 105403 , a foreground cluster at @xmath44 . a complete description of the fires observations , reduction procedures , and the construction of photometric catalogs is presented in detail in @xcite and @xcite for hdf - s and ms 105403 , respectively . briefly , the fires hdf - s and ms 105403 ( hereafter fh and fms , respectively ) are @xmath45 band limited multicolor source catalogs down to @xmath46 and @xmath47 , for a total of 833 and 1858 sources over fields of @xmath48 and @xmath49 , respectively . the fh and fms catalogs have 90% completeness level at @xmath50 and @xmath51 , respectively . the final fh ( fms ) catalogs used in the construction of the composite sample has 358 ( 1427 ) objects over an effective area of 4.74 ( 21.2 ) arcmin@xmath52 , with @xmath53 ( 22.54 ) , which for point sources corresponds to a 10 ( 8) @xmath54 signal - to - noise ratio ( @xmath55 ) in the custom isophotal aperture . lccccrr[!t ] fires - hdfs & u@xmath56b@xmath57v@xmath58i@xmath59j@xmath60hk@xmath60 & 23.80 & 23.14 & 4.74 & 358 & 68 + fires - ms1054 & ubvv@xmath58i@xmath59j@xmath60hk@xmath60 & 22.85 & 22.54 & 21.2 & 1427 & 297 + goods - cdfs & b@xmath61v@xmath58i@xmath62z@xmath63jhk@xmath60 & 21.94 & 21.34 & 65.6 & 1588 & 215 + musyc & ubvrizjhk@xmath60 & 21.33 & 21.09 & 286.1 & 5507 & 116 + from the goods / eis observations of the cdf - s ( data release version 1.0 ) a @xmath45 band limited multicolor source catalog ( hereafter cdfs ) was constructed , described in s. wuyts et al . ( 2007 , in preparation ) . goods zero points were adopted for @xmath64 and @xmath45 . the @xmath65-band zero point was obtained by matching the stellar locus on a @xmath15 versus @xmath66 color - color diagram to the stellar locus in fires hdf - s and ms 105403 . the difference with the official goods @xmath65-band zero point varies across the field , but on average our @xmath65-band zero points are @xmath67 mag brighter . a total effective area of 65.6 arcmin@xmath52 is well exposed in all bands . the final catalog contains 1588 objects with @xmath68 in this area . at @xmath69 the median @xmath55 in the @xmath45 isophotal aperture is @xmath70 . musyc consists of optical and nir imaging of four independent @xmath71 fields with extensive spectroscopic follow - up @xcite . deeper nir @xmath72 imaging was obtained over four @xmath73 subfields with the ispi camera at the cerro tololo inter - american observatory ( ctio ) blanco 4 m telescope . a complete description of the deep nir musyc observations , reduction procedures , and the construction of photometric catalogs will be presented in @xcite . the 5 @xmath54 point - source limiting depths are @xmath74 , @xmath75 , and @xmath76 . the optical @xmath77 data are described in @xcite . the present work is restricted to three of the four deep fields : the two adjacent fields centered around hdf - s proper ( hereafter mh1 and mh2 ) and the field centered around the quasar sdss 1030 + 05 ( m1030 ) . the final musyc @xmath45-selected catalog used in the construction of the composite sample has 5507 objects over an effective area of 286.1 arcmin@xmath52 , with @xmath78 , which for point sources corresponds to a @xmath1410 @xmath54 @xmath55 in the isophotal aperture . table [ tab - ref1 ] summarizes the specifications of each field , including wave band coverage , @xmath79 band total magnitude 90% completeness limit ( @xmath80 ) , effective area , the @xmath79 band total magnitude limit used to construct the composite sample ( @xmath81 ) , the number of objects , and the number of sources with spectroscopic redshifts . only a few percent of the sources in the considered catalogs have spectroscopic redshift measurements . consequently , we must rely primarily on photometric redshift estimates . photometric redshifts @xmath82 for all galaxies are derived using an identical code to that presented in @xcite , but with a slightly modified template set . this code models the observed spectral energy distribution ( sed ) using nonnegative linear combinations of a set of eight galaxy templates . as in @xcite , we use the e , sbc , scd , and i m seds from @xcite , the two least reddened starburst templates from @xcite , and a synthetic template corresponding to a 10 myr old simple stellar population ( ssp ) with a @xcite stellar initial mass function ( imf ) . we also added a 1 gyr old ssp with a salpeter imf , generated with the @xcite evolutionary synthesis code . the empirical templates have been extended into the uv and the nir using models . comparing the photometric redshifts with 696 spectroscopic redshifts ( 63 at @xmath83 ) collected from the literature and from our own observations gives a scatter in @xmath84 of @xmath85 . restricting the analysis to galaxies at @xmath83 in the musyc fields gives @xmath86 , corresponding to @xmath87 at @xmath88 . approximately 5% of galaxies in this sample are `` catastrophic '' outliers . a full discussion of the quality of the photometric redshifts is given elsewhere @xcite . the effects of photometric redshift errors on the derived lfs are modeled in [ sub - zphot ] . rest - frame luminosities are computed from the observed seds and redshift information using the method extensively described in the appendix of @xcite . this method does not depend directly on template fits to the data but rather interpolates directly between the observed fluxes , using the best - fit templates as a guide . we computed rest - frame luminosities in the @xmath89 , @xmath0 , @xmath1 , and @xmath2 filters of @xcite . for these filters we use @xmath90 , @xmath91 , @xmath92 , and @xmath93 . in all cases where a spectroscopic redshift is available we computed the rest - frame luminosities fixed at @xmath94 . stars in all @xmath45-selected catalogs were identified by spectroscopy , by fitting the object seds with stellar templates from @xcite and/or inspecting their morphologies , as in @xcite . on average , approximately 10% of all the objects were classified as stars . we constructed a composite sample of high - redshift ( @xmath95 ) galaxies to be used in the estimate of the lf in [ sec - lf ] . a large composite sample with a wide range of luminosities is required to sample both the faint and the bright end of the lf well ; moreover , a large surveyed area is necessary to account for sample variance . the very deep fires allows us to constrain the faint end of the lf , while the large area of musyc allows us to sample the bright end of the lf very well . the cdfs catalog bridges the two slightly overlapping regimes and improves the number statistics . the final composite sample includes 442 , 405 , and 547 @xmath45-selected galaxies in the three targeted redshift intervals @xmath96 , @xmath97 , and @xmath98 , respectively , for a total of 989 galaxies with @xmath99 at @xmath95 . of these , @xmath144% have spectroscopic redshifts . in figure [ fig2_ref ] we show the rest - frame @xmath0-band absolute magnitude versus the redshift for the composite sample in the studied redshift range @xmath3 . to estimate the observed lf in the case of a composite sample , we have applied an extended version of the @xmath100 algorithm @xcite as defined in @xcite so that several samples can be combined in one calculation . for a given redshift interval [ @xmath101,@xmath102 , we computed the galaxy number density @xmath103 in each magnitude bin @xmath104 in the following way : @xmath105 where @xmath106 is the number of objects in the chosen bin and @xmath107 is : @xmath108 where @xmath109 is the area in units of steradians corresponding to the @xmath110th field , @xmath111 is the number of samples combined together , @xmath112 is the comoving volume element per steradian , and @xmath113 is the minimum of @xmath114 and the maximum redshift at which the @xmath115th object could have been observed within the magnitude limit of the @xmath110th sample . the poisson error in each magnitude bin was computed adopting the recipe of @xcite valid also for small numbers . the @xmath100 estimator has the advantages of simplicity and no a priori assumption of a functional form for the luminosity distribution ; it also yields a fully normalized solution . however , it can be affected by the presence of clustering in the sample , leading to a poor estimate of the faint - end slope of the lf . although field - to - field variation represents a significant source of uncertainty in deep surveys ( since they are characterized by very small areas and hence small sampled volumes ) , the majority of published cosmological number densities and related quantities do not properly account for sample variance in their quoted error budgets . our composite sample is made of several independent fields with a large total effective area of @xmath116 arcmin@xmath52 ( about a factor of 3 larger than the nominal area of the @xmath45-selected cdfs - goods catalog used in @xcite ) , which significantly reduces the uncertainties due to sample variance . also , the large number of fields considered in this work with their large individual areas allows us to empirically measure the field - to - field variations from one field to the other in the estimate of the lf with the @xmath100 method , especially at the bright end , and to properly account for it in the error budget . in order to quantify the uncertainties due to field - to - field variations in the determination of the lf , we proceeded as follows . first , for each magnitude bin @xmath104 , we measured @xmath117 for each individual @xmath110th field using equation ( [ eq-1vmax ] ) . for each magnitude bin with @xmath118 , we estimated the contribution to the error budget of @xmath119 from sample variance using : @xmath120 with @xmath111 the number of individual fields used . for the magnitude bins with @xmath121 ( usually the brightest bin and the 3 - 4 faintest ones ) , we adopted the mean of the @xmath122 with @xmath118 . the final 1 @xmath54 error associated to @xmath119 is then @xmath123 , with @xmath124 the poisson error in each magnitude bin . we also measured the observed lf using the sty method @xcite , which is a parametric maximum likelihood estimator . the sty method has been shown to be unbiased with respect to density inhomogeneities ( e.g. , @xcite ) , it has well - defined asymptotic error properties ( e.g. @xcite ) , and does not require binning of the data . the sty method assumes that @xmath119 has a universal form , i.e. , the number density of galaxies is separable into a function of luminosity times a function of position : @xmath125 . therefore , the shape of @xmath119 is determined independently of its normalization . we have assumed that @xmath119 is described by a @xcite function , @xmath126 { } \nonumber\\ \times \exp{\big [ -10^{0.4(m^{\star}-m ) } \big]},\end{aligned}\ ] ] where @xmath127 is the faint - end slope parameter , @xmath20 is the characteristic absolute magnitude at which the lf exhibits a rapid change in the slope , and @xmath8 is the normalization . the probability of seeing a galaxy of absolute magnitude @xmath128 at redshift @xmath129 in a magnitude - limited catalog is given by @xmath130 where @xmath131 and @xmath132 are the faintest and brightest absolute magnitudes observable at the redshift @xmath129 in a magnitude - limited sample . the likelihood @xmath133 ( where the product extends over all galaxies in the sample ) is maximized with respect to the parameters @xmath127 and @xmath20 describing the lf @xmath103 . the best - fit solution is obtained by minimizing @xmath134 . a simple and accurate method of estimating errors is to determine the ellipsoid of parameter values defined by @xmath135 where @xmath136 is the @xmath137-point of the @xmath138 distribution with @xmath139 degrees of freedom . parameter @xmath136 is chosen in the standard way depending on the desired confidence level in the estimate ( as described , e.g. , by @xcite ; @xcite ) : @xmath140 , 6.2 , and 11.8 to estimate @xmath141 error contours with 68% , 95% , and 99% confidence level ( 1 , 2 , and 3 @xmath54 , respectively ) . the value of @xmath8 is then obtained by imposing a normalization on the best - fit lf such that the total number of observed galaxies in the composite sample is reproduced . the 1 , 2 , and 3 @xmath54 errors on @xmath8 are estimated from the minimum and maximum values of @xmath8 allowed by the 1 , 2 , and 3 @xmath54 confidence contours in the @xmath141 parameter space , respectively . studies of high - redshift galaxies still largely rely on photometric redshift estimates . it is therefore important to understand how the photometric redshift uncertainties affect the derived lf and to quantify the systematic effects on the lf best - fit parameters . @xcite have shown that at lower redshifts ( @xmath142 ) the measurement of the lf is strongly affected by errors associated with @xmath143 . specifically , large redshift errors together with the steep slope at the bright end of the galaxy lf tend to flatten the observed lf and result in measured @xmath20 systematically brighter than the intrinsic value , since there are more intrinsically faint galaxies scattered into the bright end of the lf than intrinsically bright galaxies scattered into the faint end . using monte carlo simulations , @xcite obtained a best - fit @xmath20 that was 0.8 mag brighter than the intrinsic value in the redshift range @xmath144 . in order to quantify the systematic effect on the lf parameters @xmath127 and @xmath20 in our redshift range of interest ( @xmath3 ) , we performed a series of monte carlo simulations . the details of these simulations and the results are presented in appendix [ app-1 ] . briefly , we generated several model catalogs of galaxies of different brightness according to an input schechter lf , extracted the redshifts of the objects from a probability distribution proportional to the comoving volume per unit redshift ( @xmath112 ) , and obtained the final mock catalogs after applying a limit in the observed apparent magnitude . to simulate the errors in the redshifts , we assumed a redshift error function parametrized as a gaussian distribution function of 1 @xmath54 width @xmath145 , with @xmath146 being the scatter in @xmath147 , and we formed the observed redshift catalog by perturbing the input galaxy redshift within the redshift error function . finally , we determined the lf for the galaxies at @xmath148 using the @xmath100 and maximum likelihood methods described in [ sec-1vmaxmeth ] and [ sec - stymeth ] , respectively . as shown in appendix [ app-1 ] , the systematic effects on the measured @xmath127 and @xmath20 in the redshift interval @xmath3 are negligible with respect to the other uncertainties in the lf estimate if the errors on the photometric redshifts are characterized by a scatter in @xmath147 of @xmath149 , which is the appropriate value for the @xmath150 sample considered in this work . this is not true at @xmath151 , where we find large systematic effects on both @xmath20 and @xmath127 , consistent with @xcite . as explained in detail in appendix [ app-1 ] , the large systematic effects found at @xmath151 arise from the strong redshift dependency of both @xmath112 and @xmath152 at low-@xmath153 ; at @xmath5 these dependencies are much less steep , and this results in smaller systematic effects on the measured lf . from the monte carlo simulations we also quantified that the effects of photometric redshift errors on the estimated luminosity density are typically a few percent ( always @xmath154% ) . we conclude that the parameters of the lf and the luminosity density estimates presented in this work are not significantly affected by the uncertainties in the photometric redshift estimates we also investigated the effects of non - gaussian redshift error probability distributions . systematic outliers in the photometric redshift distribution can potentially cause systematic errors in the lf measurements , although these are much smaller than the random uncertainties in the lf estimates ( if the outliers are randomly distributed ) . ] . in order to include this contribution in the error budget , we conservatively assume a 10% error contribution to the luminosity density error budget due to uncertainties in the photometric redshift estimates . in this section we present the results of the measurement of the lf of galaxies at @xmath150 . we have measured the global lf in the rest - frame @xmath2 and @xmath1 band at redshift @xmath155 and @xmath156 , respectively . as shown in figure [ fig - filters ] , at these redshifts , the rest - frame @xmath2 and @xmath1 bands correspond approximately to the observed @xmath45 band , which is the selection band of the composite sample . we also measured the global lf in the rest - frame @xmath0 band in the redshift interval @xmath155 , to compare it with the rest - frame @xmath2-band lf , and at redshift @xmath157 , to compare it with previous studies . for each redshift interval and rest - frame band we also split the sample based on the _ observed _ @xmath15 color ( @xmath158 , drgs ; @xmath159 , non - drgs ) and the _ rest - frame _ @xmath12 color ( @xmath160 , red galaxies ; @xmath161 , blue galaxies ) . in [ subsec - lf ] we present the global lf of all galaxies , and in [ subsec - sublf ] we present the lfs of the considered subsamples ( drgs , non - drgs , red and blue galaxies ) in the rest - frame @xmath2 band . the results for the rest - frame @xmath1 and @xmath0 bands are shown in appendix [ app-2 ] ; in appendix [ app-3 ] we compare our results with those in the literature . figure [ fig - lf_bvr_all ] shows the global rest - frame @xmath2- and @xmath0-band lfs for galaxies at @xmath155 , the rest - frame @xmath1-band lf at @xmath156 , and the rest - frame @xmath0-band lf at @xmath162 . the large surveyed area of the composite sample allows the determination of the bright end of the optical lf at @xmath150 with unprecedented accuracy , while the depth of fires allows us to constrain also the faint - end slope . this is particularly important because of the well - known correlation between the two parameters @xmath127 and @xmath20 . the best - fit parameters with their 1 , 2 and 3 @xmath54 errors ( from the maximum likelihood analysis ) are listed in table [ tab-3 ] , together with the schechter parameters of the local rest - frame @xmath2-band ( from @xcite ) and @xmath0-band ( from @xcite ) lfs . lcccc[!t ] @xmath163 & @xmath2 & @xmath164 & @xmath165 & @xmath166 + @xmath38 & @xmath2 & @xmath167 & @xmath168 & @xmath169 + @xmath156 & @xmath1 & @xmath170 & @xmath171 & @xmath172 + @xmath163 & @xmath0 & @xmath173 & @xmath174 & @xmath175 + @xmath157 & @xmath0 & @xmath176 & @xmath177 & @xmath178 + @xmath38 & @xmath0 & @xmath179 & @xmath180 & @xmath181 + at redshift @xmath155 , the faint - end slope of the rest - frame @xmath2-band lf is slightly flatter than in the rest - frame @xmath0-band , although the difference is within the errors . in the two higher redshift bins , the faint - end slope of the rest - frame @xmath1-band lf is flatter ( by @xmath182 ) than in the rest - frame @xmath0-band , although the difference is only at the 1 @xmath54 level . similarly , the faint - end slopes of the rest - frame @xmath0-band global lf in the low- and high - redshift bins are statistically identical . the characteristic magnitude @xmath183 in the low-@xmath153 interval is about 0.5 mag fainter with respect to the high - redshift one , although the difference is significant only at the @xmath141.5 @xmath54 level . we therefore conclude that the rest - frame @xmath0-band global lfs in the low- and high - redshift bins are consistent with no evolution within their errors ( @xmath184 @xmath54 ) . in figure [ fig - lf_bvr_all ] we have also plotted the local ( @xmath185 ) rest - frame @xmath2-band ( from @xcite ) and @xmath0-band ( from @xcite ) lfs . the faint - end slope of the @xmath2-band lf at @xmath155 is very similar to the faint - end slope of the local lf ; the characteristic magnitude is instead significantly ( @xmath186 @xmath54 ) brighter than the local value ( by @xmath7 mag ) , and the characteristic density is a factor of @xmath187 smaller than the local value . the rest - frame @xmath0-band lf at @xmath162 is characterized by a faint - end slope consistent with the local @xmath0-band lf ; the characteristic magnitude is significantly brighter ( @xmath186 @xmath54 ) than the local value by @xmath188 mag , while the characteristic density is a factor of @xmath189 smaller with respect to the local value . in this section we present the results of the lfs for different subsamples , by splitting the composite sample based on the observed @xmath15 color ( @xmath158 , drgs ; @xcite ) and on the rest - frame @xmath12 color ( by defining the red galaxies as those having @xmath160 , which is the median value of @xmath12 of the composite sample at @xmath155 ) . in figure [ fig3_ref ] , we show the rest - frame @xmath12 color versus the observed @xmath15 color for the composite sample at @xmath3 . in figure [ lf_r_lowz.ps ] we show the rest - frame @xmath2-band lf at @xmath155 of drgs versus non - drgs and red versus blue galaxies , together with the 1 , 2 , and 3 @xmath54 contour levels in the @xmath190 parameter space from the sty analysis . the lfs of the different subsamples in the rest - frame @xmath1 band at @xmath156 and in the rest - frame @xmath0 band at @xmath155 and @xmath162 are shown in appendix [ app-2 ] in figures . [ lf_v_highz.ps ] , [ lf_b_lowz.ps ] , and [ lf_b_highz.ps ] , respectively . in table [ tab-4 ] the best - fit parameters and their 1 , 2 , and 3 @xmath54 errors from the sty method are listed for all the considered rest - frame bands and redshift intervals . cccccc[!t ] @xmath163 & @xmath2 & @xmath158 & @xmath191 & @xmath192 & @xmath193 + & & @xmath159 & @xmath194 & @xmath195 & @xmath196 + & & @xmath160 & @xmath197 & @xmath198 & @xmath199 + & & @xmath161 & @xmath200 & @xmath201 & @xmath202 + @xmath156 & @xmath1 & @xmath158 & @xmath203 & @xmath204 & @xmath205 + & & @xmath159 & @xmath206 & @xmath207 & @xmath208 + & & @xmath160 & @xmath209 & @xmath210 & @xmath211 + & & @xmath161 & @xmath212 & @xmath213 & @xmath214 + @xmath163 & @xmath0 & @xmath158 & @xmath215 & @xmath216 & @xmath217 + & & @xmath159 & @xmath218 & @xmath219 & @xmath220 + & & @xmath160 & @xmath221 & @xmath222 & @xmath223 + & & @xmath161 & @xmath224 & @xmath225 & @xmath226 + @xmath162 & @xmath0 & @xmath158 & @xmath227 & @xmath228 & @xmath229 + & & @xmath159 & @xmath230 & @xmath231 & @xmath232 + & & @xmath160 & @xmath233 & @xmath234 & @xmath235 + & & @xmath161 & @xmath236 & @xmath237 & @xmath238 + as shown in figure [ lf_r_lowz.ps ] , the rest - frame @xmath2-band lf at @xmath155 of drgs is significantly ( @xmath186 @xmath54 ) different from that of non - drgs . the faint - end slope of the non - drg lf is much steeper , indicating that the contribution of drgs to the global luminosity and number density at faint luminosities is very small compared to that of non - drgs . the bright end of the drg lf is instead very similar to that of non - drgs , with the two subsamples contributing equally to the global lf . splitting the composite sample based on the rest - frame @xmath12 color , we find a qualitatively similar result , with the faint - end slope of the blue galaxy lf being much steeper than that of red galaxies ( although the red galaxies clearly dominate the bright end of the lf ) . the difference between the lfs of drgs ( red galaxies ) and non - drgs ( blue galaxies ) is mainly driven by the different faint - end slopes . a similar result holds in the rest - frame @xmath1 band at @xmath156 , although it is slightly less significant ( at the 2 - 3 @xmath54 level ) : the non - drg ( blue galaxy ) lf is very similar to that of drgs ( red galaxies ) at the bright end , while at the faint end , the lf of non - drgs ( blue galaxies ) is steeper than that of drgs ( red galaxies ) . in the rest - frame @xmath0 band , the differences between the lfs of drgs / red galaxies and non - drgs / blue galaxies become even less significant . although drgs / red galaxies are always characterized by lfs with flatter faint - end slopes , the significance of this result is only marginal ( @xmath184 @xmath54 ) , especially in the higher redshift interval . within our sample , there is marginal evidence for evolution with redshift : the rest - frame @xmath0-band non - drg / blue galaxy lfs in the two targeted redshift bins are characterized by similar ( within the errors ) faint - end slopes , while the characteristic magnitude is brighter by @xmath239 mag in the higher redshift bin . the lf of drgs / red galaxies tends to get steeper from low to high redshifts and @xmath20 gets brighter by @xmath240 mag . however , because of the large uncertainties ( especially for drgs and red galaxies ) on the measured schechter parameters , the differences in the rest - frame @xmath0 band between the high- and the low - redshift bins are at most at the 2 @xmath54 significance level . we note that the uncertainties on the estimated schechter parameters mainly arise from the small number statistics at the very faint end , which is probed only by fires . very deep ( down to the deepest fires ) nir imaging over large spatially disjoint fields is required for further progress in our understanding of the lowest luminosity galaxies at @xmath5 . @xcite computed the rest - frame optical ( @xmath1 band ) lf of @xmath26 lbgs using the distribution of optical @xmath241 magnitudes ( i.e. , the rest - frame uv lf ) and the distribution of @xmath241-@xmath45 colors as a function of @xmath241 magnitude . the rest - frame uv lf of lbgs was taken from @xcite with best - fit schechter parameters @xmath242 , @xmath243 mag , and @xmath244 mpc@xmath245 in our adopted cosmology . @xcite detected a correlation with 98% confidence between @xmath241-@xmath45 color and @xmath241 magnitude , such that fainter galaxies have redder @xmath241-@xmath45 colors . this trend was included in their lf analysis by using the relationship implied by the best - fit regression slope to the correlation , @xmath246 ( the scatter around this regression is very large ) . the schechter function was then fitted to the average lf values , obtaining best - fit schechter parameters @xmath247 , @xmath248 mag , and @xmath249 mpc@xmath245 . the overall shape of the rest - frame optical lf of lbgs is determined by the way in which the @xmath241-@xmath45 distribution as a function of @xmath241 magnitude redistributes @xmath241 magnitudes into @xmath45 magnitudes . therefore , as a result of the detected positive correlation between @xmath241 and @xmath241-@xmath45 , the faint - end slope of the lbg rest - frame optical lf is steeper than that of the uv lf @xcite . in figure [ v_drg_lbg.ps ] we compare the rest - frame @xmath1-band lf of blue galaxies at @xmath156 and the lbg lf from @xcite in the same rest - frame band and redshift interval . the blue galaxy lf estimated with the @xmath100 method appears consistent within the errors with the average lf values of lbgs ( shown as stars in figure [ v_drg_lbg.ps ] ) . however , the best - fit schechter parameters from the maximum likelihood analysis are only marginally consistent , with the faint - end slope of the lbg lf being significantly steeper than the one of blue galaxies , as shown in the inset of figure [ v_drg_lbg.ps ] . the same result is obtained if the rest - frame @xmath1-band lf of non - drgs ( rather than rest - frame blue galaxies ) is compared to that of lbgs . in appendix [ app-3 ] we compare our results with previously published lfs . specifically , we have compared our rest - frame @xmath0-band lf with that derived by @xcite , @xcite , and @xcite in the redshift intervals @xmath155 and @xmath157 , and our rest - frame @xmath2-band lf with the rest - frame @xmath250-band lf derived by @xcite at @xmath155 . we also compared our rest - frame @xmath0-band lfs of red and blue galaxies at @xmath157 with those measured by @xcite . the estimates of the number density @xmath251 obtained by integrating the best - fit schechter function to the faintest observed rest - frame luminosity are listed in table [ tab - ndall ] . for completeness , we also list @xmath252 , calculated by integrating the best - fit schechter lf to the rest - frame magnitude limits of the nir musyc , and @xmath253 , calculated by integrating the best - fit schechter lf to 2 mag fainter than the faintest observed luminosities . we find that the contribution of drgs ( red galaxies ) to the total number density down to the faintest probed rest - frame luminosities is 13%-25% ( 18%-29% ) depending on the redshift interval . by integrating the rest - frame @xmath0-band lf down to a fixed rest - frame magnitude limit ( @xmath254 ) , we find a hint of an increase of the contribution of blue galaxies from the low redshift bin ( 62% ) to the higher bin ( 74% ) , but the differences are not significant . if only the bright end of the lf is considered ( integrating the lf down to the fixed nir musyc limit , @xmath255 ) , the increase of the contribution of the blue galaxy population becomes significant at the 2 @xmath54 level , going from 42% in the low-@xmath153 bin to 66% in the high-@xmath153 bin . lccccc @xmath163 & @xmath2 & all & @xmath256 & @xmath257 & @xmath258 + & & @xmath158 & @xmath259 & @xmath260 & @xmath261 + & & @xmath159 & @xmath262 & @xmath263 & @xmath264 + & & @xmath160 & @xmath265 & @xmath266 & @xmath267 + & & @xmath161 & @xmath268 & @xmath269 & @xmath270 + @xmath156 & @xmath1 & all & @xmath271 & @xmath272 & @xmath273 + & & @xmath158 & @xmath274 & @xmath275 & @xmath276 + & & @xmath159 & @xmath277 & @xmath278 & @xmath279 + & & @xmath160 & @xmath280 & @xmath281 & @xmath282 + & & @xmath161 & @xmath283 & @xmath284 & @xmath285 + @xmath163 & @xmath0 & all & @xmath286 & @xmath287 & @xmath288 + & & @xmath158 & @xmath289 & @xmath290 & @xmath291 + & & @xmath159 & @xmath292 & @xmath293 & @xmath294 + & & @xmath160 & @xmath295 & @xmath296 & @xmath297 + & & @xmath161 & @xmath298 & @xmath299 & @xmath300 + @xmath162 & @xmath0 & all & @xmath301 & @xmath302 & @xmath303 + & & @xmath158 & @xmath304 & @xmath305 & @xmath306 + & & @xmath159 & @xmath307 & @xmath308 & @xmath309 + & & @xmath160 & @xmath310 & @xmath311 & @xmath312 + & & @xmath161 & @xmath313 & @xmath314 & @xmath315 + we determine the field - to - field variance in the density by fixing the parameters @xmath127 and @xmath20 to the best - fit values measured using the composite sample , and estimating @xmath316 for each @xmath110th field separately by imposing a normalization on the lf such that the total number of observed galaxies in each field is reproduced . in table [ tab - ndcomp ] , the derived @xmath316 of drgs and non - drgs in each field are listed for the three targeted redshift intervals and compared to @xmath8 measured from the composite sample . the results in the redshift range @xmath162 are plotted in figure [ fig - phistar ] . we find an overdensity of drgs in the m1030 field at all redshifts , with the excess ( as compared to the characteristic density of the composite sample ) varying from a factor of @xmath7 in the lowest redshift bin up to a factor of @xmath188 in the redshift interval @xmath162 . we also find an underdensity of drgs ( a factor of 0.82 - 0.86 ) in the goods - cdfs field , although only at @xmath317 . the value of @xmath8 for drgs in m1030 is a factor of @xmath212.4 larger than that in the goods - cdfs field at @xmath317 , although they are similar at @xmath155 . cccccccc + @xmath163 & @xmath318 & @xmath319 & @xmath320 & @xmath321 & @xmath322 & @xmath323 & @xmath324 + @xmath156 & @xmath325 & @xmath326 & @xmath327 & @xmath328 & @xmath329 & @xmath330 & @xmath331 + @xmath157 & @xmath332 & @xmath333 & @xmath334 & @xmath335 & @xmath336 & @xmath337 & @xmath338 + @xmath163 & @xmath339 & @xmath340 & @xmath341 & @xmath342 & @xmath343 & @xmath344 & @xmath345 + @xmath156 & @xmath346 & @xmath347 & @xmath348 & @xmath349 & @xmath350 & @xmath351 & @xmath352 + @xmath157 & @xmath353 & @xmath354 & @xmath355 & @xmath356 & @xmath357 & @xmath358 & @xmath359 + these results are qualitatively consistent with @xcite , who showed that the goods - cdfs field is underdense in massive ( @xmath360 @xmath35 ) galaxies at @xmath36 , with a surface density that is about 60% of the mean and a factor of 3 lower than that of their highest density field ( m1030 ) . however , our results seem to show systematically smaller underdensities for the goods - cdfs field compared to their work . in order to understand the origin of the smaller underdensity of drgs found for the goods - cdfs field in our work compared to that of massive galaxies in @xcite , we have estimated the surface density of drgs in the redshift range @xmath36 down to @xmath361 . we find that the surface density of drgs in the goods - cdfs field is @xmath362% of the mean and a factor of @xmath363 lower than that of the m1030 field , in good agreement with the values in @xcite . therefore , the smaller underdensities of drgs found for the goods - cdfs field in our work appear to arise mainly from the different targeted redshift ranges . the approach adopted in this work to quantify field - to - field variance by comparing the @xmath316 of the individual fields might also mitigate field - to - field differences , especially at the bright end . we note that there are significant differences in the observed characteristic densities even within the musyc fields , although they have areas of @xmath364 arcmin@xmath52 . for example , the observed @xmath8 of drgs in the mh1 field is consistent with the one derived from the composite sample , but it is 0.61 - 0.76 times the value in the m1030 field . these results demonstrate that densities inferred from individual @xmath364 arcmin@xmath52 fields should be treated with caution . in this section we present estimates of the luminosity density . because of the coupling between the two parameters @xmath127 and @xmath20 , the luminosity density ( obtained by integrating the lf over all magnitudes ) is a robust way to characterize the contribution to the total lf from the different subpopulations and to characterize the evolution of the lf with redshift . the luminosity density @xmath365 is calculated using : @xmath366 which assumes that the schechter parametrization of the observed lf is a good approximation and valid also at luminosities fainter than probed by our composite sample . table [ tab-5 ] lists @xmath365 with the corresponding 1 , 2 , and 3 @xmath54 errors errors of the luminosity densities were calculated by deriving the distribution of all the values of @xmath365 allowed within the 1 , 2 , and 3 @xmath54 solutions , respectively , of the schechter lf parameters from the maximum likelihood analysis . the contribution from the uncertainties in the photometric redshift estimates derived in appendix [ app-1 ] was added in quadrature . ] for all of the considered samples . we also list @xmath367 , the luminosity density calculated to the faintest probed rest - frame luminosity , and @xmath368 , the luminosity density calculated to the rest - frame magnitude limits of the deep nir musyc . while the difference between @xmath365 and @xmath367 is very small ( negligible for drgs and red galaxies , and @xmath67 dex on average for non - drgs and blue galaxies ) , the difference between @xmath365 and @xmath368 is significant , especially for non - drgs and blue galaxies ( @xmath369 dex on average ) . cccccc[!t ] @xmath163 & @xmath2 & all & @xmath370 & @xmath371 & @xmath372 + & & @xmath158 & @xmath373 & @xmath373 & @xmath374 + & & @xmath159 & @xmath375 & @xmath376 & @xmath377 + & & @xmath160 & @xmath378 & @xmath378 & @xmath379 + & & @xmath161 & @xmath380 & @xmath381 & @xmath382 + @xmath156 & @xmath1 & all & @xmath383 & @xmath384 & @xmath385 + & & @xmath158 & @xmath386 & @xmath387 & @xmath388 + & & @xmath159 & @xmath389 & @xmath390 & @xmath391 + & & @xmath160 & @xmath392 & @xmath393 & @xmath394 + & & @xmath161 & @xmath395 & @xmath396 & @xmath397 + @xmath163 & @xmath0 & all & @xmath398 & @xmath399 & @xmath400 + & & @xmath158 & @xmath401 & @xmath402 & @xmath403 + & & @xmath159 & @xmath404 & @xmath405 & @xmath406 + & & @xmath160 & @xmath407 & @xmath408 & @xmath409 + & & @xmath161 & @xmath410 & @xmath411 & @xmath412 + @xmath162 & @xmath0 & all & @xmath413 & @xmath414 & @xmath415 + & & @xmath158 & @xmath416 & @xmath417 & @xmath418 + & & @xmath159 & @xmath419 & @xmath420 & @xmath421 + & & @xmath160 & @xmath422 & @xmath423 & @xmath424 + & & @xmath161 & @xmath425 & @xmath426 & @xmath427 + in the top panel of figure [ lumdensbr_z.ps ] we have plotted the total rest - frame @xmath0-band luminosity density @xmath365 versus the redshift , including a compilation of results from the literature . only the results from the literature which are not significantly affected by field - to - field variations , or that have taken these into account , are plotted . our measurement of the total rest - frame @xmath0-band luminosity density is the only one at @xmath5 that is not significantly affected by field - to - field variance . from figure [ lumdensbr_z.ps ] , there is an indication of a possible increase of the total luminosity density in the highest redshift bin , significant at the @xmath428 @xmath54 level . the measurement at @xmath155 is consistent with the one at @xmath429 from @xcite . in figure[lumdensbr_z.ps ] we have also plotted the computed @xmath0-band rest - frame luminosity density as a function of @xmath153 predicted from large - scale @xmath430cdm hydrodynamical simulations from @xcite and from a semianalytical model taken from @xcite . while the predicted luminosity densities match the measurements at @xmath431 well , they clearly overpredict them at larger redshifts . only the prediction at @xmath26 from @xcite is consistent with our measurement , although their model still overpredicts significantly the luminosity densities in the range @xmath432 . the bottom panel of figure [ lumdensbr_z.ps ] shows the total rest - frame @xmath2-band luminosity density @xmath365 versus the redshift , including a compilation of results from the literature as in the top panel . as for the rest - frame @xmath0 band , our measurement of the rest - frame @xmath2-band @xmath365 is the first one at @xmath5 for which sample variance does not significantly contribute to the error budget . our point at @xmath5 is consistent with the trend observed at @xmath22 of decreasing luminosity densities with increasing redshifts , although the point at @xmath433 from the lf analysis of the goods - cdfs survey by @xcite is only @xmath434% of our measurement at @xmath435 . from table [ tab - ndcomp ] we see that in the redshift range @xmath155 , the cdfs field is underdense in non - drgs . the value of @xmath8 for all galaxies in the cdfs field is @xmath436 mpc@xmath437 mag@xmath30 ( estimated as in [ sec - numdens ] ) , which is a factor of @xmath438 the value of @xmath8 from the composite sample ( see table [ tab-3 ] ) . therefore , the lower value of the rest - frame @xmath2-band @xmath365 from @xcite at @xmath439 could be due to an underdensity of galaxies at @xmath27 . in figure [ lumdensbr_z.ps ] we have also plotted the computed @xmath2-band rest - frame luminosity density as function of @xmath153 predicted from large - scale @xmath430cdm hydrodynamical simulations from @xcite . as for the rest - frame @xmath0 band , the predictions match the observations well enough at @xmath440 , but at larger redshifts they significantly overpredict them . we can also compare our estimated luminosity densities in the rest - frame @xmath0 and @xmath1 bands with those from @xcite , who presented the evolution of the rest - frame optical luminosity and stellar mass densities at @xmath441 . the luminosity density in @xcite was computed by simply adding up the luminosities of all galaxies in the targeted redshift bins with rest - frame @xmath1-band luminosities @xmath442 l@xmath29 . if we integrate our measured lf down to the same limit adopted in @xcite , we obtain @xmath443 and @xmath444 for the @xmath1 and @xmath0 band , respectively , at @xmath26 , and @xmath445 for the @xmath0 band at @xmath435 ( units in erg s@xmath30 hz@xmath30 mpc@xmath245 ) , in excellent agreement with their estimates ( @xmath446 , @xmath447 , and @xmath448 , respectively ) . in [ subsec - sublf ] we showed that the contribution of drgs and red galaxies to the global lf is comparable to ( or larger than ) that of non - drgs and blue galaxies at the bright end , but it becomes significant smaller at the faint end , where non - drgs and blue galaxies dominate the global lf . in [ sec - numdens ] , the contribution of drgs to the global number density has been shown to be 13%-25% down to the faintest probed rest - frame luminosities . the contribution of drgs to the global luminosity density is 19%-29% depending on the considered rest - frame band and redshift interval ( see table [ tab-50 ] ) . their contribution increases up to 31%-44% if we cut the composite sample to the rest - frame absolute brightness limit of musyc ( @xmath449 , @xmath450 , @xmath451 of the global lf in the rest - frame @xmath2 , @xmath1 , and @xmath0 band , respectively ) , which reflects the increasing importance of drgs at the bright end . a similar result holds if we consider the red galaxy subsample . their contribution to the global luminosity density is 29%-52% down to the faintest observed luminosities and increases up to 36%-69% if we limit the analysis to the musyc brightness limits . from table [ tab-50 ] we conclude that the total luminosity density is dominated by non - drgs / blue galaxies , especially in the bluer rest - frame optical bands , although drgs / red galaxies contribute about 50% at the bright end . ccccc[!t ] @xmath163 & @xmath2 & @xmath158 & @xmath452 @xmath453 @xmath454 & @xmath455 @xmath456 @xmath457 + & & @xmath160 & @xmath458 @xmath459 @xmath460 & @xmath461 @xmath462 @xmath463 + @xmath156 & @xmath1 & @xmath158 & @xmath464 @xmath465 @xmath466 & @xmath467 @xmath468 @xmath469 + & & @xmath160 & @xmath470 @xmath471 @xmath466 & @xmath472 @xmath473 @xmath474 + @xmath163 & @xmath0 & @xmath158 & @xmath475 @xmath476 @xmath477 & @xmath478 @xmath479 @xmath480 + & & @xmath160 & @xmath481 @xmath482 @xmath483 & @xmath484 @xmath485 @xmath486 + @xmath162 & @xmath0 & @xmath158 & @xmath487 @xmath488 @xmath489 & @xmath490 @xmath491 @xmath492 + & & @xmath160 & @xmath493 @xmath494 @xmath495 & @xmath496 @xmath497 @xmath498 + although non - drgs and blue galaxies represent the major contribution to the total luminosity and number densities in the rest - frame optical bands , it has been shown that drgs usually have larger mass - to - light ratios than non - drgs ( e.g. , @xcite ; @xcite ; @xcite ) , as is generally true for red versus blue galaxies ( e.g. , @xcite ; @xcite ) . it is therefore interesting to quantify the contribution of drgs ( red galaxies ) to the total stellar mass density . following the method described in @xcite , we estimated the stellar mass density from the measured global luminosity density modulo the mass - to - light ratio @xmath13 . for each subsample we have measured the median rest - frame @xmath12 color color of the sample , @xcite computed the global @xmath12 color from the relation : @xmath499 , where @xmath500 and @xmath501 are computed by adding the luminosities of the individual galaxies . the two methods return very similar values for the @xmath12 colors . ] , estimated the corresponding @xmath13 ratio from the relation between @xmath12 color and @xmath13 ratio obtained from stellar population synthesis models , and multiplied the estimated @xmath13 ratio by the measured luminosity density to obtain the stellar mass density . to convert between the measured rest - frame @xmath12 color and the mass - to - light ratio @xmath13 , we have generated stellar population synthesis models with the evolutionary synthesis code developed by g. bruzual and s. charlot @xcite . we selected the `` padova 1994 '' evolutionary tracks , which are preferred by bruzual & charlot over the more recent `` padova 2000 '' tracks because the latter may be less reliable and predict a hotter red giant branch leading to worse agreement with observed galaxy colors . we used the solar metallicity set of tracks . the metallicities of the drgs are poorly known , with evidence for solar and supersolar metallicities for luminous drgs @xcite . these drgs appear more metal - rich than the five lbgs at @xmath26 studied by @xcite and similar to the seven uv - selected star - forming `` bx / md '' objects at @xmath27 for which @xcite inferred solar , and possibly supersolar , metallicities . in all cases , however , the determinations rely on limited samples and suffer from large uncertainties . as shown by @xcite adopting subsolar metallicity ( @xmath502 ) , the estimated @xmath13 ratios are systematically lower by a factor of @xmath503 on average . therefore , if non - drgs ( or blue galaxies ) are characterized by lower metallicities with respect to drgs ( red galaxies ) , the differences in @xmath13 ( and stellar mass densities ) would be even larger than what is estimated assuming solar metallicities for both subsamples . for the star formation history ( sfh ) we used three different prescriptions : a constant star formation history ( csf model ) , an exponentially declining in time sfh characterized by the parameter @xmath504 ( tau - model ) , and an instantaneous burst model ( ssp model ) . several values of @xmath504 were used , from @xmath505 gyr ( the resulting model being similar to the ssp model ) to 6 gyr ( closer to the csf model ) . we adopted the @xcite imf with lower and upper imf mass cutoffs @xmath506 @xmath35 and @xmath507 @xmath35 , respectively . adopting a different imf would result in different derived mass - to - light ratios , which strongly depend on the shape and cutoff of the low - mass imf ( for example , assuming a salpeter [ 1955 ] imf , the estimated @xmath13 ratio would be systematically larger by a factor of @xmath508 ) . however , since we are interested only in the relative contribution of drgs ( red galaxies ) to the global stellar mass density , the results do not depend on the adopted imf as long as all galaxies are characterized by the same imf . we assumed that the interstellar extinction by dust within the objects followed the attenuation law of @xcite derived empirically from observations of local uv - bright starburst galaxies under the formalism of a foreground screen of obscuring dust . we plot in figure [ fig - mlrr ] the relation between the rest - frame @xmath12 color and the mass - to - light ratio in the rest - frame @xmath2 band , @xmath509 , for the generated model tracks in the two cases with no extinction and with @xmath510 . it is seen that dust extinction moves the tracks roughly parallel to the model tracks . as emphasized by @xcite , dust is a second - order effect for estimating stellar @xmath13 ratios . dust extinguishes light from the stellar population , making it dimmer ; however , dust also reddens the stellar population , making it appear to have a somewhat larger stellar @xmath13 ratio . to first order , these effects cancel out , leaving a dust - reddened galaxy on the same color stellar @xmath13 ratio correlation . using the relation between color and @xmath13 , we convert the estimated median rest - frame @xmath12 color and the measurements of the luminosity densities @xmath365 to stellar mass density estimates @xmath511 . specifically , we adopted the median value of @xmath13 within the family of considered model tracks with @xmath510 ( blue dashed line plotted in figure [ fig - mlrr ] for the rest - frame @xmath2 band ) ; the error on the @xmath13 was chosen as half of the difference between the upper and lower envelope of the model tracks . the median rest - frame @xmath12 color of each subsample and the corresponding @xmath13 in the rest - frame @xmath0 , @xmath1 , and @xmath2 bands are listed in table [ tab-60 ] . cclcc @xmath163 & @xmath2 & @xmath158 & @xmath512 ( @xmath513 ) & @xmath514 ( @xmath515 ) + & & @xmath159 & @xmath516 ( @xmath517 ) & @xmath518 ( @xmath519 ) + & & @xmath520 & @xmath521 ( @xmath522 ) & @xmath523 ( @xmath524 ) + & & @xmath161 & @xmath525 ( @xmath526 ) & @xmath527 ( @xmath528 ) + @xmath156 & @xmath1 & @xmath158 & @xmath529 ( @xmath529 ) & @xmath530 ( @xmath530 ) + & & @xmath159 & @xmath531 ( @xmath532 ) & @xmath533 ( @xmath534 ) + & & @xmath520 & @xmath535 ( @xmath529 ) & @xmath536 ( @xmath530 ) + & & @xmath161 & @xmath537 ( @xmath532 ) & @xmath534 ( @xmath534 ) + @xmath163 & @xmath0 & @xmath158 & @xmath512 ( @xmath513 ) & @xmath538 ( @xmath538 ) + & & @xmath159 & @xmath516 ( @xmath517 ) & @xmath539 ( @xmath540 ) + & & @xmath520 & @xmath521 ( @xmath522 ) & @xmath541 ( @xmath542 ) + & & @xmath161 & @xmath525 ( @xmath526 ) & @xmath543 ( @xmath539 ) + @xmath162 & @xmath0 & @xmath158 & @xmath544 ( @xmath544 ) & @xmath545 ( @xmath545 ) + & & @xmath159 & @xmath546 ( @xmath547 ) & @xmath543 ( @xmath548 ) + & & @xmath520 & @xmath529 ( @xmath529 ) & @xmath545 ( @xmath545 ) + & & @xmath161 & @xmath549 ( @xmath547 ) & @xmath543 ( @xmath548 ) + although the values of the stellar mass densities of the individual subsamples might be affected by very large uncertainties , the relative contribution to the global stellar mass density of drgs ( red galaxies ) and non - drgs ( blue galaxies ) should be more robust . the contribution @xmath550 of the drgs ( red galaxies ) to the global stellar mass density is listed in table [ tab-50 ] . adopting the same assumptions for the stellar population synthesis models of drgs ( red galaxies ) and non - drgs ( blue galaxies ) ( i.e. , the median value of the considered model track ) , we see from table [ tab-60 ] that drgs ( red galaxies ) have @xmath13 ratios systematically higher than non - drgs ( blue galaxies ) by a factor of @xmath9 - 11 depending on the rest - frame band ( higher in the bluer bands ) . the differences in @xmath13 are smaller when the brighter samples ( down to the musyc limit ) are considered , with the @xmath13 of drgs ( red galaxies ) being a factor of @xmath551 - 9 ( 4 - 9 ) larger with respect to non - drgs ( blue galaxies ) . for comparison , from the analysis of _ spitzer_-irac imaging on hdf - s , @xcite found that the average mass - to - light ratio of drgs in the rest - frame @xmath4 band is about a factor of @xmath551 larger than lbgs , finding a correlation between @xmath552 and rest - frame @xmath12 color . consistently , from sed modeling of the @xmath3 fires galaxies , @xcite found that the median rest - frame @xmath1-band @xmath13 of drgs is @xmath7 - 2.3 m@xmath29 l@xmath553 ( @xmath554 for lbgs ) . the higher values of @xmath13 for drgs agree very well also with the results from sed fitting of individual galaxies in @xcite , who found for drgs a median value @xmath555 @xmath556 . finally , our estimated @xmath557 are in excellent agreement with those estimated in @xcite . because of the systematically larger mass - to - light ratios , drgs ( red galaxies ) dominate the global stellar mass density , with contributions in the range 66%-82% ( 69%-89% ) down to the faintest probed rest - frame luminosities . the contribution of drgs ( red galaxies ) increases up to 67%-85% ( 79%-92% ) if the brightest sample is considered ( down to the rest - frame magnitude limits of the deep nir musyc ) . these numbers are consistent with the results of @xcite . from a complete mass - selected sample ( @xmath558 @xmath35 ) constructed with musyc , fires , and goods - cdfs , they estimated that drgs in the redshift interval @xmath36 make up 77% of the total stellar mass , in very good agreement with our results . our results are in qualitative good agreement also with the results from @xcite , who found that the drgs contribute 64% of the stellar mass density at @xmath559 and 30%-50% at @xmath27 . we stress that the estimated contributions of drgs and red galaxies to the global stellar mass density are very uncertain and need confirmation from detailed sed analysis of mass - limited ( rather than luminosity - limited ) samples . in this paper we have measured the rest - frame optical ( @xmath2 , @xmath1 , and @xmath0 band ) luminosity functions of galaxies at redshifts @xmath3 from a composite sample constructed with the deep nir musyc , the ultra - deep fires , and the goods - cdfs . the large surveyed area ( @xmath560 arcmin@xmath52 , 76% of which comes from the deep nir musyc ) of the composite sample and the large range of luminosities spanned allows us to measure the bright end of the lf and to constrain the faint - end slope . moreover , the several independent fields and their large area enabled us to largely reduce uncertainties due to sample variance , especially at the bright end . we have used monte carlo simulations to show that the uncertainties in the photometric redshift estimates do not significantly affect the measured parameters of the lf in the studied redshift regimes . there is a hint for a steepening in the faint - end slope of the lf from the rest - frame @xmath2 band to the @xmath0 band , although the differences are not significant . the measured lf faint - end slopes at @xmath5 are consistent , within the errors , with those in the local lfs . the characteristic magnitudes are significantly brighter than the local ones ( e.g. , @xmath7 mag in the rest - frame @xmath2 band ) , while the measured values for @xmath8 are typically a factor of @xmath9 smaller with respect to the local values . the large number of objects in the composite sample allowed the first measurement of the lf of drgs ( defined based on their observed @xmath15 color ) , which we compared to that of non - drgs in the same redshift range . the drg population is characterized by a very different lf than that of non - drgs , especially at the faint end . while at the bright end the lf of drgs is similar to that non - drgs , at the faint end the latter one has a significantly steeper faint - end slope , especially in the rest - frame @xmath2 band . the significance of the difference between the lfs of drgs and non - drgs decreases going to bluer rest - frame bands and to higher redshifts , although this is mainly caused by decreasing constraints on the faint end of the lf of drgs . qualitatively similar results are found if we compare the lfs of red ( rest - frame @xmath160 ) and blue ( rest - frame @xmath561 ) galaxies in the same redshift intervals , with the former equally contributing ( or even dominating ) at the bright end and the latter dominating the faint end . in the rest - frame @xmath1 band we have also compared the lfs of blue galaxies ( non - drgs ) with those of lbgs in the same redshift range . although the two lfs agree very well at the bright end , the faint - end slope estimated by @xcite is much steeper than the one measured in this paper . as the rest - frame optical lf of lbgs was estimated in @xcite from the rest - frame uv lf and the observed distribution of @xmath241-@xmath45 colors as a function of @xmath241 magnitude , their steeper slope could be a result of an overestimate of the regression slope of the correlation between @xmath241-@xmath45 and @xmath241 and/or of the faint - end slope of the adopted rest - frame uv lf of lbgs . supporting the former possibility is the work of @xcite , who do not find any positive correlation between @xmath2 magnitudes and @xmath562 colors in their deeper sample . support for the latter comes from the very recent work of @xcite , who measured the rest - frame uv lf for @xmath563 lbgs from the keck deep fields ( kdf ; @xcite ) and find a faint - end slope @xmath564 , significantly shallower than the one adopted in @xcite . alternatively , our blue @xmath45-selected ( i.e. , rest - frame optical selected ) galaxies might simply constitute a different population than the @xmath26 lbgs ( rest - frame uv selected ) , with different characterizations of the lf at the faint end . we also caution that our measurements of the faint - end slopes still have significant uncertainties due to small number statistics . we generally find good agreement between our measured rest - frame @xmath0-band lfs at @xmath157 and those previously published by @xcite , @xcite , and @xcite . in the redshift range @xmath155 , the agreement between our rest - frame @xmath0- and @xmath2-band lfs and those measured by @xcite from the fdf survey is less good , especially for the @xmath2 band . their schechter parameters @xmath127 and @xmath20 are consistent with ours only at the 2 @xmath54 level in the rest - frame @xmath2 band , while their estimated @xmath8 is larger than ours by a factor of @xmath141.3 - 1.6 in the rest - frame @xmath0 and @xmath2 band , respectively . we have shown that this disagreement may be due to the spectroscopically confirmed existence of a ( proto)cluster at @xmath565 in the single field fdf survey . from the measured lfs we have estimated the number and luminosity densities of the global population of high-@xmath153 galaxies and of various subsamples . the contribution of drgs ( red galaxies ) to the global number density is only @xmath566%-25% ( 18%-29% ) down to the faintest probed rest - frame luminosities . however , we have shown that field - to - field variations can be very significant ( up to a factor of @xmath551 ) , especially for relatively bright samples , in accord with the highly clustered nature of high luminosity and red galaxies ( @xcite ; @xcite ; @xcite ) . the contribution of drgs ( red galaxies ) to the global luminosity density is @xmath567%-30% ( 30%-50% ) , higher in the redder rest - frame bands ( which are less affected by extinction and better tracers of the underlying stellar mass ) and at lower redshifts . with respect to the lower @xmath153 luminosity density estimates from the literature , we confirm the trend of slowly decreasing rest - frame @xmath2-band luminosity densities beyond @xmath568 , with @xmath569 at @xmath570 being a factor of @xmath11 smaller than the local one . in the rest - frame @xmath0 band , the measured global luminosity density at @xmath570 is similar to the local value . at @xmath571 , the estimated global luminosity density may be a factor of @xmath11 higher , similar to the values around @xmath568 . finally , using stellar population synthesis models , we have derived the mass - to - light ratios of the considered subsamples by converting the estimated median rest - frame @xmath12 color into @xmath13 . in the rest - frame @xmath2 and @xmath1 bands , the mass - to - light ratios of drgs ( red galaxies ) are a factor of @xmath9 larger than non - drgs ( blue galaxies ) , consistent with previous works . in the rest - frame @xmath0 band the difference in @xmath13 is higher , up to a factor of @xmath572 . using the estimated @xmath13 , we have quantified the contribution of drgs and red galaxies to the global stellar mass density , finding that the total stellar mass budget is dominated by drgs ( red galaxies ) , whose contribution is of order @xmath573%-80% of the global value . we caution that our @xmath13 ratios estimates are very rough and characterized by very large uncertainties and need confirmation from detailed sed analysis . the main limitation of this work is the small number statistics at the very faint end of the lf , which is probed only by the ultra - deep fires . the faint - end slopes of the drg and red galaxy subsamples are especially uncertain . to make further progress in the determination of the lf of different galaxy populations at @xmath5 and to better constrain the global lfs , it is crucial to better probe the faint end of the lfs . this can only be achieved with ultradeep nir imaging with high - quality optical data over many spatially disjoint fields , in order to improve the statistics at the faint end and to mitigate the effect of field - to - field variations . although we have shown that well - behaved photometric redshift errors do not affect significantly the measurement of the lf , the heavy reliance on photometric redshifts is another limitation of this work , since `` catastrophic '' failures and systematic errors could potentially affect the lf measurements . obtaining large numbers of spectroscopic redshifts for @xmath4-selected high-@xmath153 sources has proven difficult and extremely time consuming . even though the success rate for measuring spectroscopic redshift for bright galaxies is high with nir spectroscopy @xcite , only the use of multiobject nir spectrographs will make it possible to construct a large sample of high-@xmath153 @xmath4-selected galaxies with spectroscopic redshift measurements . further advances can be expected from further extension of the wavelength range into the red . scheduled _ spitzer _ irac observations on the deep nir musyc fields will allow us to ( 1 ) separate old and passively evolving galaxies from heavily obscured and strongly active star - forming galaxies ( see @xcite ) , making it possible to study the lf of physically different types of galaxies ; ( 2 ) extend the robust measurement of the lf at redshift @xmath36 into the rest - frame nir , which is much closer to a selection by stellar mass ; ( 3 ) convert the measured rest - frame nir luminosity function into a mass function and study the evolution of the stellar mass density ; and ( 4 ) extend the study of the rest - frame optical lfs to even higher redshifts . we thank all the members of the musyc collaboration for their contribution to this research . musyc has greatly benefited from the support of fundacin andes and the yale astronomy department . d.m . is supported by nasa ltsa nng04ge12 g . the authors acknowledge support from nsf carrer ast-0449678 . e.g. is supported by nsf fellowship ast-0201667 . is supported by fondecyt grant # 1040719 . we thank the anonymous referee for comments and suggestions which helped improve the paper . in order to quantify the systematic effects on the lf parameters @xmath127 and @xmath20 due to the uncertainties in the photometric redshift estimates , we performed a series of monte carlo simulations . first , we generated several model catalogs of 25,000 galaxies with redshifts between @xmath574 and @xmath575 and with luminosities drawn from an input schechter lf . while in the monte carlo simulations of @xcite the redshifts of the objects in the mock catalogs were extracted from a random uniform distribution , we took into account the fact that , under the assumption of no evolution in the number density , the probability of a galaxy existing at the redshift @xmath153 is proportional to the volume : @xmath576 where @xmath577 is the luminosity distance . since in a real survey galaxies are selected down to a limiting apparent magnitude , the final mock catalogs are obtained after applying a limit in the observed apparent magnitude . the effect of a limiting apparent magnitude is that , at a fixed observed magnitude , intrinsically fainter sources are systematically excluded from the catalog at higher redshift . next , as done by @xcite , we assumed a redshift error function parameterized as a gaussian distribution function of 1 @xmath54 width @xmath145 , with @xmath146 the scatter in @xmath147 , and we formed an observed redshift catalog by perturbing the input galaxy redshift within the redshift error function . finally , we determined the lf for the galaxies at @xmath148 using the @xmath100 and maximum likelihood methods described in [ sec - lf ] . note that we ignore @xmath4-correction in our monte carlo simulations . we first studied the effects of the photometric redshift uncertainties at @xmath151 , by using @xmath578 , @xmath579 , and assuming an input schechter lf with parameters @xmath580 and @xmath581 ( as in @xcite ) . in order to compare our results with those from the monte carlo simulations in @xcite , we used @xmath582 and we measured the lf in the same redshift range @xmath144 . we find that the median measured @xmath20 is brighter than the intrinsic value by @xmath554 mag and that the measured @xmath127 is steeper on average than the intrinsic value by @xmath67 . this result is shown in figure [ fig - chen4v ] : in the left panel , the input schechter lf is compared to the median monte carlo realization ; in the right panel , the 100 monte carlo realizations are plotted in the @xmath141 plane and compared to the best - fit values and the corresponding 1 , 2 , and 3 @xmath54 contour levels of the lf measured on the redshift - unperturbed mock catalog . as shown in figure [ fig - chen4v ] , the measured systematic effects on @xmath127 and @xmath20 caused by the redshift uncertainties arise from an excess of sources at both the faint and the bright end of the lf . a careful analysis of the monte carlo simulations reveals the origin of these excesses . because of the uncertainties in the redshifts , sources can scatter from high to low redshifts , and vice versa . as the probability of a galaxy existing at redshift @xmath153 is proportional to @xmath112 ( which peaks at @xmath583 ) , the number of sources that scatter from higher redshifts to lower ones is much larger than vice versa . this is evident in figure [ fig - mc_mvsz ] , where the rest - frame absolute magnitudes of the mock catalog are plotted versus redshift : at a fixed magnitude , the number of sources is larger at higher redshifts . since an object in the mock catalog is characterized by a fixed apparent magnitude , a new redshift estimate translates into a new rest - frame absolute magnitude ; e.g. , when a source `` scatters '' from high to low redshift , the estimated absolute brightness is fainter than the intrinsic value , and the object moves , in figure [ fig - mc_mvsz ] , from right to left along a line parallel to the black solid line . moreover , at any redshift , there are more sources at faint magnitudes than at bright ones because of the shape of the lf . therefore , the sources scattering from high redshifts into the considered redshift bin ( plotted in fig . [ fig - mc_mvsz ] in cyan ) preferentially end up at fainter magnitudes ( plotted in red ) , producing the excess at the faint end of the lf with respect to the input lf . the excess at the bright end is instead mainly produced by those sources that scatter into the considered redshift range coming from lower redshifts ( represented in fig . [ fig - mc_mvsz ] with cyan and green filled circles ) ; since at the bright end the number of objects is very small , even a handful of new sources can significantly increase the measured density with respect to the intrinsic one . because at low redshifts the dependency of @xmath152 with redshift is strong , even small uncertainties in the redshift estimate have large effects on the rest - frame absolute magnitude . for example , at @xmath584 , a @xmath585 results in a @xmath586 mag ( only 0.1 - 0.2 mag at @xmath587 ) . the systematic effect on @xmath20 that we derive ( @xmath588 mag ) is about half the effect found by @xcite . also almost no systematic effect in @xmath127 was found in their work . it seems very likely that these differences might be due to the fact that the redshifts are drawn from a random uniform distribution in @xcite while in our monte carlo simulations they are extracted from the probability function specified in eq . [ eq - pz ] . in the former case , there would be a much larger number of low-@xmath153 sources that can scatter into the considered redshift bin , resulting in a larger excess of sources at the bright end and therefore a larger systematic effect in @xmath20 . if we repeat our monte carlo simulations extracting the redshifts of the sources from a random uniform distribution , we obtain @xmath589 mag and @xmath590 , consistent with the result of @xcite as shown in the right panel of figure [ fig - chen4v ] . next , we repeated our monte carlo simulations at higher redshift by generating model catalogs with galaxies at redshifts between @xmath591 and @xmath592 , since the goal of this paper is to measure the lf of galaxies in the redshift intervals @xmath155 and @xmath157 . in figure [ fig - mock7wv ] we plot the results of our monte carlo simulations at @xmath155 , assuming an input schechter lf with @xmath593 and @xmath594 and for @xmath595 ( which corresponds to the photometric redshift errors in the deep nir musyc for @xmath596 objects ) . the systematic effects on the measured @xmath127 and @xmath20 are now very small , @xmath597 mag and @xmath598 , and negligible with respect to the other uncertainties on the estimated best - fit parameters . similar results are obtained in the redshift bin @xmath157 . at @xmath3 , the effect due to @xmath599 is much smaller than at @xmath151 , since @xmath600 peaks at @xmath601 and then decreases very slowly , so that the number of high- and low-@xmath153 sources scattering into the considered redshift bin is similar to the number of sources scattering out . also , at @xmath5 , the error on the rest - frame absolute magnitude corresponding to a redshift error is significantly smaller than at @xmath151 ; e.g. , @xmath602 mag for @xmath603 at @xmath587 . therefore , the measured lf is similar to the input one and the systematic effects on @xmath127 and @xmath20 are negligible compared to the uncertainties in the lf estimates for reasonable values of @xmath146 ( @xmath604 ) . we repeated the above monte carlo simulations assuming different input @xmath127 ( @xmath605 , @xmath606 , and @xmath607 ) to study the behavior of the systematic effects as function of the faint - end slope . no significant differences are found : @xmath608 mag for @xmath609 and @xmath607 ( depending on the considered redshift bin ) ; for @xmath610 , the systematic effect is slightly larger ( @xmath611 mag for @xmath155 and @xmath162 , respectively ) , but also the uncertainties on the best - fit @xmath20 increase with steeper faint - end slopes ( since the observed lf appears more like a power law ) so that the systematic effects on the measured best - fit schechter parameters remain very small with respect to the uncertainties on the best - fit values . finally , we investigated the effects of non - gaussian redshift error probability distributions . first , using a model catalog with galaxies at redshifts between @xmath612 and @xmath592 , we simulated the effect of a 5% `` catastrophic '' outliers by assigning random redshifts to 5% of the mock catalog . adopting the input lf with @xmath593 and @xmath613 and assuming @xmath595 , we find larger systematic effect in both @xmath127 and @xmath20 by a factor of almost 2 . next , we built the mean redshift probability distribution of the deep nir musyc by averaging the individual redshift probability distribution for each galaxy calculated by the used photometric redshift code ( for details see @xcite ) . the average musyc redshift probability distribution is well modeled by a lorentzian function , rather than a gaussian function . we find a larger systematic effect in @xmath127 , twice as much as the corresponding effect assuming a gaussian parametrization for the redshift probability distribution , but similar systematic effect in @xmath20 , although in the opposite direction . to summarize , although the systematic effects in @xmath127 and @xmath20 expectedly get larger when we simulate `` catastrophic '' outliers or we adopt a redshift error function with broader wings compared to the gaussian model , they remain much smaller than the random uncertainties in the lf estimates . we also quantified the systematic effect on the luminosity density estimates . we find that the effect is of the order of a few percent ( always @xmath6146% ) depending on the considered redshift interval and on the input schechter lf . in order to include this contribution in the error budget , we conservatively assume a 10% error contribution to the luminosity density error budget due to uncertainties in the photometric redshift estimates . in the body of the paper we showed the rest - frame @xmath2 band lfs of drgs , non - drgs , red and blue galaxies ( see fig . [ lf_r_lowz.ps ] ) . for completeness , we show here the comparison of the lfs of drgs ( red galaxies ) and non - drgs ( blue galaxies ) discussed in [ subsec - sublf ] in the rest - frame @xmath1 band at @xmath97 ( fig . [ lf_v_highz.ps ] ) and in the rest - frame @xmath0 band at @xmath96 ( fig . [ lf_b_lowz.ps ] ) and at @xmath98 ( fig . [ lf_b_highz.ps ] ) . the corresponding best - fit schechter parameters are listed in table [ tab-4 ] . here we compare our results to previous rest - frame optical lf studies at @xmath5 , which were based on smaller samples . we note that these studies are affected by significant uncertainties due to field - to - field variance ( as they are based on a single field or on a very small total surveyed area ) and by small number statistics at the bright end . @xcite analyzed a sample of 138 @xmath4-selected galaxies down to @xmath615 to construct the rest - frame @xmath0-band lf in the redshift range @xmath616 . the total area of their composite sample is @xmath617 arcmin@xmath52 , a factor of @xmath618 smaller than the area sampled in this work . @xcite repeated the analysis in @xcite with an improved composite sample ( although with the same area ) and allowing the schechter parameters @xmath8 and @xmath20 to vary with the redshift , while @xmath127 is kept constant at the low - redshift value ( @xmath151 ) . a direct comparison between the lfs of @xcite and @xcite and the lf measured in this work is shown in figure [ fig - poli ] ( _ left panel _ ) for the redshift range @xmath619 . first , we note in figure [ fig - poli ] how much better the bright end of the lf is constrained from our work : the large area of the composite sample ( @xmath620% of which comes from the deep nir musyc alone ) allows us to sample the lf up to @xmath0-band magnitudes @xmath621 mag brighter than done in @xcite and @xcite . our measurements of the lf using the @xmath100 method are consistent within the errors with those in both @xcite and @xcite . the best - fit lf estimated with the maximum likelihood analysis in @xcite is consistent , within the errors , with our best - fit solution . however , the best - fit lf estimated with the maximum likelihood analysis in @xcite is significantly different from ours , as clearly shown in the inset of figure [ fig - poli ] ( _ left panel _ ) . while their faint - end slope is consistent with our best - fit @xmath127 , their lack of constraints on the bright end of the lf results in a much fainter @xmath183 ( by @xmath240 mag ) . @xcite analyzed a sample of 5558 @xmath39-selected galaxies down to @xmath622 ( 50% compleness limit ) from the fdf survey @xcite to study the evolution of the rest - frame @xmath0- and @xmath250-band lf over the redshift range @xmath623 . the total area of their sample is @xmath624 arcmin@xmath52 over a single field , a factor of @xmath625 smaller than the total area sampled in this work . a direct comparison between the @xmath0-band lf from @xcite and ours is possible in the redshift range @xmath98 , and it is shown in the left panel of figure [ fig - poli ] ( _ red symbols _ ) . the best - fit schechter parameters @xmath127 , @xmath20 , and @xmath8 are consistent with ours within the errors . in the redshift range @xmath96 we compared our estimated @xmath0-band lf with the one defined by their best - fit schechter parameters ( @xmath127 , @xmath20 , and @xmath8 ) estimated at @xmath626 . while their best - fit schechter parameters @xmath127 and @xmath20 are consistent with ours , their best - fit @xmath8 is a factor of @xmath627 larger . this difference can be entirely accounted for by field - to - field variations ( see [ sec - numdens ] ) . in fact , the fdf survey consists of a single pointing of only @xmath624 arcmin@xmath52 and thus it is potentially strongly affected by sample variance . moreover , @xcite spectroscopically identified an overdensity of galaxies at @xmath565 ( possibly a [ proto]cluster , with more than 10 identical redshifts ) , which can potentially strongly bias the estimate of @xmath8 in this redshift bin . in the right panel of figure [ fig - poli ] we compare our measured @xmath2-band lf with that measured by @xcite in the redshift range @xmath628 . the agreement between the two lfs is much worse than for the rest - frame @xmath0-band . their best - fit schechter parameters @xmath127 and @xmath20 are now consistent with ours only at the @xmath11 @xmath54 level , with their @xmath20 about 0.3 mag brighter than ours ; their best - fit @xmath8 is a factor of @xmath21 larger than our best - fit value . as mentioned above , an overdensity of galaxies was spectroscopically found in this redshift interval in the fdf . although it is hard to quantify the effect of the presence of a ( proto)cluster at this redshift on the measured lf , it is interesting to make a connection with the work of @xcite , who spectroscopically identified a protocluster at @xmath629 in the hs 1700 + 643 field . in the spectroscopically identified galaxy sample , @xcite found 19 ( out of 55 ) objects at @xmath630 in the redshift range @xmath631 . within the @xmath632 field of view ( similar to the fdf field of view ) over which the protocluster objects are distributed , the protocluster and `` field '' galaxy sky distributions are the same . assuming that the distribution of the spectroscopically identified galaxies is representative of the whole sample , the estimated density would be a factor of @xmath141.3 - 1.6 that of `` field '' galaxies only , consistent with the difference found in @xmath8 between our composite sample and the fdf in the redshift range @xmath631 . @xcite measured the rest - frame @xmath0-band lf of red and blue galaxies at redshift @xmath633 using a sample of 138 @xmath4-selected galaxies in the redshift range @xmath616 down to @xmath615 . their red and blue populations were defined on the basis of an `` s0 color track '' . from their figure 1 , the average rest - frame @xmath12 color of their model at redshift @xmath633 is @xmath634 , very similar to our definition of blue and red galaxies ( @xmath161 and @xmath160 , respectively ) . in figure [ fig - bgiallongo ] we have compared the lfs of red and blue galaxies from this work to those presented in @xcite . as for the lf of all galaxies , we are able to constrain the bright end of the lf much better by sampling the lf to luminosities @xmath621 mag brighter . for the red galaxy population , their measurements of the lf with the @xmath100 method are consistent within the errors with ours ; their best - fit faint - end slope is also consistent , within the errors , with our estimate , although @xmath127 was fixed at the local value in @xcite ; our @xmath20 is @xmath554 mag brighter than their best - fit value , but still consistent within the errors at the 1 @xmath54 level . the best - fit normalization of @xcite is a factor of @xmath621 smaller than ours , only marginally consistent at the 2 @xmath54 level . for the blue population , the @xmath100 measurements of the lf from @xcite are consistent with ours . however , the agreement gets worse when we compare our lf estimated using the sty method with theirs , especially at the bright end , where they lack information . while their assumed faint - end slope is consistent with our best - fit solution , the characteristic magnitude from @xcite is @xmath635 mag fainter than ours . their best - fit solution is , however , very uncertain , characterized by very large error bars and still consistent with ours at the @xmath19 @xmath54 level . adelberger , k. l. , erb , d. k. , steidel , c. c. , reddy , n. a. , pettini , m. , & shapley , a. e. 2005 , , 620 , l75 adelberger , k. l. , & steidel , c. c. 2000 , , 544 , 218 adelberger , k. l. , steidel , c. c. , shapley , a. e. , hunt , m. p. , erb , d. k. , reddy , n. a. , & pettini , m. 2004 , , 607 , 226 avni , y. 1976 , , 210 , 642 avni , y. , & bahcall , j. n. 1980 , , 235 , 694 bell , e. f. , de jong , r. s. 2001 , , 550 , 212 bessell , m. s. 1990 , , 102 , 1181 blanton , m. r. , et al . 2001 , , 121 , 2358 blanton , m. r. , et al . 2003 , , 592 , 819 brown , w. r. , geller , m. j. , fabricant , d. g. , kurtz , m. j. 2001 , , 2001 , 122 , 714 bruzual , g. , & charlot , s. 2003 , , 344 , 1000 calzetti , d. , armus , l. , bohlin , r. c. , kinney , a. l. , koornneef , j. , & storchi - bergmann , t. 2000 , , 533 , 682 chabrier , g. 2003 , , 115 , 763 chen , h .- w , et al . 2003 , , 586 , 745 cole , s. , et al . 2001 , , 326 , 255 coleman , g. d. , wu , c .- c . , weedman , d. w. 1980 , , 43 , 393 daddi , e. , et al . 2003 , , 588 , 50 dahlen , t. , mobasher , b. , somerville , r. s. , moustakas , l. a. , dickinson , m. , ferguson , h. c. , giavalisco , m. 2005 , , 631 , 126 davis , m. , et al . 2003 , proc . spie , 4834 , 161 efstathiou , g. , ellis r. s. , & peterson , b. a. 1988 , , 232 , 431 ellis r. s. , colless , m. , broadhurst , t. , heyl , j. , glazebrook , k. 1996 , , 280 , 235 frster schreiber , n. m. , et al . 2004 , , 616 , 40 frster schreiber , n. m. , et al . 2006 , , 131 , 1891 franx , m. , et al . 2003 , , 587 , l79 gabasch , a. , et al . 2004 , , 421 , 41 gabasch , a. , et al . 2006 , , 448 , 101 gawiser , e. , et al . 2006 , , 162 , 1 gehrels , n. 1986 , , 303 , 336 giallongo , e. , salimbeni , s. , menci , n. , zamorani , g. , fontana , a. , dickinson , m. , cristiani , s. , pozzetti , l. 2005 , , 622 , 116 giavalisco , m. , et al . 2004 , , 600 , l93 hauschildt , p. h. , allard , f. , & baron , e. 1999 , , 512 , 377 heidt , j. , et al . 2003 , , 398 , 49 ilbert , o. , et al . 2005 , , 439 , 86 kauffmann , g. , et al . 2003 , , 341 , 33 kendall , m. g. , & stuart , a. 1961 , the advanced theory of statistics , vol . 2 ( london : griffin & griffin ) kinney , a. l. , calzetti , d. , bohlin , r. c. , mcquade , k. , storchi - bergmann , t. , & schmitt , h. r. 1996 , , 467 , 38 knudsen , k. k. , et al . 2005 , , 632 , l9 kochanek , c. s. , et al . 2001 , , 560 , 566 kriek , m. et al . 2006 , , 649 , l71 labb , i. , et al . 2003 , , 125 , 1107 labb , i. , et al . 2005 , , 624 , l81 labb , i. , et al . 2006 , , submitted lampton , m. , margon , b. , & bowyer , s. 1976 , , 208 , 177 le fvre , et al . 2004 , , 417 , 839 lilly , s.j . , tresse , l. , hammer , f. , crampton , d. , le fvre , o. 1995 , , 455 , 108 lin , h. , yee , h. k. c. , carlberg , r. g. , ellingson , e. 1997 , , 475 , 494 nagamine , k. , cen , r. , & ostriker , j. p. 2000 , , 541 , 25 nagamine , k. , fukugita , m. , cen , r. , & ostriker , j. p. 2001 , , 327 , l10 norberg , p. , et al . 2002 , , 336 , 907 papovich , c. , et al . 2006 , , 640 , 92 pettini , m. , shapley , a. e. , steidel , c. c. , cuby , j .- g . , dickinson , m. , moorwood , a. f. m. , adelberger , k. l. , & giavalisco , m. 2001 , , 554 , 981 poli , f. , et al . 2003 , , 593 , l1 quadri , r. , et al . 2007a , , 654 , 138 quadri , r. , et al . 2007b , , submitted ( astro - ph/0612612 ) reddy , n. a. , erb , d. k. , steidel , c. c. , shapley , a. e. , adelberger , k. l. , pettini , m. 2005 , , 633 , 748 reddy , n. a. , steidel , c. c. , fadda , d. , yan , l. , pettini , m. , shapley , a. e. , erb , d. k. , adelberger , k. l. 2006 , , 644 , 792 rubin , k. h. r. , van dokkum , p. g. , coppi , p. , johnson , o. , frster schreiber , n. m. , franx , m. , & van der werf , p. 2004 , , 613 , l5 rudnick , g. , et al . 2001 , , 122 , 2205 rudnick , g. , et al . 2003 , , 599 , 847 rudnick , g. , et al . 2006 , , 650 , 624 salpeter , e. e. 1955 , , 121 , 161 sandage , a. , tammann , g. a. , & yahil , a. 1979 , , 232 , 352 saracco , p. , giallongo , e. , cristiani , s. , dodorico , s. , fontana , a. , iovino , a. , poli , f. , & vanzella , e. 2001 , , 375 , 1 sawicki , m. , & thompson , d. 2005 , , 635 , 100 sawicki , m. , & thompson , d. 2006 , , 642 , 653 schechter , p. 1976 , , 203 , 297 schmidt , m. 1968 , , 151 , 393 shapley , a. e. , erb , d. k. , pettini , m. , steidel , c. c. , & adelberger , k. l. 2004 , , 612 , 108 shapley , a. e. , steidel , c. c. , adelberger , k. l. , dickinson , m. , giavalisco , m. , & pettini , m. 2001 , , 562 , 95 shapley , a. e. , steidel , c. c. , erb , d. k. , reddy , n. a. , adelberger , k. l. , pettini , m. , barmby , p. , & huang , j. 2005 , , 626 , 698 steidel , c. c. , adelberger , k. l. , giavalisco , m. , dickinson , m. , & pettini , m. 1999 , , 519 , 1 steidel , c. c. , adelberger , k. l. , shapley , a. e. , erb , d. k. , reddy , n. a. , & pettini , m. 2005 , , 626 , l44 steidel , c. c. , adelberger , k. l. , shapley , a. e. , pettini , m. , dickinson , m. , & giavalisco , m. 2003 , , 592 , 728 steidel , c. c. , giavalisco , m. , pettini , m. , dickinson , m. , & adelberger , k. l. 1996 , , 462 , l17 steidel , c. c. , shapley , a. e. , pettini , m. , adelberger , k. l. , erb , d. k. , reddy , n. a. , & hunt , m. p. 2004 , , 604 , 534 van dokkum , p. g. , et al . 2003 , , 587 , l83 van dokkum , p. g. , et al . 2004 , , 611 , 703 van dokkum , p. g. , et al . 2006 , , 638 , 59 wolf , c. , meisenheimer , k. , rix , h .- w . , borch , a. , dye , s. , kleinheinrich , m. 2003 , , 401 , 73 zucca , e. , et al . 2006 , , 445 , 879

we present the rest - frame optical ( @xmath0 , @xmath1 , and @xmath2 band ) luminosity functions ( lfs ) of galaxies at @xmath3 , measured from a @xmath4-selected sample constructed from the deep nir musyc , the ultradeep fires , and the goods - cdfs . this sample is unique for its combination of area and range of luminosities . the faint - end slopes of the lfs at @xmath5 are consistent with those at @xmath6 . the characteristic magnitudes are significantly brighter than the local values ( e.g. , @xmath7 mag in the @xmath2 band ) , while the measured values for @xmath8 are typically @xmath9 times smaller . the @xmath0-band luminosity density at @xmath10 is similar to the local value , and in the @xmath2 band it is @xmath11 times smaller than the local value . we present the lf of distant red galaxies ( drgs ) , which we compare to that of non - drgs . while drgs and non - drgs are characterized by similar lfs at the bright end , the faint - end slope of the non - drg lf is much steeper than that of drgs . the contribution of drgs to the global densities down to the faintest probed luminosities is 14%-25% in number and 22%-33% in luminosity . from the derived rest - frame @xmath12 colors and stellar population synthesis models , we estimate the mass - to - light ratios ( @xmath13 ) of the different subsamples . the @xmath13 ratios of drgs are @xmath9 times higher ( in the @xmath2 and @xmath1 bands ) than those of non - drgs . the global stellar mass density at @xmath3 appears to be dominated by drgs , whose contribution is of order @xmath1460%-80% of the global value . qualitatively similar results are obtained when the population is split by rest - frame @xmath12 color instead of observed @xmath15 color .

the jaynes cummings ( jc ) model is a fundamental building block in quantum optics ; it describes the interaction of a qubit with a quantum electromagnetic field under long wave and rotating wave approximations . it is exactly solvable @xcite and has proven useful to describe phenomena as rabi oscillations @xcite and collapse and revivals of the atomic inversion @xcite , among others ; see @xcite for a review on the model . if the number of qubits increases , the model , known as the dicke or tavis cummings model , shows many - body phenomena in the form of a superradiant phase @xcite . the dicke model is also exactly solvable @xcite and is known to show super - fluorescence and amplified spontaneous emission ; see @xcite for a recent review . in recent years , a general dicke hamiltonian , including quadratic self - interactions on both the field and qubit ensemble was introduced to study the effect of the nonlinearities and their relation to the stark shift , in units of @xmath0 , @xmath1 in this model the frequencies for the field and two - level system transitions are given by @xmath2 and @xmath3 , the quadratic interactions are assumed to be equal and given by @xmath4 , while the coupling between field and qubit is given by the parameter @xmath5 . an exact solution to this system was found by quantum inverse methods involving bethe anzats @xcite . the importance of the dicke model and its generalizations lies in its ability to describe more than atoms interacting with the quantized field of a cavity ; i.e. lasers . for example , it may describe open dynamical cavity - qed systems @xcite , ion trap systems @xcite , circuit - qed systems @xcite , and bose - einstein condensates interacting with classical or quantized electromagnetic fields @xcite . in this contribution , we present an exact solution , up to the roots of a polynomial , to a more general dicke hamiltonian by considering non - identical nonlinear interactions in ( [ eq : hbogoliubov ] ) . in the following , we will discuss our general dicke hamiltonian and the physical systems it can describe . we then show how a novel right unitary transform involving susskind glogower operators helps us diagonalize the hamiltonian in the field basis . with this at hand , it is simple to diagonalize the resulting tridiagonal hamiltonian in the dicke basis via orthogonal polynomials satisfying a three - term recurrence relation . in order to verify the validity of our exact solution , we recover the time evolution for a system involving just the single qubit . finally , we study the time evolution of different ensemble sizes to illustrate the simplicity of our approach and the results it yields ; we focus on the population inversion dynamics of the qubit ensemble as well as the evolution of the entropy and q - function of the field . let us consider a system composed by an ensemble of @xmath6 identical two - level systems ( ` qubits ' ) that interact with each other . these qubitas are in the presence of a quantized field and a kerr medium . for the sake of simplicity , we move into the frame defined by the transformation @xmath7 , where the excitation number operator is given by @xmath8 , and work with the hamiltonian in units of @xmath0 , @xmath9 the qubits ensemble is described by collective dicke operators satisfying the @xmath10 algebra , @xmath11 = 2 \hat{j}_{z}$ ] , @xmath12 = \pm \hat{j}_{\pm}$ ] , while the annihilation and creation operators for a single mode field satisfy @xmath13=1 $ ] . the transition frequency of each qubit , @xmath14 , and the frequency of the field , @xmath15 , are summarized by the detuning @xmath16 . the kerr medium is described by the parameter @xmath17 , while the qubit - qubit and ensemble - field couplings are given by @xmath4 and @xmath18 , in that order . the hamiltonian ( [ eq : hamiltonian ] ) describes the @xmath6-atom maser in general . in the special case of equal self - interactions , @xmath19 , it can be transformed into the @xmath6-atom maser including , kerr nonlinearity and stark shift as discussed in @xcite . different parameter sets describe particular physical models ; e.g.,@xmath20 delivers the kerr model @xcite , @xmath21 yield the dicke or tavis cummings model @xcite and @xmath22 gives the micromaser with kerr nonlinearity @xcite . furthermore , the general hamiltonian ( [ eq : hamiltonian ] ) and its reductions are experimentally feasible in cavity- and circuit - qed as well as trapped ions . it may also be possible to realize some of these models with two - mode bose - einstein condensates coupled to radiation fields @xcite . the case of equal - self interactions , @xmath19 , has been solved by inverse quantum methods in the past @xcite . this solution involves the bethe ansatz method . the general hamiltonian ( [ eq : hamiltonian ] ) can also be solved by extending our right unitary approach to the quantum landau zener problem for a single two - level system presented in @xcite , which delivers an evolution operator with the form @xmath23 where the auxiliary hamiltonians are given by @xmath24 where the ket @xmath25 is a dicke state , the operator @xmath26 is the density matrix for the pure state of the field with @xmath27 photons , the operator @xmath28 is the photon number operator and the symbol @xmath29 represents kronecker delta . these auxiliary hamiltonians are diagonal in the field basis ; i.e. they are given in terms of the photon number functions @xmath30^{1/2 } \left [ \hat{n } + 1 + \frac{n}{2 } - j \right]^{1/2}.\end{aligned}\ ] ] there is , however , a simpler approach to solve this general radiation - matter interaction model . in order to present a simpler approach to solve hamiltonian ( [ eq : hamiltonian ] ) , let us define the right unitary transformation @xmath31 where we have used the susskind glogower operators , @xmath32 which act as lowering and raising ladder operators on the fock state basis , @xmath33 and @xmath34 in that order , and are right - unitary , @xmath35 and @xmath36 , where @xmath26 is the density matrix for the pure state of the field with @xmath27 photons . again , the ket @xmath25 is a dicke or angular momentum state . then , it is possible to write the general hamiltonian ( [ eq : hamiltonian ] ) as : @xmath37 where the auxiliary hamiltonian is given by , @xmath38 we have used the notation @xmath39 to bring forward that this hamiltonian is _ semi - classical_-like because it is only expressed in terms of the number operator , @xmath40^{1/2 } \left [ \hat{n } + 1 - \frac{n}{2 } -j \right]^{1/2}. \end{aligned}\ ] ] it is possible to express the dynamics of this model as the evolution operator @xmath41 where powers of the form @xmath42 are needed . these powers can be obtained by realizing from ( [ eq : rightunitary ] ) and ( [ eq : scham ] ) that @xmath43 leads to @xmath44 by means of @xmath45 and @xmath46 for @xmath47 and @xmath48 . thus , the evolution operator in the reduced form is given by the expression @xmath49 the hamiltonian @xmath39 is diagonal in the field basis and is symmetric tridiagonal in the dicke basis ; i.e. it is diagonalizable in the angular momentum basis . the eigenvalues of this hamiltonian can be found by the method of minors and are given by the roots of the characteristic polynomial @xmath50 p_{n}\left(\nu\right ) - g^2\left(-\frac{n}{2}+1,\hat{n}\right ) p_{n-1}\left(\nu\right)\end{aligned}\ ] ] with @xmath51 p_{j-1}\left(\nu\right ) + \nonumber \\ & & - g^2\left(\frac{n}{2 } + 2 -j , \hat{n}\right ) p_{j-2}\left(\nu\right ) , \quad j \ge 2 \nonumber \\\end{aligned}\ ] ] the eigenvectors are calculated from the eigenvalue equation for the hamiltonian and give @xmath52 where the amplitudes answer to the following recurrence relations , @xmath53 c_{\frac{n}{2}}^{\left(j\right ) } + g\left ( \frac{n}{2 } , \hat{n } \right ) c_{\frac{n}{2 } - 1}^{\left(j\right)}&= & 0 , \\ \left [ f\left(j , \hat{n } \right ) - \nu_{j } \right ] c_{k}^{\left(j\right ) } + g \left(j , \hat{n } \right ) c_{k- 1}^{\left(j\right ) } + g \left(j+1 , \hat{n } \right ) c_{k + 1}^{\left(j\right ) } & = & 0 , \\ \left [ f\left(-\frac{n}{2 } , \hat{n } \right ) - \nu_{j } \right ] c_{-\frac{n}{2}}^{\left(j\right ) } + g\left(-\frac{n}{2}+1 , \hat{n } \right ) c_{\frac{n}{2 } + 1}^{\left(j\right)}&= & 0 . \end{aligned}\ ] ] the time evolution given in the previous section accounts for the full dynamics of the system and helps calculating any given quantity of interest . as an example , we will focus on the time evolution of the reduced density matrix for the field where the initial state is given by a pure state @xmath54 , @xmath55 the notation @xmath56 is used to describe the components of the _ semi - classical _ time evolution operator . this allows us to calculate the mean photon number evolution , @xmath57 and in consequence the population inversion @xmath58 . other interesting quantities are the purity of the field , @xmath59 and von neumann entropy , @xmath60 , \end{aligned}\ ] ] which are a good measure of the degree of mixedness of the reduced system . let us consider a system with just the single qubit , @xmath61 the _ semi - classical _ hamiltonian is given by @xmath62 and it is possible to give a closed form time evolution operator as @xmath63 } \left\ { \cos \frac{\omega(\hat{n } ) t}{2 } - \frac{i \left[\beta(\hat{n } ) \hat{\sigma}_{z } + 2 \lambda \sqrt{n } \hat{\sigma}_{x } \right]}{\omega(\hat{n } ) } \sin \frac{\omega(\hat{n})t}{2 } \right\},\\ \beta(\hat{n } ) & = & \delta + \kappa \left(1 - 2 \hat{n } \right ) , \\ \omega(\hat{n } ) & = & \sqrt { \left [ \beta(\hat{n } ) \right]^2 + 4 \hat{n } \lambda^2 } \end{aligned}\ ] ] it is trivial to apply the operator @xmath64 ( @xmath65 ) to any given initial state ket ( bra ) and then apply the _ semi - classical _ exponential . figure [ fig : fig1 ] shows the time evolution of the mean population inversion ( first row ) , entropy of the reduced field ( second row ) and husimi s q function of the field ( third row ) for a single qubit as given by a jaynes cummings model ( left column ) and a jaynes cummings kerr model ( right column ) . our results are in accordance with those in the literature @xcite and we can proceed to sample the dynamics of ensembles . , under the jaynes - cummings model , left column ( a , c , e , f ) , and under a jaynes - cummings - kerr model , right column ( b , d , g , h ) . the initial state for both cases is @xmath66 with @xmath67 . ] for an ensemble of qubits , the task of finding a closed form expression for the time evolution becomes cumbersome but it is possible to numerically diagonalize the _ semi - classical _ hamiltonian and implement the time evolution of any given initial state . as an example , we consider the evolution of ensembles of three , fig . [ fig : fig2 ] , and twenty five , fig . [ fig : fig3 ] , qubits . the information about the particular initial conditions and parameter values can be found in the figures and their captions . at the time , it is not our goal to report and in - depth analysis of the dynamics of generalized dicke models but just to present our diagonalization scheme to obtain an exact solution via susskind - glogower operators . for this reason , we will just briefly comment some basic characteristics of the dynamics . by considering an initial state given by the separable state consisting of a coherent field and the ensemble in its ground state , @xmath68 , it is possible to see that the dicke model presents strong collapse and revivals of the population inversion as long as the mean photon number is larger than the number of qubits in the system . a clear collapse of the population inversion is seen in any case studied here , up to @xmath69 . the strength of the oscillations in the population inversion diminishes as the number of qubits in the system gets close to the mean photon number of the coherent state but they become ever - present at smaller times as we get larger ensemble sizes for a fixed value of the coherent state parameter . meanwhile , the purity and entropy of such a dicke model signals an ever - present entangled state between the field and the ensemble as the number of qubits gets close to or equal to the mean number of photons ; i.e. the plots change from strong , well - defined , unmodulated dips in the functions to a strongly modulated flat - liner close to the value of a mixed reduced density matrix @xcite . the q - function for the reduced field behaves as expected . for @xmath70 , @xmath71 well - defined phase blobs appear and evolve half of them clock - wise and the other half counter - clock - wise as time goes by . the revivals in the population inversion are associated to the overlapping of these phase blobs ; a stronger revival corresponding to a better overlapping . however , when an interacting ensemble of qubits is considered under dicke kerr dynamics , the collapse and revivals of the population inversion are always weak but well defined and periodical . purity and entropy functions point a return to a quasi - separable state on the first revival for the cases analyzed with the number of qubits less or equal to the mean photon number of the field . the mean value of these functions gradually increases with time and some dips appear periodically due to the constructive interference of the wavefunction components , leading to revivals in the population inversion . under dicke kerr dynamics the phase blobs seem heavily defined by the kerr process and for @xmath67 four phase blobs appear and two of them evolve clockwise while the other two do it counter - clockwise . this process produces an overlap of two and two of the phase blobs leading to a weak local minimum in the purity / entropy but does not register in the population inversion . it is only when the four phase blobs overlap that a pronounced local minimum and a revival of the population inversion appears . , with three qubits under the dicke model , left column ( a , c , e , f ) , and with three interacting qubits under a dicke - kerr model , right column ( b , d , g , h ) . the initial state for both cases is @xmath72 with @xmath67 . ] , with twenty five qubits under the dicke model , left column ( a , c , e , f ) , and with twenty five interacting qubits under a dicke - kerr model , right column ( b , d , g , h ) . the initial state for both cases is @xmath73 with @xmath67 . ] we have considered the general @xmath6-atom maser model which can be described by the dicke model plus dipople dipole interactions and kerr nonlinearity . as a side result , we extend a previous result based on susskind glogower operators that gives the exact dynamics of a jaynes cummings model as the product of two time evolution operators . our main result is a different and simpler approach involving susskind glogower operators and right unitary transformations that allow us to represent our generalized dicke model as a transformed _ semi - classical_-like hamiltonian which is diagonal in the field basis and tridiagonal in the dicke basis ; thus , the diagonalization of this _ semi - classical _ hamiltonian is known up to the roots of its characteristic polynomial . the transformed _ semi - classical_-like hamiltonian gives the time evolution of the system and provides access to the dynamics of any quantity of interest . we use our result to derive a closed analytical form for the time evolution operator of a single qubit interacting with a quantized field in the presence of a kerr medium , a jaynes - cummings - kerr model . also , we present the time evolution of the population inversion , reduced field entropy and husimi s q - function of the field for ensembles consisting of three and twenty - five interacting two - level systems under a dicke - kerr model where the interaction and kerr nonlinearity are equal . this is done to show how simple it is to deal with many atoms with our partial diagonalization approach . it is possible that one could follow the dynamics of hundreds and maybe a few thousands of qubits with our approach in a simple workstation with efficient programming ; e.g. , this is of importance in the description of realistic micromasers and may be relevant to the study of fields interacting with bose - einstein condensates in the two - mode approximation . some systems , e.g. circuit - qed and open - dynamical systems , may deliver a strong coupled version of the general dicke hamiltonian in ( [ eq : hamiltonian ] ) , @xmath74 notice that the @xmath75 term @xcite has been kept for the sake of generality . the presence of the strong interaction term deters the use of the approach presented above . here , we want to show two things . the first is that we can get rid of the second order nonlinearity , @xmath76 , if it is weak compared to the frequency of the field . this allows us to use a squeezed states basis for the field , described by the transformation , @xmath77 that helps us get rid of the @xmath76 term . the second thing we want to show is that a _ small rotation _ @xcite , @xmath78 has an effect similar to that of the rotating wave approximation . this _ small rotation _ leads to just a dicke hamiltonian including a kerr medium and dipole - dipole interactions between the qubit ensemble components , @xmath79 after we have moved to a frame defined by the total excitation number @xmath80 rotating at the frequency of the field and defined the parameters @xmath81 , @xmath82 and @xmath83 . note that we have taken the self - interaction nonlinearities @xmath17 and @xmath84 a couple orders of magnitude smaller than the transition frequency @xmath14 in order to neglect products of couplings and nonlinearities . we want to emphasize that , while we can not deal with the strong - coupling regime , this _ small rotation _ method may be valid in the regime where phase transitions appear @xmath85 @xcite . 10 url # 1#1urlprefix[2][]#2 jaynes e t and cummings f w 1963 _ proc . ieee _ * 51 * 89 109 rodrguez - lara b m and lee r k 2012 classical dynamics of a two - species bose - einstein condensate in the presence of nonlinear maser processes _ spontaneous symmetry breaking , self - trapping , and josephson oscillations in nonlinear systems _ progress in optical science and photonics ( springer berlin heidelberg )

we show a right unitary transformation approach based on susskind glogower operators that diagonalizes a generalized dicke hamiltonian in the field basis and delivers a tridiagonal hamiltonian in the dicke basis . this tridiagonal hamiltonian is diagonalized by a set of orthogonal polynomials satisfying a three - term recurrence relation . our result is used to deliver a closed form , analytic time evolution for the case of a jaynes cummings kerr model and to study the time evolution of the population inversion , reduced field entropy , and husimi s q - function of the field for ensembles of interacting two - level systems under a dicke kerr model .

preprocessing and the analysis of preprocessed data are ubiquitous components of statistical inference , but their treatment has often been informal . we aim to develop a theory that provides a set of formal statistical principles for such problems under the banner of multiphase inference . the term `` multiphase '' refers to settings in which inferences are obtained through the application of multiple procedures in sequence , with each procedure taking the output of the previous phase as its input . this encompasses settings such as multiple imputation ( mi , @xcite ) and extends to other situations . in a multiphase setting , information can be passed between phases in an arbitrary form ; it need not consist of ( independent ) draws from a posterior predictive distribution , as is typical with multiple imputation . moreover , the analysis procedure for subsequent phases is not constrained to a particular recipe , such as rubin s mi combining rules ( @xcite ) . the practice of multiphase inference is currently widespread in applied statistics . it is widely used as an analysis technique within many publications any paper that uses a `` pipeline '' to obtain its final inputs or clusters estimates from a previous analysis provides an example . furthermore , projects in astronomy , biology , ecology , and social sciences ( to name a small sampling ) increasingly focus on building databases for future analyses as a primary objective . these projects must decide what levels of preprocessing to apply to their data and what additional information to provide to their users . providing all of the original data clearly allows the most flexibility in subsequent analyses . in practice , the journey from raw data to a complete model is typically too intricate and problematic for the majority of users , who instead choose to use preprocessed output . unfortunately , decisions made at this stage can be quite treacherous . preprocessing is typically irreversible , necessitating assumptions about both the observation mechanisms and future analyses . these assumptions constrain all subsequent analyses . consequently , improper processing can cause a disproportionate amount of damage to a whole body of statistical results . however , preprocessing can be a powerful tool . it alleviates complexity for downstream researchers , allowing them to deal with smaller inputs and ( hopefully ) less intricate models . this can provide large mental and computational savings . two examples of such trade - offs come from nasa and high - throughput biology . when nasa satellites collect readings , the raw data are usually massive . these raw data are referred to as the `` level 0 '' data ( @xcite ) . the level 0 data are rarely used directly for scientific analyses . instead , they are processed to levels 1 , 2 , and 3 , each of which involves a greater degree of reduction and adjustment . level 2 is typically the point at which the processing becomes irreversible . @xcite provide an excellent illustration of this process for the atmospheric infrared sounder ( airs ) experiment . this processing can be quite controversial within the astronomical community . several upcoming projects , such as the advanced technology solar telescope ( atst ) will not be able to retain the level 0 or level 1 data ( @xcite ) . this inability to obtain raw data and increased dependence on preprocessing has transformed low - level technical issues of calibration and reduction into a pressing concern . high - throughput biology faces similar challenges . whereas reproducibility is much needed ( e.g. , @xcite ) , sharing raw datasets is difficult because of their sizes . the situation within each analysis is similar . confronted with an overwhelming onslaught of raw data , extensive preprocessing has become crucial and ubiquitous . complex models for genomic , proteomic , and transcriptomic data are usually built upon these heavily - processed inputs . this has made the intricate details of observation models and the corresponding preprocessing steps the groundwork for entire fields . to many statisticians , this setting presents something of a conundrum . after all , the ideal inference and prediction will generally use a complete correctly - specified model encompassing the underlying process of interest and all observation processes . then , why are we interested in multiphase ? we focus on settings where there is a natural separation of knowledge between analysts , which translates into a separation of effort . the first analyst(s ) involved in preprocessing often have better knowledge of the observation model than those performing subsequent analyses . for example , the first analyst may have detailed knowledge of the structure of experimental errors , the equipment used , or the particulars of various protocols . this knowledge may not be easy to encapsulate for later analysts the relevant information may be too large or complex , or the methods required to exploit this information in subsequent analyses may be prohibitively intricate . hence , the practical objective in such settings is to enable the best possible inference given the constraints imposed and provide an account of the trade - offs and dangers involved . to borrow the phrasing of @xcite and @xcite , we aim for achievable practical efficiency rather than theoretical efficiency that is practically unattainable . multiphase inference currently represents a serious gap between statistical theory and practice . we typically delineate between the informal work of preprocessing and feature engineering and formal , theoretically - motivated work of estimation , testing , and so forth . however , the former fundamentally constrains what the latter can accomplish . as a result , we believe that it represents a great challenge and opportunity to build new statistical foundations to inform statistical practice . we present two examples that show both the impetus for and perils of undertaking multiphase analyses in place of inference with a complete , joint model . the first concerns microarrays , which allow the analysis of thousands of genes in parallel . we focus on expression microarrays , which measure the level of gene expression in populations of cells based upon the concentration of rna from different genes . these are typically used to study changes in gene expression between different experimental conditions . in such studies , the estimand of interest is typically the log - fold change in gene expression between conditions . however , the raw data consist only of intensity measurements for each probe on the array , which are grouped by gene along with some form of controls . these intensities are subject to several forms of observation noise , including additive background variation and additional forms of interprobe and interchip variation ( typically modeled as multiplicative noise ) . to deal with these forms of observation noise , a wide range of background correction and normalization strategies have been developed ( for a sampling , see @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ) . later analyses then focus on the scientific question of interest without , for the most part , addressing the underlying details of the observation mechanisms . background correction is a particularly crucial step in this process , as it is typically the point at which the analysis moves from the original intensity scale to the log - transformed scale . as a result , it can have a large effect on subsequent inferences about log - fold changes , especially for genes with low expression levels in one condition ( @xcite , @xcite ) . one common method ( mas5 ) , provided by one microarray manufacturer , uses a combination of background subtraction and truncation at a fixed lower threshold for this task ( @xcite ) . other more sophisticated techniques use explicit probability models for this de - convolution . a model with normally - distributed background variation and exponentially distributed expression levels has proven to be the most popular in this field ( @xcite , @xcite ) . unfortunately , even the most sophisticated available techniques pass only point estimates onto downstream analyses . this necessitates ad - hoc screening and corrections in subsequent analyses , especially when searching for significant changes in expression ( e.g. , @xcite ) . retaining more information from the preprocessing phases of these analyses would allow for better , simpler inference techniques with greater power and fewer hacks . the motivation behind the current approach is quite understandable : scientific investigators want to focus on their processes of interest without becoming entangled in the low - level details of observation mechanisms . nevertheless , this separation can clearly compromise the validity of their results . the role of preprocessing in microarray studies extends well beyond background correction . normalization of expression levels across arrays , screening for data corruption , and other transformations preceding formal analysis are standard . each technique can dramatically affect downstream analyses . for instance , quantile normalization equates quantiles of expression distributions between arrays , removing a considerable amount of information . this mutes systematic errors ( @xcite ) , but it can seriously compromise analyses in certain contexts ( e.g. , mirna studies ) . another example of multiphase inference can be found in the estimation of correlations based upon indirect measurements . this appears in many fields , but astrophysics provides one recent and striking case . the relationships between the dust s density , spectral properties , and temperature are of interest in studies of star - forming dust clouds . these characteristics shed light on the mechanisms underlying star formation and other astronomical processes . several studies ( e.g. , @xcite , @xcite , @xcite , @xcite ) have investigated these relationships , finding negative correlations between the dust s temperature and spectral index . this finding is counter to previous astrophysical theory , but it has generated many alternative explanations . such investigations may , however , be chasing a phantasm . these correlations have been estimated by simply correlating point estimates of the relevant quantities ( temperature @xmath0 and spectral index @xmath1 ) based on a single set of underlying observations . as a result , they may conflate properties of this estimation procedure with the underlying physical mechanisms of interest . this has been noted in the field by @xcite , but the scientific debate on this topic continues . @xcite provide a particularly strong argument , using a cohesive hierarchical bayesian approach , that improper multiphase analyses have been a pervasive issue in this setting . improper preprocessing led to incorrect , negative estimates of the correlation between temperature and spectral index , according to @xcite . these incorrect estimates even appeared statistically significant with narrow confidence intervals based on standard methods . on a broader level , this case again demonstrates some of the dangers of multiphase analyses when they are not carried out properly . those analyzing this data followed an intuitive strategy : estimate what we want to work with ( @xmath0 and @xmath1 ) , then use it to estimate the relationship of interest . unfortunately , such intuition is not a recipe for valid statistical inference . multiphase inference has wide - ranging connections to both the theoretical and applied literatures . it is intimately related to previous work on multiple imputation and missing data ( @xcite ( @xcite ) , @xcite , @xcite , @xcite ) . in general , the problem of multiphase inference can be formulated as one of missing data . however , in the multiphase setting , missingness arises from the preprocessing choices made , not a probabilistic response mechanism . thus , we can leverage the mathematical and computational methods of this literature , but many of its conceptual tools need to be modified . multiple imputation addresses many of the same issues as multiphase inference and is indeed a special case of the latter . concepts such as congeniality between imputation and analysis models and self - efficiency ( @xcite ) have natural analogues and roles to play in the analysis of multiphase inference problems . multiphase inference is also tightly connected to work on the comparison of experiments and approximate sufficiency , going back to @xcite ( @xcite ) and continuing through @xcite and @xcite , among others . this literature has addressed the relationship between decision properties and the probabilistic structure of experiments , the relationship between different notions of statistical information , and notions of approximate sufficiency all of these are quite relevant for the study of multiphase inference . we view the multiphase setting as an extension of this work to address a broader range of real - world problems , as we will discuss in section [ sec : riskmonotone ] . the literature on bayesian combinations of experts also informs our thinking on multiphase procedures . @xcite provides an excellent review of the field , while @xcite provides the core formalisms of interest for the multiphase setting . overall , this literature has focused on obtaining coherent ( or otherwise favorable ) decision rules when combining information from multiple bayesian agents , in the form of multiple posterior distributions . we view this as a best - case scenario , focusing our theoretical development towards the mechanics of passing information between phases . we also focus on the sequential nature of multiphase settings and the challenges this brings for both preprocessors and downstream analysts , in contrast to the more `` parallel '' or simultaneous focus of the literature mentioned above . there are also fascinating links between multiphase inference and the signal processing literature . there has been extensive research on the design of quantizers and other compression systems ; see for example @xcite . such work is often focused on practical questions , but it has also yielded some remarkable theory . in particular , the work of @xcite on the relationship between surrogate loss functions in quantizer design and @xmath2-divergences suggests possible ways to develop and analyze a wide class of multiphase procedures , as we shall discuss in section [ sec : future ] . to formalize the notion of multiphase inference , we begin with a formal model for two - phase settings . the first phase consists of the data generation , collection , and preprocessing , while the second phase consists of inference using the output from the first phase . we will call the first - phase agent the `` preprocessor '' and the second - phase agent the `` downstream analyst '' . the preprocessor observes the raw data @xmath3 . this is a noisy realization of @xmath4 , variables of interest that are not directly obtainable from a given experiment , e.g. , gene expression from sequencing data , or stellar intensity from telescopic observations . we assume that the joint density of @xmath4 and @xmath3 with respect to product measure @xmath5 can be factored as @xmath6 here , @xmath7 encapsulates the underlying process of interest and @xmath8 encapsulates the observation process . we assume that @xmath9 is of fixed dimension in all asymptotic settings . in practice , the preprocessor should be able to postulate a reasonable `` observation model '' @xmath10 , but will not always know the true `` scientific model '' @xmath11 . this is analogous to the mi setting , where the imputer does not know the form of the final analysis . from the original data generating process and outputs @xmath0 , with @xmath4 as missing data . the downstream analyst observes the preprocessor s output @xmath0 and has both @xmath4 and @xmath3 missing . ] using this model , the preprocessor provides the downstream analyst with some output @xmath12 , where @xmath13 is a ( possibly stochastic ) additional input . when @xmath14 is stochastic ( e.g. , an mcmc output ) , the conditional distribution @xmath15 is its theoretical description instead of its functional form . however , for simplicity , we will present our results when @xmath0 is a deterministic function of @xmath3 only , but many results generalize easily . given such @xmath0 , downstream analysts can carry out their inference procedures . figure [ fig : models ] depicts our general model setup . this model incorporates several restrictions . first , it is markovian with respect to @xmath3 , @xmath4 , and @xmath9 ; @xmath3 is conditionally independent of @xmath9 given @xmath4 ( and @xmath16 ) . second , the parameters governing the observation process ( @xmath16 ) and those governing the scientific process ( @xmath9 ) are distinct . in bayesian settings , we further assume that @xmath16 and @xmath9 are independent _ a priori_. the parameters @xmath16 are nuisance from the perspective of all involved ; the downstream analyst wants to draw inferences about @xmath4 and @xmath9 , and the preprocessor wants to pass forward information that will be useful for said inferences . if downstream inferences are bayesian with respect to @xmath16 , then @xmath17 ( which holds under ( [ e : model ] ) ) is sufficient for all inference under the given model and prior . hence , this conditional density is frequently of interest in our theoretical development , as is the corresponding marginalized model @xmath18 . we will compare results obtained with a fixed prior to those obtained in a more general setting to better understand the effects of nuisance parameters in multiphase inference . these restrictions are somewhat similar to those underlying rubin s ( @xcite ) definition of `` missing at random '' ; however , we do not have missing data mechanism ( mdm ) in this setting _ per se_. the distinction between missing and observed data ( @xmath4 and @xmath3 ) is fixed by the structure of our model . in place of mdm , we have two imposed patterns of missingness : one for the data - generating process , and one for the inference process . the first is @xmath10 , which creates a noisy version of the desired scientific variables . here , @xmath4 can be considered the missing data and @xmath3 the observed . for the inference process , the downstream analyst observes @xmath0 in place of @xmath3 but desires inference for @xmath9 based upon @xmath19 . hence , @xmath3 and @xmath4 are both missing for the downstream analyst . neither pattern is entirely intrinsic to the problem both are fixed by choice . the selection of scientific variables @xmath4 for a given marginal likelihood @xmath20 is a modeling decision . the selection of preprocessing @xmath21 is a design decision . this contrasts with the typical missing data setting , where mdm is forced upon the analyst by nature . with multiphase problems , we seek to design and evaluate engineered missingness . thus the investigation of multiphase inference requires tools and ideas from design , inference , and computation in addition to the established theory of missing data . with this model in place , we turn to formally defining multiphase procedures . this is more subtle than it initially appears . in the mi setting , we focus on complete - data procedures for the downstream analyst s estimation and do not restrict the dependence structure between missing data and observations . in contrast , we restrict the dependence structure as in ( [ e : model ] ) , but place far fewer constraints on the analysts procedures . here , we focus our definitions and discussion on the two - phase case of a single preprocessor and downstream analyst . this provides the formal structure to describe the interface between any two phases in a chain of multiphase analyses . in our multiphase setting , downstream analysts need not have any complete - data procedure in the sense of one for inferring @xmath9 from @xmath4 and @xmath3 ; indeed , they need not formally have one based only upon @xmath4 for inferring @xmath9 . we require only that they have a set of procedures for their desired inference using the quantities provided from earlier phases as inputs ( @xmath0 ) , not necessarily using direct observations of @xmath4 or @xmath3 . such situations are common in practice , as methods are often built around properties of preprocessed data such as smoothness or sparsity that need not hold for the actual values of @xmath4 . for the preprocessor , the input is @xmath3 and the output is @xmath0 . here @xmath0 could consist of a vector of means with corresponding standard errors , or , for discrete @xmath3 , @xmath0 could consist of carefully selected cross - tabulations . in general , @xmath0 clearly needs to be related to @xmath4 to capture inferential information , but its actual form is influenced by practical constraints ( e.g. , aggregation to lower than desired resolutions due to data storage capacity ) . for the downstream analyst , the input is @xmath0 and the output is an inference for @xmath22 . this analyst can obviously adapt . for example , suppose @xmath23 for each entry @xmath24 of @xmath4 . if the preprocessor provides @xmath25 , the analyst may simply use an unweighted mean to estimate @xmath9 . if the preprocessor instead gives the analyst @xmath26 , where @xmath27 contains standard errors , the latter could instead use a weighted mean to estimate @xmath9 . this adaptation extends to an arbitrary number of possible inputs @xmath28 , each of which corresponds to a set of constraints facing the preprocessor . to formalize this notion of adaptation , we first define an index set @xmath29 with one entry for each such set of constraints . this maps between forms of input provided by the preprocessor and estimators selected by the downstream analyst . in this way , @xmath29 captures the downstream analyst s knowledge of previous processing and the underlying probability model . thus , this index set plays an central role in the definition of multiphase inference problems , far beyond that of a mere mathematical formality ; it regulates the amount of mutual knowledge shared between the preprocessor and the downstream analyst . now , we turn to the estimators themselves . we start with point estimation as a foundation for a broader class of problems . testing begins with estimating rejection regions , interval estimation with estimating coverage , classification with estimating class membership , and prediction with estimating future observations and , frequently , intermediate parameters . the framework we present therefore provides tools that can be adapted for more than estimation theory . we define multiphase estimation procedures as follows : a _ multiphase estimation procedure _ @xmath30 is a set of estimators @xmath31 indexed by the set @xmath29 , where @xmath28 corresponds to the output of the @xmath32th first - phase method ; that is , @xmath30 is a family of estimators with different inputs . when clear , we will drop the subscripts @xmath32 and index the estimators in @xmath30 by their inputs . this definition provides enough flexibility to capture many practical issues with multiphase inference , and it can be iterated to define procedures for analyses involving a longer sequence of preprocessors and analysts . it also encompasses the definition of a missing data procedure used by @xcite . such procedures can not , of course , be arbitrarily constructed if they are to deliver results with general validity . hence , having defined these procedures , we will cull many of them from consideration in section [ sec : riskmonotone ] . the obvious choice of our estimand , suggested by our notation thus far , is the parameter for the scientific model , @xmath9 . this is very amenable to mathematical analysis and relevant to many investigations . hence , it forms the basis for our results in section [ sec : theory ] . however , for multiphase analyses , other classes of estimands may prove more useful in practice . in particular , functions of @xmath4 , future scientific variables @xmath33 , or future observations @xmath34 may be of interest . prediction of such quantities is a natural focus in the multiphase setting because such statements are meaningful to both the preprocessor and downstream analyst . such estimands naturally encompass a broad range of statistical problems including prediction , classification , and clustering . however , there is often a lack of mutual knowledge about @xmath35 , so the preprocessor can not expect to `` target '' estimation of @xmath9 in general , as we shall discuss in section [ sec : remarks ] . it is not automatic for multiphase estimation procedures to produce better results as the first phase provides more information . to obtain a sensible context for theoretical development , we must regulate the way that the downstream analyst adapts to different inputs . for instance , they should obtain better results ( in some sense ) when provided with higher - resolution information . this carries over from the mi setting ( @xcite , @xcite , @xcite , @xcite ) , where notions such as self - efficiency are useful for regulating the downstream analyst s procedures . we define a similar property for multiphase estimation procedures , but without restricting ourselves to the missing data setting . specifically , let @xmath36 indicate @xmath37 is a deterministic function of @xmath38 . in practice , @xmath37 could be a subvector , aggregation , or other summary of @xmath38 . a multiphase estimation procedure @xmath30 is _ risk monotone _ with respect to a loss function @xmath39 if , for all pairs of outputs @xmath40 , @xmath36 implies @xmath41 . an asymptotic analogue of risk monotonicity is defined as would be expected , scaling the relevant risks at an appropriate rate to obtain nontrivial limits . this is a natural starting point for regulating multiphase estimation procedures ; stronger notions may be required for certain theoretical results . note that this definition does not require that `` higher - quality '' inputs necessarily lead to lower risk estimators . risk monotonicity requires only that estimators based upon a larger set of inputs perform no worse than those with strictly less information ( in a deterministic sense ) . however , risk monotonicity is actually quite tight in another sense . it requires that additional information can not be misused by the downstream analyst , imposing a strong constraint on mutual knowledge . for an example , consider the case of unweighted and weighted means . to obtain better results when presented with standard errors , the downstream analyst must know that they are being given ( the correct ) standard errors and to weight by inverse variances . this definition is related to the comparison of experiments , as explored by @xcite ( @xcite ) , but diverges on a fundamental level . our ordering of experiments , based on deterministic functions , is more stringent than that of @xcite , but they are related . indeed , our @xmath42 relation implies that of @xcite . in the latter work , an experiment @xmath43 is defined as more informative than experiment @xmath1 , denoted @xmath44 , if all losses attainable from @xmath1 are also attainable from @xmath43 . this relation is also implied when @xmath43 is sufficient for @xmath1 . our stringency stems from our broader objectives in the multiphase setting . from a decision - theoretic perspective , the partial ordering of experiments investigated by blackwell and others deal with which risks are attainable given pairs of experiments , allowing for arbitrary decision procedures . in contrast , our criterion restricts procedures based on whether such risks are actually attained , with respect to a particular loss function . this is because , in the multiphase setting , it is not generally realistic to expect downstream analysts to be capable of obtaining optimal estimators for all forms of preprocessing . the conceptually - simplest way to generate such a procedure is to begin with a complete probability model for @xmath45 . under traditional asymptotic regimes , all procedures consisting of bayes estimators based upon such a model will ( with full knowledge of the transformations involved in each @xmath28 and a fixed prior ) be risk monotone . the same is true asymptotically under the same regimes ( for squared - error loss ) for procedures consisting of mles under a fixed model . under some other asymptotic regimes , however , these principles of estimation do not guarantee risk - monotonicity ; we explore this further in section [ sec : missinfo ] . but such techniques are not the only way to generate risk monotone procedures from probability models . this is analogous to self - efficiency , which can be achieved by procedures that are neither bayesian nor mle ( @xcite , @xcite ) . and @xmath38 form the basis set of statistics . each of these has three descendants ( @xmath46 from @xmath37 and @xmath47 from @xmath38 ) . these descendants are deterministic functions of their parent , but they are not deterministic functions of any other basis statistics . given correctly - specified models for @xmath37 and @xmath38 , a risk monotone procedure can be constructed for all statistics ( @xmath48 ) shown here as described in the text . ] a risk monotone procedure can be generated from any set of probability models for distinct inputs that `` span '' the space of possible inputs . suppose that an analyst has a set of probability models , all correctly specified , for @xmath49 , where @xmath50 ranges over a subset @xmath51 of the relevant index set @xmath29 . we also assume that this analyst has a prior distribution @xmath52 for each such basis models . these priors need not agree between models ; the analyst can build a risk - monotone procedure from an inconsistent set of prior beliefs . suppose that the inputs @xmath53 are not deterministic functions of each other and all other inputs can be generated as nontrivial deterministic transformations of one of these inputs . formally , we require @xmath54 for all distinct @xmath55 and , for each @xmath56 there exists a unique @xmath57 such that @xmath58 ( each output is uniquely descended from a single @xmath59 ) , as illustrated in figure [ fig : risk - monotone ] . this set can form a basis , in a sense , for the given procedure . using the given probability models with a single loss function and set of priors ( potentially different for each model ) , the analyst can derive a bayes rule under each model . for each @xmath57 , we require @xmath60 to be an appropriate bayes rule on said model . as @xmath61 for some function @xmath62 , we then have the implied @xmath63 , yielding the bayes rule for estimating @xmath9 based on @xmath28 , which is no less risky than @xmath60 . the requirement that each output @xmath28 derives from a unique @xmath59 means that each basis component @xmath59 has a unique line of descendants . within each line , each descendant is comparable to only a single @xmath59 in the sense of deterministic dependence . between these lines , such comparisons are not possible . this ensures the overall risk - monotonicity . biology provides an illustration of such bases . a wide array of methodological approaches have been used to analyze high - throughput gene expression data . one approach , builds upon order and rank statistics ( @xcite , @xcite , @xcite ) . another common approach uses differences in gene expression between conditions or experiments , often aggregating over pathways , replicates , and so forth . each class of methods is based upon a different form of preprocessing : ranks transformations for the former , normalization and aggregation for the latter . taking procedures based on rank statistics and aggregate differences in expression as a basis , we can consider constructing a risk - monotone procedure as above . thus , the given formulation can bring together apparently disparate methods as a first step in analyzing their multiphase properties . such constructions are , unfortunately , not sufficient to generate all possible risk monotone procedures . obtaining more general conditions and constructions for risk monotone procedures is a topic for further work . by casting the examples in section [ sec : examples ] into the formal structure just established , we can clarify the practical role of each mathematical component and see how to map theoretical results into applied guidance . we also provide an example that illustrates the boundaries of the framework s utility , and another that demonstrates its formal limits . these provide perspective on the trade - offs made in formalizing the multiphase inference problem . the case of microarray preprocessing presented previously fits quite nicely into the model of section [ sec : model ] . there , @xmath3 corresponds to the observed probe - level intensities , @xmath4 corresponds to the true expression level for each gene under each condition , and @xmath9 corresponds to the parameters governing the organism s patterns of gene expression . in the microarray setting , @xmath8 would characterize the relationship between expression levels and observed intensities , governed by @xmath16 . these nuisance parameters could include chip - level offsets , properties of any additive background , and the magnitudes of other sources of variation . the assumptions of a markovian dependence structure and distinct parameters for each part of the model appear quite reasonable in this case , as ( 1 ) the observation @xmath3 can only ( physically ) depend upon the sample preparation , experimental protocol , and rna concentrations in the sample and ( 2 ) the distributions @xmath7 and @xmath8 capture physically distinct portions of the experiment . background correction , normalization , and the reduction of observations to log - fold changes are common examples of preprocessing @xmath21 . as discussed previously , estimands based upon @xmath4 may be of greater scientific interest than those based upon @xmath9 . for instance , we may want to know whether gene expression changed between two treatments in a particular experiment ( a statement about @xmath4 ) than whether a parameter regulating the overall patterns of gene expression takes on a particular value . for the astrophysical example , the fit is similarly tidy . the raw astronomical observations correspond to @xmath3 , the true temperature , density , and spectral properties of each part of the dust cloud become @xmath4 , and the parameters governing the relationship between these quantities ( e.g. , their correlation ) form @xmath9 . the @xmath8 distribution governs the physical observation process , controlled by @xmath16 . this process typically includes the instruments response to astronomical signals , atmospheric distortions , and other earthbound phenomena . as before , the conditional independence of @xmath9 and @xmath3 given @xmath4 and @xmath16 is sensible based upon the problem structure , as is the separation of @xmath9 and @xmath16 . here @xmath4 corresponds to signals emitted billions or trillions of miles from earth , whereas the observation process occurs within ground- or space - based telescopes . hence , any non - markovian effects are quite implausible . preprocessing @xmath21 corresponds to the ( point ) estimates of temperature , density , and spectral properties from simple models of @xmath3 given @xmath4 and @xmath16 . the multiphase framework encompasses a broad range of settings , but it does not shed additional light on all of them . if @xmath0 is a many - to - one transformation of @xmath3 , then our framework implies that the preprocessor and downstream analyst face structurally different inference ( and missing data ) problems . this is the essence of multiphase inference , in our view . settings where @xmath64 is degenerate or @xmath0 is a one - to - one function of @xmath3 are boundary cases where our multiphase interpretation and framework add little . for a concrete example of these cases , consider a time - to - failure experiment , with the times of failure @xmath65 , @xmath66 . now , suppose that the experimenters actually ran the experiment in @xmath67 equally - sized batches . they observe each batch only until its first failure ; that is , they observe and report @xmath68 for each batch @xmath50 . subsequent analysts have access only to @xmath69 . this seems to be a case of preprocessing , but it actually resides at the very edge of our framework . we could take the complete observations to be @xmath4 and the batch minima to be @xmath3 . this would satisfy our markov constraint , with a singular , and hence deterministic , observation process @xmath70 simply selecting a particular order statistic within each batch . however , @xmath21 is one - to - one ; the preprocessor observes only the order statistics , as does the downstream analyst . there is no separation of inference between phases ; the same quantities are observed and missing to both the preprocessor and the downstream analyst . squeezing this case into the multiphase framework is technically valid but unproductive . the framework we present is not , however , completely generic . consider a chemical experiment involving a set of reactions . the underlying parameters @xmath9 describe the chemical properties driving the reactions , @xmath4 are the actual states of the reaction , and @xmath3 are the ( indirectly ) measured outputs of the reactions . the measurement process for these experiments , as described by @xmath64 , could easily violate the structure of our model in this case . for instance , the same chemical parameters could affect both the measurement and reaction processes , violating the assumed separation of @xmath9 and @xmath16 . even careful preprocessing in such a setting can create a fundamental incoherence . suppose the downstream analysis will be bayesian , so the preprocessor provides the conditional density of @xmath3 as a function of @xmath4 , latexmath:[${p_y}(y @xmath16 share components , and the preprocessor uses their prior on @xmath16 to create @xmath70 , the conditional density need not be sufficient for @xmath9 under the downstream analyst s model . because the downstream analyst s prior on @xmath9 need not be compatible with the preprocessor s prior on @xmath16 , inferences based on the preprocessor s @xmath70 can be seriously flawed in this setting . hence , we exclude such cases from our investigation for the time being . thinking bayesianly , our model ( [ e : model ] ) obviously does not exclude the possibility that the downstream analyst has more knowledge about @xmath9 than the preprocessor in the form of a prior on @xmath22 . however , _ prior _ information means that it is based on studies that do not overlap with the current one . probabilistically speaking , this means that our model permits the downstream analyst to formally incorporate another data set @xmath72 , as long as @xmath72 is conditionally independent of the scientific variables @xmath4 and observations @xmath3 given @xmath73 or @xmath9 . for example , the downstream analyst could observe completely separate experiments pertaining to the same underlying process governed by @xmath9 or the outcomes of separate calibration pertaining to @xmath16 , but not additional replicates governed by the same realization of @xmath4 . in a biological setting , this means that the downstream analyst could have access to results from samples not available to the preprocessor ( e.g. , biological replicates ) , possibly using the same equipment ; however , they could not have access to additional analyses of the same biological sample ( e.g. , technical replicates ) , as a single biological sample would typically correspond to a single realization of @xmath4 . these examples remind us that our multiphase setting does not encompass all of statistical inference . this is quite a relief to us . our work aims to open new directions for statistical research , but it can not possibly address every problem under the sun ! multiphase theory hinges on procedural constraints . consider , for example , finding the optimal multiphase estimation procedure in terms of the final estimator s bayes risk . without stringent procedural constraints , the result is trivial : compute the appropriate bayes estimator using the distribution of @xmath0 given @xmath9 . similarly , the optimal preprocessing @xmath0 will , without tight constraints , simply compute an optimal estimator using @xmath3 and pass it forward . note that both of these cases respect risk - monotonicity to the letter ; it is not sufficiently tight to enable interesting , relevant theory . more constraints , based upon careful consideration of applied problems , are clearly required . this is not altogether bad news . we need only look to the history of multiple imputation to see how rich theory can arise from stringent , pragmatic constraints . multiple imputation forms a narrow subset of multiphase procedures : @xmath4 corresponds to the complete data ( @xmath74 , in mi notation ) , @xmath3 corresponds to the observed data @xmath75 and missing data indicator @xmath76 , and @xmath0 usually consists of posterior predictive draws of the missing data together with the observed data . the markovian property depicted in figure [ fig : models ] holds when the parameter ( @xmath77 ) for the missing data mechanism @xmath78 ) is distinct from the parameter of interest ( @xmath22 ) in @xmath79 , which is a common assumption in practice . the second - phase procedure is then restricted to repeatedly applying a complete - data procedure and combining the results . these constraints were originally imposed for practical reasons in particular , to make the resulting procedure feasible with existing software . however , they have opened the door to deep theoretical investigations . in that spirit , we consider two types of practically - motivated constraints for multiphase inference : restrictions on the downstream analyst s procedure and restrictions on the preprocessor s methods . these constraints are intended to work in concert with coherence conditions ( e.g. , risk monotonicity ) , not in isolation , to enable meaningful theory . constraints on the downstream analyst are intended to reflect practical limitations of their analytic capacity . examples include restricting the downstream analyst to narrow classes of estimators ( e.g. , linear functions of preprocessed inputs ) , to specific principles of estimation ( e.g. , mles ) , or to special cases of a method we can reasonably assume the downstream analyst could handle , such as a complete - data estimator @xmath80 , available from software with appropriate inputs . estimators derived from nested families of models are often suitable for this purpose . for example , whereas @xmath80 may involve only an ordinary regression , the computation of @xmath81 may require a weighted least - squares regression.=1 another constraint on the downstream analyst pertains to nuisance parameters . such constraints are of great practical and theoretical interest , as we believe that the preprocessor will typically have better knowledge and statistical resources available to address nuisance parameters than the downstream analyst . an extreme but realistic case of this is to assume that the downstream analyst can not address nuisance parameters at all . as we shall discuss in section [ sec : theory ] , this would force the preprocessor to either marginalize over the nuisance parameters , find a pivot with respect to them , or trust the downstream analyst to use a method robust to the problematic parameters . turning to the preprocessor , we consider restricting either the form of the preprocessor s output or the mechanics of their methods . in the simplest case of the former , we could require that @xmath0 consist of the posterior mean ( @xmath82 ) and posterior covariance ( @xmath83 ) of the unknown @xmath4 under the preprocessor s model . a richer , but still realistic , class of output would be finite - dimensional real or integer vectors . restricting output to such a class would prevent the preprocessor from passing arbitrary functions onto the downstream analyst . this leads naturally to the investigation of ( finite - dimensional ) approximations to the preprocessor s conditional density , aggregation , and other such techniques . on the mechanical side , we can restrict either the particulars of the preprocessor s methods or their broader properties . examples of the former include particular computational approximations to the likelihood function or restrictions to particular principles of inference ( e.g. , summaries of the likelihood or posterior distribution of @xmath4 given @xmath84 ) . such can focus our inquiries to specific , feasible methods of interest or reflect the core statistical principles we believe the preprocessor should take into account . in a different vein , we can require that preprocessor s procedures be distributable across multiple researchers , each with their own experiments and scientific variables of interest . such settings are of interest for both the accumulation of scientific results for later use and for the development of distributed statistical computation . this leads to preprocessing based upon factored `` working '' models for @xmath4 , as we explore further in section [ sec : sufficiency ] . nuisance parameters play an important role in these constraints , narrowing the class of feasible methods ( e.g. , marginalization over such parameters may be exceedingly difficult ) and largely determining the extent to which preprocessing can be distributed . we explore these issues in more detail throughout section [ sec : theory ] . we now present a few steps towards a theory of multiphase inference . in this , we endeavor to address three basic questions : ( 1 ) how can we determine what to retain , ( 2 ) what limits the performance of multiphase procedures , and ( 3 ) what are some minimal requirements for being an ideal preprocessor ? we find insight into the first question from the language of classical sufficiency . we leverage and specialize results from the missing - data literature to address the second . for the third question , we turn to the tools of decision theory . suppose we have a group of researchers , each with their own experiments . they want to preprocess their data to reduce storage requirements , ease subsequent analyses , and ( potentially ) provide robustness to measurement errors . this group is keenly aware of the perils of preprocessing and want to ensure that the output they provide will be maximally useful for later analyses . their question is , `` which statistics should we retain ? '' if each of these researchers was conducting the final analysis themselves , using only their own data , they would be in a single - phase setting . the optimal strategy then is to keep a minimal sufficient statistic for each researcher s model . similarly , if the final analysis were planned and agreed upon among all researchers , we would again have a single - phase setting , and it is optimal to retain the sufficient statistics for the agreed - upon model . we use the term _ optimal _ here because it achieves maximal data reduction without losing information about the parameters of interest . such lossless compression in the general sense of avoiding statistical redundancy is often impractical , but it provides a useful theoretical gold standard . in the multiphase setting , especially with multiple researchers in the first phase , achieving optimal preprocessing is far more complicated even in theory . if @xmath21 is the output of the _ entire _ preprocessing phase , then in order to retain all information we must require @xmath21 to be a sufficient statistics for @xmath85 under model ( [ e : model ] ) ; that is , @xmath86 where @xmath39 denotes a likelihood function ; or at least in the ( marginal ) bayesian sense , @xmath87 where @xmath88 is the posterior of @xmath22 given data @xmath89 with the likelihood given by ( [ e : model ] ) . note that ( [ e : con1 ] ) implies ( [ e : con2 ] ) , and ( [ e : con2 ] ) is useful when the downstream analyst wants only a bayesian inference of @xmath22 . in either case the construction of the sufficient statistic generally depends on the joint model for @xmath3 as implied by ( [ e : model ] ) , requiring more knowledge than individual researchers typically possess . often , however , it is reasonable to assume the following conditional independence . let @xmath90 be the specification of @xmath91 for researcher @xmath92 , where @xmath93 forms a _ partition _ of @xmath3 . we then assume that @xmath94 note in the above definition implicitly we also assume the baseline measure @xmath95 is a product measure @xmath96 , such as lebesgue measure . the assumption ( [ e : obsm ] ) holds , for example , in microarray applications , when different labs provide conditionally - independent observations of probe - level intensities . the preceding discussion suggests that this assumption is necessary for ensuring ( [ e : con1 ] ) or even ( [ e : con2 ] ) , but obviously it is far from sufficient because it says nothing about the model on @xmath4 . it is reasonable or at least more logical than not to assume each researcher has the best knowledge to specify his / her own observation model @xmath97 ( @xmath98 . but , for the scientific model @xmath99 used by the downstream analyst , the best we can hope is that each researcher has _ a working model _ @xmath100 that is in some way related to @xmath99 . the notation @xmath101 reflects our hope to construct a common working parameter @xmath102 that can ultimately be _ linked _ to the scientific parameter @xmath22 . given this working model , the @xmath24th researcher can obtain the corresponding ( minimal ) sufficient statistic @xmath103 for @xmath104 with respect to @xmath105 when one has a prior @xmath106 for @xmath107 , one could alternately decide to retain the ( bayesian ) sufficient statistic @xmath108 with respect to the model @xmath109 our central interest here is to determine when the collection @xmath110 will satisfy ( [ e : con1 ] ) and when @xmath111 will satisfy ( [ e : con2 ] ) . this turns out to be an exceedingly difficult problem if we seek a necessary and sufficient condition for _ when _ this occurs . however , it is not difficult to identify sufficient conditions that can provide useful practical guidelines . we proceed by first considering cases where @xmath112 forms a partition of @xmath4 . compared to the assumption on partitioning @xmath3 , this assumption is less likely to hold in practice because different researchers can share common parts of @xmath4 s or even the entire scientific variable @xmath4 . however , as we shall demonstrate shortly , we can extend our results formally to all models for @xmath4 , as long as we are willing to put tight restrictions on the allowed class of working models . specifically , the following condition describes a class of working models that are ideal because they permit separate preprocessing yet retain joint information . note again that an implicit assumption here is that the baseline measure @xmath113 is a product measure @xmath114 . [ ( distributed separability condition ( dsc ) ) ] a set of working models @xmath115 is said to satisfy the _ distributed separability condition _ with respect to @xmath116 if there exists a probability measure @xmath117 such that @xmath118\,{\mathrm{d}}p_{\eta}({\eta}| { \theta } ) . \label{eq : dsc}\ ] ] [ thm : dsc ] under the assumptions ( [ e : obsm ] ) and ( [ eq : dsc ] ) , we have the collection of individual sufficient statistics from ( [ e : prob ] ) , that is , @xmath119 , is jointly sufficient for @xmath85 in the sense that ( [ e : con1 ] ) holds . under the additional assumption that @xmath120 forms a partition of @xmath77 and @xmath121 , both @xmath21 corresponding to ( [ e : prob ] ) and @xmath122 corresponding to ( [ e : prom ] ) are bayesianly sufficient for @xmath22 in the sense that ( [ e : con2 ] ) holds . by the sufficiency of @xmath123 for @xmath124 , we can write @xmath125 this implies that , @xmath126 } & = & \int _ { x } \biggl [ \prod_{i=1}^{r } { p_y}(y_i | x_i , { \xi}_i ) \biggr ] \\ & & { } \times\biggl [ \int_{\eta } \biggl [ \prod _ { i=1}^{r } { \tilde{p}_x}\bigl(x_i | g_i({\eta})\bigr ) \biggr]\,{\mathrm{d}}p_{\eta}({\eta}| { \theta } ) \biggr]\,{\mathrm{d}}\mu _ x(x ) , \\ { [ \mbox{by factorization of } \mu_x ] } & = & \int _ { \eta } \prod_{i=1}^{r } \biggl [ \int_{x_i } { p_y}(y_i | x_i , { \xi}_i ) { \tilde{p}_x}\bigl(x_i | g_i ( { \eta})\bigr)\,{\mathrm{d}}\mu_{x_i}(x_i ) \biggr]\,{\mathrm{d}}p_{\eta}({\eta}| { \theta } ) , \\ { \bigl[\mbox{by } ( \ref{eq : a1})\bigr ] } & = & \biggl [ \prod _ { i=1}^{r } h_i(y_i ) \biggr ] \biggl [ \int_{\eta } \prod_{i=1}^{r } f_i\bigl(t_i;g_i({\eta } ) , { \xi}_i\bigr)\,{\mathrm{d}}p_{\eta}({\eta}| { \theta } ) \biggr ] .\end{aligned}\ ] ] this establishes ( 1 ) by the factorization theorem . assertion ( 2 ) is easily established via an analogous argument , by integrating all the expressions above with respect to @xmath127 . we emphasize that dsc does not require individual researchers to model their parts of @xmath4 in the same way as the downstream analyst would , which would make it an essentially tautological condition . rather , it requires that individual researchers understand their own problems and how they can fit into the broader analysis hierarchically . this means that the working model for each @xmath128 @xmath129 can be more saturated than the downstream analyst s model for the same part of @xmath4 . consider a simple case with @xmath130 , where the preprocessor correctly assumes the multivariate normality for @xmath4 but is unaware that its covariance actually has a block structure or is unwilling to impose such a restriction to allow for more flexible downstream analyses . clearly any sufficient statistic under the unstructured multivariate model is also sufficient for any ( nested ) structured ones . the price paid here is failing to achieve the greatest possible sufficient reduction of the data , but this sacrifice may be necessary to ensure the broader validity of downstream analyses . for example , even if downstream analysts adopt a block - structured covariance , they may still want to perform a model checking , which would not be possible if all they are given is a _ minimal _ sufficient statistic for the model to be checked . knowledge suitable for specifying a saturated model is more attainable than complete knowledge of @xmath116 , although ensuring common knowledge of its ( potential ) hierarchical structure still requires some coordination among the researchers . each of them could independently determine for which classes of scientific models their working model satisfies the dsc . however , without knowledge of the partition of @xmath4 across researchers and the overarching model(s ) of interest , their evaluations need not provide any useful consensus . this suggests the necessity of some general communications and a practical guideline for distributed preprocessing , even when we have chosen a wise division of labors that permits dsc to hold . formally , dsc is similar in flavor to de finetti s theorem , but it does not require the components of the factorized working model to be exchangeable . dsc , however , is by no means necessary ( even under ( [ e : con1 ] ) ) , as an example in section [ sec : counterexamples ] will demonstrate . its limits stem from `` unparameterized '' dependence dependence between @xmath128 s that is not controlled by @xmath9 . when such dependence is present , statistics can exist that are sufficient for both @xmath131 and @xmath9 without the working model satisfying dsc . however , a simple necessary condition for distributed sufficiency is available . unsurprisingly , it links the joint sufficiency of @xmath132 under @xmath133 to the joint sufficiency of @xmath134 under the scientific model @xmath99 , where @xmath135 is any sufficient statistic for the working model @xmath136 . [ thm : necessary ] if , for all observation models satisfying ( [ e : obsm ] ) , the collections of individual sufficient statistics from ( [ e : prob ] ) @xmath119 are jointly sufficient for @xmath137 in the sense that ( [ e : con1 ] ) holds , then any collection of individual sufficient statistics under @xmath138 , that is , @xmath139 , must be sufficient for @xmath22 under @xmath140 . the proof of this condition emerges easily by considering the trivial observation model @xmath141 , where @xmath142 is the indicator function of set @xmath143 . theorem [ thm : necessary ] holds even if we require the observation model to be nontrivial , as the case of @xmath144\}}$ ] for arbitrary @xmath145-neighborhoods of @xmath128 demonstrates . the result says that if we want distributed preprocessing to provide a lossless compression regardless of the actual form of the observation model , then even under the conditional independence assumption ( [ e : obsm ] ) , we must require the individual working models to _ collectively _ preserve sufficiency under the scientific model . note that preserving sufficiency for a model is a much weaker requirement than preserving the model itself . indeed , two models can have very different model spaces yet share the same _ form _ of sufficient statistics , as seen with i.i.d . @xmath146 and @xmath147 models , both yielding the sample average as a complete sufficient statistic . although we find this sufficiency - preserving condition quite informative about the limits of lossless distributed preprocessing , it is not a sufficient condition . as a counterexample , consider @xmath148 independent for @xmath149 , @xmath150 , where @xmath151 . for the true model , we assume @xmath152 as follows : @xmath153 , @xmath154 , and all variables are mutually independent . for the working model , we take @xmath155 as follows : @xmath156 independently , and @xmath157 with probability 1 for all @xmath158 . obviously @xmath159 is a sufficient statistic for both @xmath155 and @xmath99 because of their normality . because @xmath27 is _ minimally _ sufficient for @xmath102 , this implies that any sufficient statistic for @xmath155 must be sufficient for @xmath99 , therefore the sufficiency preserving condition holds . however , the collection of the complete sufficient statistics @xmath160 for @xmath102 under @xmath161 is not sufficient for @xmath22 under @xmath162 because the latter is no longer an exponential family . the trouble is caused by the failure of the working models to capture additional flexibility in the scientific model that is not controlled by its parameter @xmath22 . therefore , obtaining a condition that is both necessary and sufficient for lossless compression via distributed preprocessing is a challenging task . such a condition appears substantially more intricate than those presented in theorems [ thm : dsc ] and [ thm : necessary ] and may therefore be less useful as an applied guideline . below we discuss a few further subtleties . although theorem [ thm : dsc ] covers both likelihood and bayesian cases , it is important to note a subtle distinction between their general implications . in the likelihood setting ( [ e : con1 ] ) , we achieve lossless compression for all downstream analyses targeting @xmath73 . this allows the downstream analyst to obtain inferences that are robust to the preprocessor s beliefs about @xmath16 , and they are free to revise their inferences if new information about @xmath16 becomes available . but , the downstream analyst must address the nuisance parameter @xmath77 from the preprocessing step , a task a downstream analyst may not be able or willing to handle . in contrast , the downstream analyst need not worry about @xmath77 in the bayesian setting ( [ e : con2 ] ) . however , this is achieved at the cost of robustness . all downstream analyses are potentially affected by the preprocessors beliefs about @xmath77 . furthermore , because @xmath122 is required only to be sufficient for @xmath22 , it may not carry any information for a downstream analyst to check the preprocessor s assumptions about @xmath16 . fortunately , as it is generally logical to expect the preprocessor to have better knowledge addressing @xmath77 than the downstream analyst , such robustness may not be a serious concern from a practical perspective . theoretically , the trade - off between robustness and convenience is not clear - cut ; they can coincide for other types of preprocessing , as seen in section [ sec : missinfo ] below . as discussed earlier , ( conditional ) dependencies among the observation variables @xmath163 across different @xmath24 s will generally rule out the possibility of achieving lossless compression by collecting individual sufficient statistics . this points to the importance of appropriate separation of labors when designing distributed preprocessing . in contrast , dependencies among @xmath128 s are permitted , at the expense of redundancy in sufficient statistics . we first consider deterministic dependencies , and for simplicity , take @xmath164 and constrain attention to the case of sufficiency for @xmath9 . suppose we have @xmath165 and @xmath166 forming a partition of @xmath4 , with a working model @xmath167 that satisfied the dsc for some @xmath168 . imagine we need to add a common variable @xmath72 to both @xmath165 and @xmath166 that is conditionally independent of @xmath169 given @xmath22 and has density @xmath170 , with the remaining model unchanged . however , the two researchers are unaware of the sharing of @xmath72 , so they set up @xmath171 and @xmath172 , with @xmath173 does not correspond to the scientific variable @xmath174 of interest . however , we notice that if we can force @xmath175 in @xmath176 , then we can recover @xmath4 . this forcing is not a mere mathematical trick . rather , it reflects an extreme yet practical strategy when researchers are unsure whether they share some components of their @xmath177 with others . the strategy is simply to retain statistics sufficient for the entire part that they may _ suspect _ to be common , which in this case means that both researchers will retain statistics sufficient for the @xmath178s @xmath179 in their entirety . mathematically , this corresponds to letting @xmath180 , where @xmath181 . it is then easy to verify that dsc holds , if we take @xmath182 , where @xmath183 . this is because when @xmath184 , both sides of ( [ eq : dsc ] ) are zero . when @xmath175 , we have ( adopting integration over @xmath185 functions ) @xmath186\,{\mathrm{d}}p_{\eta } ' \bigl ( { \eta } ' | { \theta}\bigr ) \\ & & \quad= \int_\eta \int_{\zeta _ 1 } \biggl [ \prod_{i=1}^2 \tilde p_{x_i}(x_i |\eta_i ) \delta_{\ { z=\zeta_i\ } } \biggr]\,{\mathrm{d}}p_{\eta } ( { \eta}| { \theta } ) \delta_{\{\zeta_1=\zeta_2\}}\,{\mathrm{d}}p_{z}(\zeta_1|\theta ) \\ & & \quad = \biggl [ \int_\eta\prod_{i=1}^2 \tilde p_{x_i}(x_i |\eta_i)\,{\mathrm{d}}p_{\eta } ( { \eta}| { \theta } ) \biggr ] \int_{\zeta_1 } \delta_{\{\zeta_1=z\}}\,{\mathrm{d}}p_{z}(\zeta_1|\theta ) \\ & & \quad = p_{x_{}}(x_1 , x_2|\theta)p_{z}(z| \theta)=p_{x}(x|\theta).\end{aligned}\ ] ] this technique of expanding @xmath102 to include shared parts of the @xmath4 allows the dsc and theorem [ thm : dsc ] to be applied to all models @xmath140 , not only those with with distinct @xmath128 s . however , this construction also restricts working models to those with deterministic relationships between parts of @xmath102 and each @xmath128 . the derivation above demonstrates both the broader applications of dsc as a theoretical condition and its restrictive nature as a practical guideline . retaining sufficient statistics for both @xmath187 and @xmath188 can create redundancy . if each preprocessor observes @xmath72 without noise , then only one of them actually needs to retain and report their observation of @xmath72 . however , if each observes @xmath72 with independent noise , then both of their observations are required to obtain a sufficient statistic for @xmath22 . the noise - free case also provides a straightforward counterexample to the necessity of dsc . assuming both preprocessors observe @xmath72 directly , as long as one of the copies of @xmath72 is retained via the use of the saturated @xmath185 density , the other copy can be modeled in any way and hence can be made to violate dsc without affecting their joint sufficiency for @xmath22 . regardless of the dependencies among the @xmath128 s , there is always a safe option open to the preprocessors for data reduction : retain @xmath123 sufficient for @xmath189 under @xmath190 . this will preserve sufficiency for @xmath9 under any scientific model @xmath116 : [ thm : safe ] if @xmath191 is correctly specified and satisfies ( [ e : obsm ] ) , then any collection of individual sufficient statistics @xmath192 with each @xmath123 sufficient for @xmath189 is jointly sufficient for @xmath73 in the sense of ( [ e : con1 ] ) for all @xmath116 . by the factorization theorem , we have @xmath193 for any @xmath24 . hence , by ( [ e : obsm ] ) , @xmath194 \int_x [ \prod_{i=1}^r p_t(t_i | x_i , { \xi}_i ) ] { p_x}(x @xmath195 is sufficient for @xmath22 , by the factorization theorem for sufficiency . theorem [ thm : safe ] provides a universal , safe strategy for sufficient preprocessing and a lower bound on the compression attainable from distributed sufficient preprocessing . as all minimal sufficient statistics for @xmath22 are functions of any sufficient statistic for @xmath196 , retaining minimal sufficient statistics for each @xmath189 results in less compression than any approach properly using knowledge of @xmath140 . however , the compression achieved relative to retaining @xmath3 itself may still be significant . minimal sufficient statistics for @xmath22 provide an upper bound on the attainable degree of compression by the same argument . achieving this compression generally requires that each preprocessor knows the true scientific model @xmath140 . between these bounds , the dsc ( [ eq : dsc ] ) shows a trade - off between the generality of preprocessing ( with respect to different scientific models ) and the compression achieved : the smaller the set of scientific models for which a given working model satisfies ( [ eq : dsc ] ) , the greater the potential compression from its sufficient statistics . more generally , stochastic dependence among @xmath128 s reduces compression and increases redundancy in distributed preprocessing . these costs are particularly acute when elements of @xmath9 control dependence among @xmath128 s , as seen in the following example where @xmath197 here @xmath198 is a column vector with @xmath199 @xmath200 s as its components , and @xmath201 is the usual kronecker product . if @xmath202 is known , then each researcher can reduce their observations @xmath163 to a scalar statistic @xmath203 and preserve sufficiency for @xmath204 . if @xmath202 is unknown , then each researcher must retain all of @xmath205 ( but not @xmath206 for @xmath207 ) in addition to these sums to ensure sufficiency for @xmath208 , because the minimal sufficient statistic for @xmath209 requires the computation of @xmath210 . thus , the cost of dependence here is @xmath89 additional pieces of information per preprocessor . dependence among the @xmath128 s forces the preprocessors to retain enough information to properly combine their individual contributions in the final analysis , downweighting redundant information . this is true even if they are interested only in efficient estimation of @xmath204 , leading to less reduction of their raw data and less compression from preprocessing than the independent case . from this investigation , we see that it is generally not enough for each researcher involved in preprocessing to reduce data based on even a correctly - specified model for their problem at hand . we instead need to look to other models that include each experimenter s data hierarchically , explicitly considering higher - level structure and relationships . however , significant reductions of the data are still possible despite these limitations . each @xmath123 need not be sufficient for each @xmath128 , nor must @xmath0 be sufficient for @xmath4 overall . this often implies that much less data need to be retained and shared than retaining sufficient statistics for each @xmath128 would demand . for instance , if a working model with @xmath211 satisfies the dsc for a given model @xmath116 and @xmath212 , then only means and covariance matrices of @xmath206 within each experiment @xmath24 need to be retained . the discussions above demonstrate the importance of involving downstream analysts in the design of preprocessing techniques . their knowledge of @xmath116 is extremely useful in determining what compression is appropriate , even if said knowledge is imperfect . constraining the scientific model to a broad class may be enough to guarantee effective preprocessing . for example , suppose we fix a working model and consider all scientific models that can be expressed as ( [ eq : dsc ] ) by varying the choices of @xmath117 . this yields a very broad class of hierarchical scientific models for downstream analysts to evaluate , while permitting effective distributed preprocessing based on the given working model.=1 practically , we see two paths to distributed preprocessing : coordination and caution . coordination refers to the downstream analyst evaluating and guiding the design of preprocessing as needed . such guidance can guarantee that preprocessed outputs will be as compact and useful as possible . however , it is not always feasible . it may be possible to specify preprocessing in detail in some industrial and purely computational settings . accomplishing the same in academic research or for any research conducted over time is an impractical goal . without such overall coordination , caution is needed . it is not generally possible to maintain sufficiency for @xmath9 without knowledge of the possible models @xmath116 unless the retained summaries are sufficient for @xmath4 itself . preprocessors should therefore proceed cautiously , carefully considering which scientific models they effectively exclude through their preprocessing choices . this is analogous to the oft - repeated guidance to include as many covariates and interactions as possible in imputation models ( @xcite , @xcite ) . having considered the lossless preprocessing , we now turn to more realistic but less clear - cut situations . we consider a less careful preprocessor and a sophisticated downstream analyst . the preprocessor selects an output @xmath0 , which may discard much information in @xmath3 but nevertheless preserves the identifiability of @xmath22 , and the downstream analyst knows enough to make the best of whatever output they are given . that is , the index set @xmath29 completely and accurately captures all relevant preprocessing methods @xmath213 . this does not completely capture all the practical constraints discussed in section [ sec : concepts ] . however , it is important to establish an upper bound on the performance of multiphase procedures before incorporating such issues . this upper bound is on the fisher information , and hence a lower bound on the asymptotic variances of estimators @xmath214 of @xmath9 . as we will see , nuisance parameters ( @xmath16 ) play a crucial role in these investigations . when using a lossy compression , an obvious question is how much information is lost compared to a lossless compression . this question has a standard asymptotic answer when the downstream analyst adopts an mle or bayes estimator , so long as nuisance parameters behave appropriately ( as will be discussed shortly ) . if the downstream analyst adopts some other procedures , such as an estimating equation , then there is no guarantee that the procedure based on @xmath3 is more efficient than the one based on @xmath0 . that is , one can actually obtain a more efficient estimator with less data when one is not using _ probabilistically principled _ methods , as discussed in detail in @xcite . therefore , as a first step in our theoretical investigations , we will focus on mles ; the results also apply to bayesian estimators under the usual regularity conditions to guarantee the asymptotic equivalence between mles and bayesian estimators . specifically , let @xmath215 and @xmath216 be the mles of @xmath217 based respectively on @xmath3 and @xmath0 under model ( [ e : model ] ) . we place standard regularity conditions for the joint likelihood of @xmath73 , assuming bounded third derivatives of the log - likelihood , common supports of the observation distributions with respect to @xmath73 , full rank for all information matrices at the true parameter value @xmath218 , and the existence of an open subset of the parameter space that contains @xmath218 . these conditions imply the first and second bartlett identities . however , the most crucial assumption here is a sufficient accumulation of information , indexed by an _ information size _ @xmath219 , to constrain the behavior of remainder terms in quadratic approximations of the relevant score functions . independent identically distributed observations and fixed - dimensional parameters would satisfy this requirement , in which case @xmath219 is simply the data size of @xmath3 , but weaker conditions can suffice ( for an overview , see @xcite ) . in general , this assumption requires that the dimension of both @xmath220 and @xmath77 are bounded as we accumulate more data , preventing the type of phenomenon revealed in @xcite . for multiphase inferences , cases where these dimensions are unbounded are common ( at least in theory ) and represent interesting settings where preprocessing can actually improve asymptotic efficiency , as we discuss shortly . to eliminate the nuisance parameter @xmath77 , we work with the observed fisher information matrices based on the profile likelihoods for @xmath9 , denoted by @xmath221 and @xmath222 respectively . let @xmath223 be the limit of @xmath224 , the so - called _ fraction of missing information _ ( see @xcite ) , as @xmath225 . the proof of the following result follows the standard asymptotic arguments for mles , with the small twist of applying them to profile likelihoods instead of full likelihoods . ( we can also invoke the more general arguments based on decomposing estimating equations , as given in @xcite . ) [ thm : missinfo ] under the conditions given above , we have asymptotically as @xmath225 , @xmath226^{-1 } \rightarrow f\ ] ] and @xmath227^{-1 } \rightarrow i - f.\ ] ] this establishes the central role of the fraction of missing information @xmath223 in determining the asymptotic efficiency of multiphase procedures under the usual asymptotic regime . as mentioned above , this is an ideal - case bound on the performance of multiphase procedures , and it is based on the usual squared - error loss ; both the asymptotic regime and amount of knowledge held by the downstream analyst are optimistic . we explore these issues below , focusing on ( 1 ) mutual knowledge and alternative definitions of efficiency , ( 2 ) the role of reparameterization , ( 3 ) asymptotic regimes and multiphase efficiency , and ( 4 ) the issue of robustness in multiphase inference . in practice , downstream analysts are unlikely to have complete knowledge of @xmath8 . therefore , even if they were given the entire @xmath3 , they would not be able to produce the optimal estimator @xmath228 , making the @xmath223 value given by theorem [ thm : missinfo ] an unrealistic yardstick . nevertheless , theorem [ thm : missinfo ] suggests a direction for a more realistic standard . the classical theory of estimation focuses on losses of the form @xmath229 , where @xmath230 denotes the truth . risk based on this type of loss , given by @xmath231 $ ] , is a raw measure of performance , using the truth as a baseline . an alternative is regret , the difference between the risk of a given estimator and an ideal estimator @xmath232 ; that is , @xmath233 . regret is popular in the learning theory community and forms the basis for oracle inequalities . it provides a more adaptive baseline for comparison than raw risk , but we can push further . consider evaluating loss with respect to an estimator rather than the truth . for mean - squared error , this yields @xmath234 .\ ] ] can this provide a better baseline , and what are its properties ? for mles , @xmath235 behaves the same ( asymptotically ) as additive regret because theorem [ thm : missinfo ] implies that , as @xmath225 under the classical asymptotic regime , @xmath236 \\[-8pt ] \nonumber & = & r\bigl({\hat{{\theta}}}(t ) , { \theta}_0\bigr)-r\bigl({\hat{{\theta}}}(y ) , { \theta}_0\bigr ) .\end{aligned}\ ] ] for inefficient estimators , ( [ eq : same ] ) does not hold in general because @xmath237 is no longer guaranteed to be asymptotically uncorrelated with @xmath238 . in such cases , this is precisely the reason @xmath81 can be more efficient than @xmath238 or , more generally , there exists a constant @xmath239 such that @xmath240 is ( asymptotically ) more efficient than @xmath238 . in the terminology of @xcite , the estimation procedure @xmath241 is not _ self - efficient _ if ( [ eq : same ] ) does not hold , viewing @xmath3 as the complete data @xmath242 and @xmath0 as the observed data @xmath243 . indeed , if @xmath244 , @xmath245 may actually be _ larger _ for a _ better _ @xmath81 because of the inappropriate baseline @xmath238 ; it is a measure of difference , not dominance , in such cases . hence , some care is needed in interpreting this measure . therefore , we can view ( [ eq : risk ] ) as a generalization of the usual notion of regret , or the relative regret if we divide it by @xmath246 . this generalization is appealing for the study of preprocessing : we are evaluating the estimator based on preprocessed data directly against what could be done with the complete raw data , sample by sample , and we no longer need to impose the restriction that the downstream analysts must carry out the most efficient estimation under a model that captures the actual preprocessing . this direction is closely related to the idea of strong efficiency from @xcite and @xcite , which generalizes the idea of asymptotic decorrelation beyond the simple ( but instructive ) setting covered here . such ideas from the theory of missing data provide a strong underpinning for the study of multiphase inference and preprocessing . theorem [ thm : missinfo ] also emphasizes the range of effects that preprocessing can have , even in ideal cases . consider the role that @xmath223 plays under different transformations of @xmath9 . although the eigenvalues of @xmath223 are invariant under one - to - one transformations of the parameters , submatrices of @xmath223 can change substantially . formally , if @xmath208 is transformed to @xmath247 , then the fraction of missing information for @xmath248 can be very different from that for @xmath204 . these changes mean that changes in parameterization can reallocate the fractions of missing information among resulting subparameters in unexpected and sometimes very unpleasant ways . this is true even for linear transformations ; a given preprocessing technique can preserve efficiency for @xmath204 and @xmath202 individually while performing poorly for @xmath249 . such issues have arisen in , for instance , the work of @xcite when attempting to characterize the behavior of multiple imputation estimators under uncongeniality . on a fundamental level , theorem [ thm : missinfo ] is a negative result for preprocessing , at least for mles . reducing the data from @xmath3 to @xmath0 can only hinder the downstream analyst . formally , this means that @xmath250 ( asymptotically ) in the sense that @xmath251 is positive semi - definite . as a result , @xmath238 will dominate @xmath81 in asymptotic variance for any preprocessing @xmath0 . thus , the only justification for preprocessing appears to be pragmatic ; if the downstream analyst could not make use of @xmath8 for efficient inference or such knowledge could not be effectively transmitted , preprocessing provides a feasible way to obtain the inferences of interest . however , this conclusion depends crucially on the assumed behavior of the nuisance parameter @xmath16 . the usual asymptotic regime is not realistic for many multiphase settings , particularly with regards to @xmath16 . in many problems of interest , @xmath252 does not tend to zero as @xmath219 increases , preventing sufficient accumulation of information on the nuisance parameter @xmath77 . a typical regime of this type would accumulate observations @xmath163 from individual experiments @xmath24 , each of which brings its own nuisance parameter @xmath253 . such a process could describe the accumulation of data from microarrays , for instance , with each experiment corresponding to a chip with its own observation parameters , or the growth of astronomical datasets with time - varying calibration . in such a regime , preprocessing can have much more dramatic effects on asymptotic efficiency . in the presence of nuisance parameters , inference based on @xmath0 can be more robust and even more efficient than inference based on @xmath3 . it is well - known that the mle can be inefficient and even inconsistent in regimes where @xmath254 ( going back to at least @xcite ) . bayesian methods provide no panacea either . marginalization over the nuisance parameter @xmath16 is appealing , but resulting inferences are typically sensitive to the prior on @xmath16 , even asymptotically . in many cases ( such as the canonical neyman scott problem ) , only a minimal set of priors provide even consistent bayes estimators . careful preprocessing can , however , enable principled inference in such regimes . such phenomena stand in stark contrast to the theory of multiple imputation . in that theory , complete data inferences are typically assumed to be valid . thus , under traditional missing data mechanisms , the observed data ( corresponding to @xmath0 ) can not provide better inferences than @xmath3 . this is not necessarily true in multiphase settings . if the downstream analyst is constrained to particular principles of inference ( e.g. , mle or bayes ) , then estimators based on @xmath0 can provide lower asymptotic variance than those based on @xmath3 . this occurs , in part , because the mechanisms generating @xmath3 and @xmath0 from @xmath4 are less restricted in the multiphase setting compared to the traditional missing - data framework . principled inferences based on @xmath4 would , in the multiphase setting , generally dominate those based on either @xmath3 or @xmath0 . however , such a relationship need not hold between @xmath3 and @xmath0 without restrictions on the behavior of @xmath16 . we emphasize that this does not contradict the general call in @xcite to follow the probabilistically - principled methods ( such as mle and bayes recipes ) to prevent violations of self - efficiency , precisely because the well - established principles of single - phase inference may need to be `` re - principled '' before they can be equally effective in the far more complicated multiphase setting . in the simplest case , if a @xmath0 can be found such that it is a pivot with respect to @xmath16 and remains dependent upon @xmath9 , then sensitivity to the behavior of @xmath77 can be eliminated by preprocessing . in such cases , an mle or bayes rule based on @xmath0 can dominate that based on @xmath3 even asymptotically . one such example would be providing @xmath255-statistics from each of a set of experiments to the downstream analyst . this clearly limits the range of feasible downstream inferences . with these @xmath255-statistics , detection of signals via multiple testing ( e.g. , @xcite ) would be straightforward , but efficient combination of information across experiments could be difficult . this is a ubiquitous trade - off of preprocessing : reductions that remove nuisance parameters and improve robustness necessarily reduce the amount of information available from the data . these trade - offs must be considered carefully when designing preprocessing techniques universal utility is unattainable without the original data . a more subtle case involves the selection of @xmath0 as a `` partial pivot '' . in some settings , there exists a decomposition of @xmath16 as @xmath256 such that @xmath257 for some fixed @xmath89 and all @xmath219 , and the distribution of @xmath0 is free of @xmath258 for all values of @xmath259 . many normalization techniques used in the microarray application of section [ sec : examples ] can be interpreted in this light . these methods attempt to reduce the unbounded set of experiment - specific nuisance parameters affecting @xmath0 to a bounded , manageable size . for example , suppose each processor @xmath24 observes @xmath260 , @xmath261 . the downstream analyst wants to estimate @xmath262 , considering @xmath263 and @xmath264 as nuisance parameters . in our previous notation , we have @xmath265 and @xmath266 . suppose each preprocessor reduces her data to @xmath267 , where @xmath268 is the ols estimator of @xmath269 based on @xmath270 . the distribution of each @xmath123 depends on @xmath264 but is free of @xmath269 . hence , @xmath271 is a partial pivot as defined above , with @xmath272 and @xmath273 . such pivoting techniques can allow @xmath81 to possess favorable properties even when @xmath238 is inconsistent or grossly inefficient . as mentioned before , this kind of careful preprocessing can dominate bayesian procedures in the presence of nuisance parameters when @xmath274 can grow with @xmath275 . in these regimes , informative priors on @xmath16 can affect inferences even asymptotically . however , reducing @xmath3 to @xmath0 so only the @xmath276-part of @xmath16 is relevant for @xmath0 s distribution allows information to accumulate on @xmath259 , making inferences far more robust to the preprocessor s beliefs about @xmath16 . these techniques share a common conceptual framework : invariance . invariance has a rich history in the bayesian literature , primarily as a motivation for the construction of noninformative or reference priors ( e.g. , @xcite , @xcite , @xcite , @xcite , @xcite ) . it is fundamental to the pivotal methods discussed above and arises in the theory of partial likelihood ( @xcite ) . we see invariance as a core principle of preprocessing , although its application is somewhat different from most bayesian settings . we are interested in finding functions of the data whose distributions are invariant to subsets of the parameter , not priors invariant to reparameterization . for instance , the rank statistics that form the basis for cox s proportional hazards regression in the absence of censoring ( @xcite ) can be obtained by requiring a statistic invariant to monotone transformations of time . indeed , cox s regression based on rank statistics can be viewed as an excellent example of eliminating an infinite dimensional nuisance parameter , i.e. , the baseline hazard , via preprocessing , which retains only the rank statistics . the relationship between invariance in preprocessing , modeling , and prior formulation is a rich direction for further investigation . an interesting practical question arises from this discussion of robustness : how realistic is it to assume efficient inference with preprocessed data ? this may seem unrealistic as preprocessing is frequently used to simplify problems so common methods can be applied . however , preprocessing can make many assumptions more appropriate . for example , aggregation can make normality assumptions more realistic , normalization can eliminate nuisance parameters , and discretization greatly reduces reliance on parametric distributional assumptions altogether . it may therefore be more appropriate to assume that efficient estimators are generally used with preprocessed data than with raw data . the results and examples explored here show that preprocessing is a complex topic in even large - sample settings . it appears formally futile ( but practically useful ) in standard asymptotic regimes . under other realistic asymptotic regimes , preprocessing emerges as a powerful tool for addressing nuisance parameters and improving the robustness of inferences . having established some of the formal motivation and trade - offs for preprocessing , we discuss further extensions of these ideas into more difficult settings in section [ sec : future ] . in some cases , effective preprocessing techniques are quite apparent . if @xmath10 forms an exponential family with parameter @xmath4 or @xmath196 , then we have a straightforward procedure : retain a minimal sufficient statistic . to be precise , we mean that one of the following factorizations holds for a sufficient statistic @xmath21 of bounded dimension : @xmath277 retaining this sufficient statistic will lead to a lossless compression , assuming that the first - phase model is correct . unfortunately , such nice cases are rare . even the bayesian approach offers little reprieve . integrating @xmath10 with respect to a prior @xmath278 typically removes the observation model from the exponential family consider , for instance , a normal model with unknown variance becoming a @xmath279 distribution . if @xmath280 is approximately quadratic as a function of @xmath4 , then retaining its mode and curvature would seem to provide much of the information available from the data to downstream analysts . however , such intuition can be treacherous . if a downstream analyst is combining inferences from a set of experiments , each of which yielded an approximately quadratic likelihood , the individual approximations may not be enough to provide efficient inferences . approximations that hold near the mode of each experiment s likelihood need not hold away from these modes including at the mode of the joint likelihood from all experiments . thus , remainder terms can accumulate in the combination of such approximations , degrading the final inference on @xmath9 . furthermore , the requirement that @xmath280 be approximately quadratic in @xmath4 is quite stringent . to justify such approximations , we must either appeal to asymptotic results from likelihood theory or confine our attention to a narrow class of observation models @xmath281 . unfortunately , asymptotic theory is often an inappropriate justification in multiphase settings , because @xmath4 grows in dimension with @xmath3 in many asymptotic regimes of interest , so there is no general reason to expect information to accumulate on @xmath4 . these issues are of particular concern as such quadratic approximations are a standard implicit justification for passing point estimates with standard errors onto downstream analysts . moving away from these cases , solutions become less apparent . no processing ( short of passing the entire likelihood function ) will preserve all information from the sample when sufficient statistics of bounded dimension do not exist . however , multiphase approaches can still possess favorable properties in such settings . we begin by considering a stubborn downstream analyst she has her method and will not consider anything else . for example , this analyst could be dead set on using linear discriminant analysis or anova . the preprocessor has only one way to affect her results : carefully designing a particular @xmath0 given to the downstream analyst . such a setting is extreme . we are saying that the downstream analyst will charge ahead with a given estimator regardless of her input with neither reflection nor judgment . we investigate this setting because it maximizes the preprocessor s burden in terms of her contribution to the final estimate s quality . formally , we consider a fixed second - stage estimator @xmath81 ; that is , the form of its input @xmath0 and the function producing @xmath214 are fixed , but the mechanism actually used to generate @xmath0 is not . @xmath0 could be , for example , a vector of fixed dimension . as we discuss below , admissible designs for the first - phase with a fixed second - phase method are given by a ( generalized ) bayes rule . this uses the known portion of the model @xmath10 to construct inputs for the second stage and assumes that any prior the preprocessor has on @xmath16 is equivalent to what a downstream analyst would have used in the preprocessor s position . formally , this describes all rules that are admissible among the class of procedures using a given second - stage method , following from previous complete class results in statistical decision theory ( e.g. , @xcite , @xcite ) . assume that the second - stage procedure @xmath81 is fixed as discussed above and we are operating under the model ( [ e : model ] ) . further assume that the preprocessor s prior on @xmath16 is the only such prior used in all bayes rule constructions . for @xmath282 , consider a smooth , strictly convex loss function @xmath39 . then , under appropriate regularity conditions ( e.g. , @xcite , @xcite ) , if @xmath81 is a smooth function of @xmath0 , then all admissible procedures for generating @xmath0 are bayes or generalized bayes rules with respect to the risk @xmath283 . the same holds when @xmath0 is restricted to a finite set . this guideline follows directly from conventional complete class results in decision theory . we omit technical details here , focusing instead on the guideline s implications . however , a sketch of its proof proceeds along the following lines . there are two ways to approach this argument : intermediate loss and geometry . the intermediate loss approach uses an intermediate loss function @xmath284 . this @xmath285 is the loss facing the preprocessor given a fixed downstream procedure @xmath81 . if @xmath285 is well - behaved , in the sense of satisfying standard conditions ( strict convexity , or a finite parameter space , and so on ) , then the proof is complete from previous results for real @xmath0 . similarly , if @xmath0 is restricted to a finite discrete set , then we face a classical multiple decision problem and can apply previous results to @xmath286 . these straightforward arguments cover a wide range of realistic cases , as @xcite has shown . otherwise , we must turn to a more intricate geometric argument . broadly , this construction uses a convex hull of risks generated by attainable rules . this guideline has direct bearing upon the development of inputs for machine learning algorithms , typically known as _ feature engineering_. given an algorithm that uses a fixed set of inputs , it implies that using a correctly - specified observation model to design these inputs is necessary to obtain admissible inferences . thus , it is conceptually similar to `` rao - blackwellization '' over part of a probability model . however , several major caveats apply to this result . first , on a practical level , deriving such bayes rules is quite difficult for most settings of interest . second , and more worryingly , this result s scope is actually quite limited . as we discussed in section [ sec : missinfo ] , even bayesian estimators can be inconsistent in realistic multiphase regimes . however , these estimators are still admissible , as they can not be dominated in risk for particular values of the nuisance parameters @xmath77 . admissibility therefore is a minimal requirement ; without it , the procedure can be improved uniformly , but with it , it can still behave badly in many ways . finally , there is the problem of robustness . an optimal input for one downstream estimator @xmath287 may be a terrible input for another estimator @xmath288 , even if @xmath289 and @xmath290 take the same form of inputs . such considerations are central to many real - world applications of preprocessing , as researchers aim to construct databases for a broad array of later analyses . however , this result does show that engineering inputs for downstream analyses using bayesian observation models can improve overall inferences . how to best go about this in practice is a rich area for further work . as befits first steps , we are left with a few loose ends and puzzles . starting with the dsc condition ( [ eq : dsc ] ) of section [ sec : sufficiency ] , we provide a simple counterexample to its necessity . suppose we have @xmath291 . let @xmath292 independent of each other . now , let @xmath293 , @xmath294 , @xmath295 , where @xmath296 , @xmath297 , @xmath298 is a vector of signs @xmath299 , or @xmath300 for @xmath165 , @xmath301 denotes the element - wise absolute value , and @xmath302 denotes the hadamard product . we fix @xmath303 . as our working model , we posit that @xmath304 independently . then , we clearly have @xmath305 as a sufficient statistic for both @xmath131 and @xmath9 . however , the dsc does not hold for this working model . we can not write the actual joint distribution of @xmath4 as a marginalization of @xmath306 with respect to some distribution over @xmath131 in such a way that @xmath307 is sufficient for @xmath131 . to enforce @xmath308 under the working model , any such model must use @xmath131 to share this information . for this example , we can obtain a stronger result : no factored working model @xmath306 exists such that ( 1 ) @xmath309 is sufficient for @xmath310 under @xmath311 and ( 2 ) the dsc holds . for contradiction , assume such a working model exists . under this working model , @xmath163 is conditionally independent of @xmath101 given @xmath309 , so we can write @xmath312 . as the dsc holds for this working model , we have @xmath313 \int_{\eta } \biggl [ \prod _ { i=1}^2 h_i\bigl(y_i^\top y_i ; g_i(\eta)\bigr ) \biggr ] p_{\eta}({\mathrm{d}}{\eta}| { \theta } ) .\ ] ] hence , we must have @xmath314 conditionally independent of @xmath315 given @xmath316 . however , this conditional independence does not hold under the true model . hence , the given working model can not both satisfy the dsc and have @xmath309 sufficient for each @xmath101 . the issue here is unparameterized dependence , as mentioned in section [ sec : sufficiency ] . the @xmath4 s have a dependence structure that is not captured by @xmath9 . thus , requiring that a working model preserves sufficiency for @xmath9 does not ensure that it has enough flexibility to capture the true distribution of @xmath3 . a weaker condition than the dsc ( [ eq : dsc ] ) that is necessary and sufficient to ensure that all sufficient statistics for @xmath131 are sufficient for @xmath9 may be possible . from sections [ sec : missinfo ] and [ sec : completeclass ] , we are left with puzzles rather than counterexamples . as mentioned previously , many optimality results are trivial without sufficient constraints . for instance , minimizing risk or maximizing fisher information naively yield uninteresting ( and impractical ) multiphase strategies : have the preprocessor compute optimal estimators , then pass them downstream . overly tight constraints bring their own issues . restricting downstream procedures to excessively narrow classes ( e.g. , point estimates with standard errors ) limits the applied utility of resulting theory and yields little insight on the overall landscape of multiphase inference . striking the correct balance with these constraints is a core challenge for the theory of multiphase inference and will require a combination of computational , engineering , and statistical insights . as we discussed in sections [ sec : concepts ] and [ sec : theory ] , we have a deep well of questions that motivate further research on multiphase inference . these range from the extremely applied ( e.g. , enhancing preprocessing in astrophysical systems ) to the deeply theoretical ( e.g. , bounding the performance of multiphase procedures in the presence of nuisance parameters and computational constraints ) . we outline a few directions for this research below . but , before we look forward , we take a moment to look back and place multiphase inference within the context of broader historical debates . such `` navel gazing '' helps us to understand the connections and implications of the theory of multiphase inference . on a historical note , the study of multiphase inference touches the long - running debate over the role of decision theory in statistics . one side of this debate , championed by wald and lehmann ( among others ) , has argued that decision theory lies at the core of statistical inference . risk - minimizing estimators and , more generally , optimal decision rules play a central role in their narrative . even subjectivists such as savage and de finetti have embraced the decision theoretic formulation to a large extent . other eminent statisticians have objected to such a focus on decisions . as noted by @xcite , fisher in particular vehemently rejected the decision theoretic formulation of statistical inference . one interpretation of fisher s objections is that he considered decision theory useful for eventual economic decision - making , but not for the growth of scientific knowledge . we believe that the study of multiphase inference brings a unifying perspective to this debate . fisher s distinction between intermediate processing and final decisions is fundamental to the problem of multiphase inference . however , we also view decision theory as a vital theoretical tool for the study of multiphase inference . passing only risk - minimizing point estimators to later analysts is clearly not a recipe for valid inference . the key is to consider the use of previously generated results explicitly in the final decision problem . in the study of multiphase inference , we do so by focusing on the separation of knowledge and objectives between agents . such separation between preprocessing and downstream inference maps nicely to fisher s distinction between building scientific knowledge and reaching actionable decisions . thus , we interpret fisher s line of objections to decision - theoretic statistics as , in part , a rejection of adopting a myopic single - phase perspective in multiphase settings . we certainly do not believe that our work will bring closure to such an intense historical debate . however , we do see multiphase inference as an important bridge between these competing schools of thought . we see a wide range of open questions in multiphase inference . can more systematic ways to leverage the potential of preprocessing be developed ? is it possible to create a mathematical `` warning system , '' alerting practitioners when their inferences from preprocessed data are subject to severe degradation and showing where additional forms of preprocessing are required ? and , can multiphase inference inform developments in distributed statistical computation and massive - data inference ( as outlined below in section [ sec : computation ] ) ? all of these problems call for a shared collection of statistical principles , theory , and methods . below , we outline a few directions for the development of these tools for multiphase inference . the mechanics of passing information between phases constitute a major direction for further research . one approach leverages the fact that the likelihood function itself is always a minimal sufficient statistic . thus , a set of ( computationally ) efficient approximations to the likelihood function @xmath317 for @xmath318 could provide the foundation for a wide range of multiphase methods . many probabilistic inference techniques for the downstream model ( e.g. , mcmc samplers ) would be quite straightforward to use given such an approximation . the study of such multiphase approximations also offers great dividends for distributed statistical computation , as discussed below . we believe these approximations are promising direction for general - purpose preprocessing . however , there are stumbling blocks . first , nuisance parameters remain an issue . we want to harness and understand the robustness benefits offered by preprocessing , but likelihood techniques themselves offer little guidance in this direction . even the work of @xcite on partial likelihood focuses on the details of estimation once the likelihood has been partitioned . we would like to identify the set of formal principles underlying techniques such as partial pivoting ( to mute the effect of infinite - dimensional nuisance parameters ) , building a more rigorous understanding of the role of preprocessing in providing robust inferences . as discussed in section [ sec : missinfo ] , invariance relationships may be a useful focus for such investigations , guiding both bayesian and algorithmic developments . second , we must consider the burden placed on downstream analysts by our choice of approximation . probabilistic , model - based techniques can integrate such information with little additional development . however , it would be difficult for a downstream analyst accustomed to , say , standard regression methods to make use of a complex emulator for the likelihood function . the burden may be substantial for even sophisticated analysts . for instance , it could require a significant amount of effort and computational sophistication to obtain estimates of @xmath4 from such an approximation , and estimates of @xmath4 are often of interest to downstream analysts in addition to estimates of @xmath9 . with these trade - offs in mind and through the formal analysis of widely - applicable multiphase techniques , we can begin to establish bounds on the error properties of such techniques in a broad range of problems under realistic constraints ( in both technical and human terms ) . more general constraints , for instance , can take the form of upper bounds on the regret attainable with a fixed amount of information passed from preprocessor to downstream analyst for fixed classes of scientific models . extensions to nonparametric downstream methods would have both practical and theoretical implications . in cases where the observation model is well - specified but the scientific model is less clearly defined , multiphase techniques can provide a useful alternative to computationally - expensive semi - parametric techniques . fusing principled preprocessing with flexible downstream inference may provide an interesting way to incorporate model - based subject - matter knowledge while effectively managing the bias - variance trade - off . the directions discussed above share a conceptual , if not technical , history with the development of congeniality ( @xcite ) . both the study of congeniality in mi and our study of multiphase inference seek to bound and measure the amount of degradation in inferences that can occur when agents attempt ( imperfectly ) to combine information . despite these similarities , the treatment of nuisance parameters are rather different . nuisance parameters lie at the very heart of multiphase inference , defining many of its core issues and techniques . for mi , the typical approaches have been to integrate them out in a bayesian analysis ( e.g. , @xcite ) or assume that the final analyst will handle them ( e.g. , @xcite ) . recent work by @xcite has shed new light on the role of nuisance parameters in mi , but the results are largely negative , demonstrating that nuisance parameters are often a stumbling block for practical mi inference . understanding the role of preprocessing in addressing nuisance parameters , providing robust analyses , and effectively distributing statistical inference represent further challenges beyond those pursued with mi . therefore , much remains to be done in the study of multiphase inference , both theoretical and methodological . we also see multiphase inference as a source for computational techniques , drawing inspiration from the history of mi . mi was initially developed as a strategy for handling missing data in public data releases . however , because mi separates the task of dealing with incomplete data from the task of making inferences , its use spread . it has frequently been used as a practical tool for dealing with missing - data problems where the joint inference of missing data and model parameters would impose excessive modeling or computational burdens . that is , increasingly the mi inference is carried out from imputation through analysis by a single analyst or research group . this is feasible as a computational strategy only because the error properties and conditions necessary for the validity of mi are relatively well - understood ( e.g. , @xcite , @xcite).=1 multiphase methods can similarly guide the development of efficient , statistically - valid computational strategies . once we have a theory showing the trade - offs and pitfalls of multiphase methods , we will be equipped to develop them into general computational techniques . in particular , our experience suggests that models with a high degree of conditional independence ( e.g. , exchangeable distributions for @xmath4 ) can often provide useful inputs for multiphase inferences , even when the true overall model has a greater degree of stochastic structure . the conditional independence structure of such models allows for highly parallel computation with first - phase procedures , providing huge computational gains on modern distributed systems compared to methods based on the joint model.=1 for example , in @xcite , a factored model was used to preprocess a massive collection of irregularly - sampled astronomical time series . the model was sophisticated enough to account for complex observation noise , yet its independence structure allowed for efficient parallelization of the necessary computation . its output was then combined and used for population - level analyses . just as markov chain monte - carlo ( mcmc ) has produced a windfall of tools for approximate high - dimensional integration ( see @xcite for many examples ) , we believe that this type of principled preprocessing , with further theoretical underpinnings , has the potential to become a core tool for the statistical analysis of massive datasets.=1 we would like to acknowledge support from the arthur p. dempster award and partial financial support from the nsf . we would also like to thank arthur p. dempster and stephen blyth for their generous feedback . this work developed from the inaugural winning submission for said award . we also thank david van dyk , brandon kelly , nathan stein , alex damour , and edo airoldi for valuable discussions and feedback , and steven finch for proofreading . finally , we would like to thank our reviewers for their thorough and thoughtful comments , which have significantly enhanced this paper.=1

preprocessing forms an oft - neglected foundation for a wide range of statistical and scientific analyses . however , it is rife with subtleties and pitfalls . decisions made in preprocessing constrain all later analyses and are typically irreversible . hence , data analysis becomes a collaborative endeavor by all parties involved in data collection , preprocessing and curation , and downstream inference . even if each party has done its best given the information and resources available to them , the final result may still fall short of the best possible in the traditional single - phase inference framework . this is particularly relevant as we enter the era of `` big data '' . the technologies driving this data explosion are subject to complex new forms of measurement error . simultaneously , we are accumulating increasingly massive databases of scientific analyses . as a result , preprocessing has become more vital ( and potentially more dangerous ) than ever before . we propose a theoretical framework for the analysis of preprocessing under the banner of multiphase inference . we provide some initial theoretical foundations for this area , including distributed preprocessing , building upon previous work in multiple imputation . we motivate this foundation with two problems from biology and astrophysics , illustrating multiphase pitfalls and potential solutions . these examples also emphasize the motivations behind multiphase analyses both practical and theoretical . we demonstrate that multiphase inferences can , in some cases , even surpass standard single - phase estimators in efficiency and robustness . our work suggests several rich paths for further research into the statistical principles underlying preprocessing . to tackle our increasingly complex and massive data , we must ensure that our inferences are built upon solid inputs and sound principles . principled investigation of preprocessing is thus a vital direction for statistical research .

electronic structure calculations have played an important role in understanding the properties of a wide range of materials systems @xcite . in particular , the kohn - sham formalism of density functional theory @xcite has been the workhorse of ground - state electronic structure calculations . however , the kohn - sham approach requires the computation of single - electron wavefunctions to compute the kinetic energy of non - interacting electrons , whose computational complexity typically scales as @xmath1 for an @xmath2-electron system , thus , limiting standard calculations to materials systems containing few hundreds of atoms . while there has been progress in developing close to linear - scaling algorithms for the kohn - sham approach @xcite , these are still limited to a few thousands of atoms , especially for metallic systems @xcite . the orbital - free approach to dft @xcite , on the other hand , models the kinetic energy of non - interacting electrons as an explicit functional of the electron density , thus circumventing the computationally intensive step of computing the single - electron wavefunctions . further , the computational complexity of orbital - free dft scales linearly with the system size as the ground - state dft problem reduces to a minimization problem in a single field the electron density . the past two decades has seen considerable progress in the development of accurate models for orbital - free kinetic energy functionals @xcite , and , in particular , for systems whose electronic - structure is close to a free electron gas ( for e.g. al , mg ) . also , orbital - free dft calculations are being increasingly used in the simulations of warm dense matter where the electronic structure is close to that of a free electron gas at very high temperatures @xcite . as the reduced computational complexity of orbital - free dft enables consideration of larger computational domains , recent studies have also focused on studying extended defects in al and mg , and have provided important insights into the energetics of these defects @xcite . the widely used numerical implementation of orbital - free dft is based on a fourier space formalism using a plane - wave discretization @xcite . a fourier space formulation provides an efficient computation of the extended interactions arising in orbital - free dft electrostatics and kinetic energy functionals through fourier transforms . further , the plane - wave basis is a complete basis and provides variational convergence in ground - state energy with exponential convergence rates . however , the fourier space formulations are restricted to periodic geometries and boundary conditions that are suitable for perfect bulk materials , but not for materials systems containing extended defects . also , the extended spatial nature of the plane - wave basis affects the parallel scalability of the numerical implementation and is also not suitable for multi - scale methods that rely on coarse - graining . in order to address the aforementioned limitations of fourier space techniques , recent efforts have focussed on developing real - space formulations for orbital - free dft and numerical implementations based on finite - element @xcite and finite difference discretizations @xcite . in the present work , we build on these prior efforts to develop an efficient real - space formulation of orbital - free dft employing the widely used non - local wang - govind - carter ( wgc ) @xcite kinetic energy functional . as in prior efforts @xcite , we reformulate the extended interactions in electrostatics and the non - local terms in the wgc kinetic energy functionals as local variational problems in auxiliary potential fields . however , the proposed reformulation of electrostatic interactions is notably different from previous works , and enables the evaluation of variational configurational forces corresponding to both internal atomic relaxations as well as external cell relaxation under a single framework . further , the proposed formulation naturally extends to all - electron orbital - free dft calculations of warm dense matter @xcite . in the proposed real - space formulation , the ground - state orbital - free dft problem is reformulated as an equivalent saddle point problem of a local functional in electron density , electrostatic potential and the auxiliary potential fields ( kernel potentials ) accounting for the extended interactions in the kinetic energy functional . we employ a higher - order finite - element basis to discretize the formulation , and demonstrate the optimal numerical convergence of both the ground - state energy and configurational forces with respect to the discretization . further , we propose an efficient numerical approach to compute the saddle point problem in electron density , electrostatic potential and kernel potentials by expressing the saddle point problem as a fixed point iteration problem , and using a self - consistent field approach to solve the fixed point iteration problem . we subsequently investigate the accuracy and transferability of the proposed real - space formulation of orbital - free dft for al and mg materials systems . to this end , we compute the bulk properties of al , mg and al - mg intermetallics , and compare it with kohn - sham dft . as orbital - free dft only admits local pseudopotentials , the kohn - sham dft calculations are conducted using both local and non - local psedupotentials . our studies indicates that the bulk properties computed using orbital - free dft for al , mg and al - mg intermetallics are in good agreement with kohn - sham dft . we further investigate the accuracy of orbital - free dft by computing the interatomic forces in al and mg , which are also in good agreement with kohn - sham dft calculations . our studies demonstrate that orbital - free dft is accurate and transferable across a wide range of properties for al , mg and al - mg intermetallics , and can be used to study properties of these materials systems that require computational domains that are not accessible using kohn - sham dft . for instance , in the present study we computed the formation energy of @xmath3 al - mg alloy containing @xmath4 atoms in a unit cell employing the proposed real - space formulation of orbital - free dft , but the same system was found to be prohibitively expensive using kohn - sham dft . we finally investigate the cell - size effects in the electronic structure of point defects , in particular a mono - vacancy in al . prior studies using fourier - based formulations of orbital - free dft have suggested that the formation energy of a mono - vacancy in al is well converged by 108 - 256 atom cell - sizes @xcite . however , coarse - grained real - space calculations have suggested that much larger cell - sizes of the order of 1,000 atoms are required for convergence of vacancy formation energies @xcite , which was also supported by asymptotic estimates @xcite . in order to understand the underpinnings of this discrepancy , we use the finite - element discretized real - space formulation of orbital - free dft and compute the vacancy formation energy using two boundary conditions : ( i ) periodic boundary conditions , equivalent to fourier - space based formulations ; ( ii ) bulk dirichlet boundary conditions , where the perturbations in the electronic structure arising due to the vacancy vanishes on the boundary of the computational domain . our study suggests that while the vacancy formation energy is well converged by 108 atom cell - size using periodic boundary conditions , the electronic fields are not well - converged by this cell - size . on the other hand the bulk dirichlet boundary conditions show well converged formation energy as well as electronic fields by cell sizes of @xmath51,000 atoms , which is consistent with prior real - space calculations . this study reveals that while periodic boundary conditions show a superior convergence in formation energies due to the variational nature of the formalism , the true cell - size effects which also measure convergence of electronic fields are provided by the bulk dirichlet boundary conditions . we note that the proposed real - space formulation with finite - element discretization are crucial to employing bulk dirichlet boundary conditions , which enable the study of isolated defects in bulk . the remainder of the paper is organized as follows . section ii provides a description of the orbital - free dft problem . section iii presents the proposed real - space formulation of the orbital - free dft problem , the configurational forces associated with structural relaxations , and the finite - element discretization of the formulation . section iv discusses the numerical implementation of the formulation and presents an efficient numerical approach for the solution of the saddle point real - space variational problem . section v presents the numerical convergence results of the finite - element discretization of the real - space formulation , the accuracy and transferability of the real - space orbital - free dft formalism for al - mg materials system , and the study of the role of boundary conditions on the cell - size effects in electronic structure calculations of point defects . we finally conclude with a summary and outlook in section vi . the ground - state energy of a charge neutral materials system containing @xmath6 nuclei and @xmath2 valence electrons in density functional theory is given by @xcite @xmath7 where @xmath8 denotes the electron - density and @xmath9 denotes the vector containing the positions of @xmath6 nuclei . in the above , @xmath10 denotes the kinetic energy of non - interacting electrons , @xmath11 is the exchange - correlation energy , @xmath12 is the hartree energy or classical electrostatic interaction energy between electrons , @xmath13 is the classical electrostatic interaction energy between electrons and nuclei , and @xmath14 denotes the electrostatic repulsion energy between nuclei . we now discuss the various contributions to the ground - state energy , beginning with the exchange - correlation energy . the exchange - correlation energy , denoted by @xmath11 , incorporates all the quantum - mechanical interactions in the ground - state energy of a materials system . while the existence of a universal exchange - correlation energy as a functional of electron - density has been established by hohenberg , kohn and sham @xcite , its exact functional form has been elusive to date , and various models have been proposed over the past decades . for solid - state calculations , the local density approximation ( lda ) @xcite and the generalized gradient approximation @xcite have been widely adopted across a range of materials systems . in particular , the lda exchange - correlation energy , which is adopted in the present work , has the following functional form : @xmath15 where @xmath16 , and @xmath17 @xmath18 and @xmath19 . in the present work , we use the ceperley and alder constants @xcite in equation . the last three terms in equation represent electrostatic interactions between electrons and nuclei . the hartree energy , or the electrostatic interaction energy between electrons , is given by @xmath20 the interaction energy between electrons and nuclei , in the case of local pseudopotentials that are adopted in the present work , is given by @xmath21 where @xmath22 denotes the pseudopotential corresponding to the @xmath23 nucleus , which , beyond a core radius is the coulomb potential corresponding to the effective nuclear charge on the @xmath23 nucleus . the nuclear repulsive energy is given by @xmath24 where @xmath25 denotes the effective nuclear charge on the @xmath26 nucleus . the above expression assumes that the core radius of the pseudopotential is smaller than internuclear distances , which is often the case in most solid - state materials systems . we note that in a non - periodic setting , representing a finite atomic system , all the integrals in equations - are over @xmath27 and the summations in equations - include all the atoms . in the case of an infinite periodic crystal , all the integrals over @xmath28 in equations - are over the unit cell whereas the integrals over @xmath29 are over @xmath27 . similarly , in equations - , the summation over @xmath30 is on the atoms in the unit cell , and the summation over @xmath31 extends over all lattice sites . henceforth , we will adopt these notions for the domain of integration and summation . the remainder of the contribution to the ground - state energy is the kinetic energy of non - interacting electrons , denoted by @xmath10 , which is computed exactly in the kohn - sham formalism by computing the single - electron wavefunctions ( eigenfunctions ) in the mean - field @xcite . the conventional solution of the kohn - sham eigenvalue problem , which entails the computation of the lowest @xmath2 eigenfunctions and eigenvalues of the kohn - sham hamiltonian , scales as @xmath32 that becomes prohibitively expensive for materials systems containing a few thousand atoms . while efforts have been focused towards reducing the computational complexity of the kohn - sham eigenvalue problem @xcite , this remains a significant challenge especially in the case of metallic systems . in order to avoid the computational complexity of solving for the wavefunctions to compute @xmath10 , the orbital - free approach to dft models the kinetic energy of non - interacting electrons as an explicit functional of electron density @xcite . these models are based on theoretically known properties of @xmath33 for a uniform electron gas , perturbations of uniform electron gas , and the linear response of uniform electron gas @xcite . as the orbital - free models for the kinetic energy functional are based on properties of uniform electron gas , their validity is often limited to materials systems whose electronic structure is close to a free electron gas , in particular , the alkali and alkali earth metals . further , as the orbital - free approach describes the ground - state energy as an explicit functional of electron - density , it limits the pseudopotentials calculations to local pseudopotentials . while these restrictions constrain the applicability of the orbital - free approach , numerical investigations @xcite indicate that recently developed orbital - free kinetic energy functionals and local pseudopotentials can provide good accuracy for al and mg , which comprise of technologically important materials systems . further , there are ongoing efforts in developing orbital - free kinetic energy models for covalently bonded systems and transition metals @xcite . in the present work , we restrict our focus to the wang - goving - carter ( wgc ) density - dependent orbital - free kinetic energy functional @xcite , which is a widely used kinetic energy functional for ground - state calculations of materials systems with an electronic structure close to a free electron gas . in particular , the functional form of the wgc orbital - free kinetic energy functional is given by @xmath34 where @xmath35 in equation , the first term denotes the thomas - fermi energy with @xmath36 , and the second term denotes the von - weizs@xmath37cker correction @xcite . the last term denotes the density dependent kernel energy , @xmath38 , where the kernel @xmath39 is chosen such that the linear response of a uniform electron gas is given by the lindhard response @xcite . in the wgc functional @xcite , the parameters are chosen to be @xmath40 and @xmath41 . for materials systems whose electronic structure is close to a free - electron gas , the taylor expansion of the density dependent kernel about a reference electron density ( @xmath42 ) , often considered to be the average electron density of the bulk crystal , is employed and is given by @xmath43 in the above equation , @xmath44 and the density independent kernels resulting from the taylor expansion are given by @xmath45 numerical investigations have suggested that the taylor expansion to second order provides a good approximation of the density dependent kernel for materials systems with electronic structure close to a free electron gas @xcite . in particular , in the second order taylor expansion , the contribution from @xmath46 has been found to dominate contributions from @xmath47 . thus , in practical implementations , often , only contributions from @xmath46 in the second order terms are retained for computational efficiency . in this section , we present the local variational real - space reformulation of orbital - free dft , the configurational forces associated with internal ionic relaxations and cell relaxation , and the finite - element discretization of the formulation . [ sec : rs - formulation ] we recall that the various components of the ground - state energy of a materials system ( cf . section [ sec : ofdft ] ) are local in real - space , except the electrostatic interaction energy and the kernel energy component of the wgc orbital - free kinetic energy functional that are extended in real - space . conventionally , these extended interactions are computed in fourier space to take advantage of the efficient evaluation of convolution integrals using fourier transforms . for this reason , fourier space formulations have been the most popular and widely used in orbital - free dft calculations @xcite . however , fourier space formulations employing the plane - wave basis result in some significant limitations . foremost of these is the severe restriction of periodic geometries and boundary conditions . while this is not a limitation in the study of bulk properties of materials , this is a significant limitation in the study of defects in materials . for instance , the geometry of a single isolated dislocation in bulk is not compatible with periodic geometries , and , thus , prior electronic structure studies have mostly been limited to artificial dipole and quadrapole arrangements of dislocations . further , numerical implementations of fourier - space formulations also suffer from limited scalability on parallel computing platforms . moreover , the plane - wave discretization employed in a fourier space formulation provides a uniform spatial resolution , which is not suitable for the development of coarse - graining techniques such as the quasi - continuum method @xcite that rely on an adaptive spatial resolution of the basis . we now propose a real - space formulation that is devoid of the aforementioned limitations of a fourier space formulation . the proposed approach , in spirit , follows along similar lines as recent efforts @xcite , but the proposed formulation differs importantly in the way the extended electrostatic interactions are treated . in particular , the proposed formulation provides a unified framework to compute the configurational forces associated with both internal ionic and cell relaxations discussed in [ sec : configurationalforces ] . we begin by considering the electrostatic interactions that are extended in the real - space . we denote by @xmath48 a regularized dirac distribution located at @xmath49 , and the @xmath26 nuclear charge is given by the charge distribution @xmath50 . defining @xmath51 and @xmath52 , the repulsive energy @xmath14 can subsequently be reformulated as @xmath53 where @xmath54 denotes the self energy of the nuclear charges and is given by @xmath55 we denote the electrostatic potential corresponding to the @xmath26 nuclear charge ( @xmath56 ) as @xmath57 , and is given by @xmath58 the self energy , thus , can be expressed as @xmath59 noting that the kernel corresponding to the extended electrostatic interactions in equations - is the green s function of the laplace operator , the electrostatic potential and the electrostatic energy can be computed by taking recourse to the solution of a poisson equation , or , equivalently , the following local variational problem : [ eq : selfenergylocal ] @xmath60 @xmath61 in the above , @xmath62 denotes the hilbert space of functions such that the functions and their first - order derivatives are square integrable on @xmath27 . we next consider the electrostatic interaction energy corresponding to both electron and nuclear charge distribution . we denote this by @xmath63 , which is given by @xmath64 we denote the electrostatic potential corresponding to the total charge distribution ( electron and nuclear charge distribution ) as @xmath65 , which is given by @xmath66 the electrostatic interaction energy of the total charge distribution , in terms of @xmath65 , is given by @xmath67 as before , the electrostatic interaction energy as well as the potential of the total charge distribution can be reformulated as the following local variational problem : [ eq : totelecenergylocal ] @xmath68 @xmath69 in the above , @xmath70 is a suitable function space corresponding to the boundary conditions of the problem . in particular , for non - periodic problems such as isolated cluster of atoms @xmath71 . for periodic problems , @xmath72 where @xmath73 denotes the unit cell and @xmath74 denotes the space of periodic functions on @xmath73 such that the functions and their first - order derivatives are square integrable . the electrostatic interaction energy in dft , comprising of @xmath75 , @xmath13 and @xmath14 ( cf . equations - ) , can be rewritten in terms of @xmath63 and @xmath54 as @xmath76 for the sake of convenience of representation , we will denote by @xmath77 the vector containing the electrostatic potentials corresponding to all nuclear charges in the simulation domain . using the local reformulation of @xmath63 and @xmath54 ( cf . equations and ) , the electrostatic interaction energy in dft can now be expressed as the following local variational problem : [ eq : elecrsreformulation ] @xmath78 @xmath79 in the above , the minimization over @xmath80 represents a simultaneous minimization over all electrostatic potentials corresponding to @xmath81 . we note that , while the above reformulation of electrostatic interactions has been developed for pseudopotential calculations , this can also be extended to all - electron calculations in a straightforward manner by using @xmath82 and @xmath25 to be the total nuclear charge in the above expressions . thus , this local reformulation provides a unified framework for both pseudopotential as well as all - electron dft calculations . we now consider the local reformulation of the extended interactions in the kernel energy component of the wgc orbital - free kinetic energy functional ( cf . ) . here we adopt the recently developed local real - space reformulation of the kernel energy @xcite , and recall the key ideas and local reformulation for the sake of completeness . we present the local reformulation of @xmath83 and the local reformulations for other kernels ( @xmath84 , @xmath47 , @xmath46 ) follows along similar lines . consider the kernel energy corresponding to @xmath83 given by @xmath85 we define potentials @xmath86 and @xmath87 given by @xmath88 taking the fourier transform of the above expressions we obtain @xmath89 following the ideas developed by choly & kaxiras @xcite , @xmath90 can be approximated to very good accuracy by using a sum of partial fractions of the following form @xmath91 where @xmath92 , @xmath93 , @xmath94 are constants , possibly complex , that are determined using a best fit approximation . using this approximation and taking the inverse fourier transform of equation , the potentials in equation ( [ eq : kerpotential_v ] ) reduce to @xmath95\,,\notag \\ v^0_{\beta}({\boldsymbol{\textbf{x}}})=\sum\limits_{j=1}^m\,[\omega^0_{\beta_j}({\boldsymbol{\textbf{x}}})+a_j \rho^{\beta}({\boldsymbol{\textbf{x}}})]\,.\end{aligned}\ ] ] where @xmath96 and @xmath97 for @xmath94 are given by the following helmholtz equations : @xmath98 we refer to these auxiliary potentials , @xmath99 and @xmath100 introduced in the local reformulation of the kernel energy as _ kernel potentials_. expressing the helmholtz equations in a variational form , we reformulate @xmath101 in ( [ eq : ker0_energy ] ) as the following local variational problem in kernel potentials : [ eq : kernel_variational ] @xmath102 @xmath103d{\boldsymbol{\textbf{x}}}\big\}\ , . \end{split}\ ] ] the variational problem in equation represents a simultaneous saddle point problem on kernel potentials @xmath104 and @xmath105 for @xmath106 . following a similar procedure , we construct the local variational reformulations for the kernel energies @xmath107 , @xmath108 and @xmath109 corresponding to kernels @xmath110 , @xmath47 and @xmath46 , respectively . we denote by @xmath111 , @xmath112 and @xmath113 the lagrangians with respective kernel potentials corresponding to kernel energies of @xmath110 , @xmath47 and @xmath46 , respectively . we refer to the supplemental material for the numerical details of the approximations for each of the kernels used in the present work . finally , using the local variational reformulations of the extended electrostatic and kernel energies , the problem of computing the ground - state energy for a given positions of atoms is given by the following local variational problem in electron - density , electrostatic potentials , and kernel potentials : @xmath114 in the above , @xmath115 denotes the index corresponding to a kernel , and @xmath116 and @xmath70 are suitable function spaces corresponding to the boundary conditions of the problem . in particular , for periodic problems , @xmath72 and @xmath117 . it is convenient to use the substitution @xmath118 , and enforce the integral constraint in @xmath116 using a lagrange multiplier . also , for the sake of notational simplicity , we will denote by @xmath119 and @xmath120 the array of kernel potentials @xmath121 and @xmath122 , respectively . subsequently , the variational problem in equation can be expressed as @xmath123 [ sec : configurationalforces ] we now turn our attention to the configurational forces corresponding to geometry optimization . to this end , we employ the approach of inner variations , where we evaluate the generalized forces corresponding to perturbations of underlying space , which provides a unified expression for the generalized force corresponding to the geometry of the simulation cell internal atomic positions , as well as , the external cell domain . we consider infinitesimal perturbations of the underlying space @xmath124 corresponding to a generator @xmath125 given by @xmath126 such that @xmath127 . we constrain the generator @xmath128 such that it only admits rigid body deformations in the compact support of the regularized nuclear charge distribution @xmath129 in order to preserve the integral constraint @xmath130 . let @xmath28 denote a point in @xmath73 , whose image in @xmath131 is @xmath132 . the ground - state energy on @xmath133 is given by @xmath134 where @xmath135 , @xmath136 , @xmath137 and @xmath138 are solutions of the saddle point variational problem given by equation evaluated over the function space @xmath139 . the subscript @xmath140 on @xmath141 is used to denote that the variational problem is solved on @xmath131 . for the sake of convenience , we will represent the integrand of the lagrangian @xmath141 in equation by @xmath142 and @xmath143 , where @xmath144 denotes the integrand whose integrals are over @xmath73 and @xmath145 denotes the integrand whose integrals are over @xmath27 . the ground - state energy on @xmath133 in terms of @xmath144 and @xmath145 can be expressed as @xmath146 transforming the above integral to domain @xmath73 , we obtain @xmath147 we now evaluate the configurational force given by the gteaux derivative of @xmath148 : @xmath149 in the above , we denote by ` @xmath150 ' the outer product between two vector , by ` @xmath151 ' the dot product between two vectors and by ` @xmath152 ' the dot product between two tensors . we note that in the above expression there are no terms involving the explicit derivatives of @xmath144 and @xmath145 with respect to @xmath153 as @xmath154 , which follows from the restriction that @xmath155 corresponds to rigid body deformations in the compact support of @xmath129 . we further note that terms arising from the inner variations of @xmath148 with respect to @xmath135 , @xmath136 , @xmath137 , @xmath138 and @xmath156 vanish as @xmath157 @xmath158 , @xmath159 , @xmath160 and @xmath161 are the solutions of the saddle point variational problem corresponding to @xmath162 . we now note the following identities @xmath163 @xmath164 using these identities in equation , and rearranging terms , we arrive at @xmath165 where @xmath166 and @xmath167 denote eshelby tensors corresponding to @xmath144 and @xmath145 , respectively . the expressions for the eshelby tensors @xmath166 and @xmath168 explicitly in terms of @xmath169 , @xmath170 , @xmath119 , @xmath120 , @xmath171 and @xmath172 are given by @xmath173 in the above , for the sake of brevity , we represented by @xmath174 the integrand corresponding to @xmath175 . we also note that the terms @xmath176 and @xmath177 do not appear in the expressions for @xmath166 and @xmath178 , respectively , as @xmath179 on the compact support of @xmath129 owing to the restriction that @xmath128 corresponds to rigid body deformations in these regions . it may appear that evaluation of the second term in equation is not tractable as it involves an integral over @xmath27 . to this end , we split this integral on a bounded domain @xmath180 containing the compact support of @xmath48 , and its complement . the integral on @xmath181 can be computed as a surface integral . thus , @xmath182 where @xmath183 denotes the outward normal to the surface @xmath184 . the last equality follows from the fact that @xmath185 on @xmath181 . the configurational force in equation provides the generalized variational force with respect to both the internal positions of atoms as well as the external cell domain . in order to compute the force on any given atom , we restrict the compact support of @xmath128 to only include the atom of interest . in order to compute the stresses associated with cell relaxation ( keeping the fractional coordinates of atoms fixed ) , we restrict @xmath128 to affine deformations . thus , this provides a unified expression for geometry optimization corresponding to both internal ionic relaxations as well as cell relaxation . we further note that , while we derived the configurational force for the case of pseudopotential calculations , the derived expression is equally applicable for all - electron calculations by using @xmath186 . [ sec : fe - discretization ] among numerical discretization techniques , the plane - wave discretization has been the most popular and widely used in orbital - free dft @xcite as it naturally lends itself to the evaluation of the extended interactions in electrostatic energy and kernel kinetic energy functionals using fourier transforms . further , the plane wave basis offers systematic convergence with exponential convergence in the number of basis functions . however , as noted previously , the plane - wave basis also suffers from notable drawbacks . importantly , plane - wave discretization is restricted to periodic geometries and boundary conditions which introduces a significant limitation , especially in the study of defects in bulk materials @xcite . further , the plane - wave basis has a uniform spatial resolution , and thus is not amenable to adaptive coarse - graining . moreover , the use of plane - wave discretization involves the numerical evaluation of fourier transforms whose scalability is limited on parallel computing platforms . in order to circumvent these limitations of the plane - wave basis , there is an increasing focus on developing real - space discretization techniques for orbital - free dft based on finite - difference discretization @xcite and finite - element discretization @xcite . in particular , the finite - element basis @xcite , which is a piecewise continuous polynomial basis , has many features of a desirable basis in electronic structure calculations . while being a complete basis , the finite - element basis naturally allows for the consideration of complex geometries and boundary conditions , is amenable to unstructured coarse - graining , and exhibits good scalability on massively parallel computing platforms . moreover , the adaptive nature of the finite - element discretization also enables the consideration of all - electron orbital - free dft calculations that are widely used in studies of warm dense matter @xcite . further , recent numerical studies have shown that by using a higher - order finite - element discretization significant computational savings can be realized for both orbital - free dft @xcite and kohn - sham dft calculations @xcite , effectively overcoming the degree of freedom disadvantage of the finite - element basis in comparison to the plane - wave basis . let @xmath187 denote the finite - element subspace of @xmath70 , where @xmath188 represents the finite - element mesh size . the discrete problem of computing the ground - state energy for a given positions of atoms , corresponding to equation , is given by the constrained variational problem : @xmath189 in the above , @xmath190 , @xmath191 , @xmath192 and @xmath193 denote the finite - element discretized fields corresponding to square - root electron - density , electrostatic potential , and kernel potentials , respectively . we restrict our finite - element discretization such that atoms are located on the nodes of the finite - element mesh . in order to compute the finite - element discretized solution of @xmath194 , we represent @xmath195 as a point charge on the finite - element node located at @xmath196 , and the finite - element discretization provides a regularization for @xmath194 . previous investigations have suggested that such an approach provides optimal rates of convergence of the ground - state energy ( cf . @xcite for a discussion ) . the finite - element basis functions also provide the generator of the deformations of the underlying space in the isoparametric formulation , where the same finite - element shape functions are used to discretize both the spatial domain as well as the fields prescribed over the domain . thus , the configurational force associated with the location of any node in the finite - element mesh can be computed by substituting for @xmath128 , in equation , the finite - element shape function associated with the node . thus , the configurational force on any finite - element node located at an atom location corresponds to the variational ionic force , which are used to drive the internal atomic relaxation . the forces on the finite - element nodes that do not correspond to an atom location represent the generalized force of the energy with respect to the location of the finite - element nodes , and these can be used to obtain the optimal location of the finite - element nodes a basis adaptation technique . we note that the local real - space variational formulation in section [ sec : rs - formulation ] , where the extended interactions in the electrostatic energy and kernel functionals are reformulated as local variational problems , is essential for the finite - element discretization of the formulation . in this section , we present the details of the numerical implementation of the finite - element discretization of the real - space formulation of orbital - free dft discussed in section [ sec : rs - ofdft ] . subsequently , we discuss the solution procedure for the resulting discrete coupled equations in square - root electron - density , electrostatic potential and kernel potentials . [ sec : fe - basis ] a finite - element discretization using linear tetrahedral finite - elements has been the most widely used discretization technique for a wide range of partial differential equations . linear tetrahedral elements are well suited for problems involving complex geometries and moderate levels of accuracy . however in electronic structure calculations , where the desired accuracy is commensurate with chemical accuracy , linear finite elements are computationally inefficient requiring of the order of hundred thousand basis functions per atom to achieve chemical accuracy . a recent study @xcite has demonstrated the significant computational savings of the order of 1000-fold compared to linear finite - elements that can be realized by using higher - order finite - element discretizations . thus , in the present work we use higher - order hexahedral finite elements , where the basis functions are constructed as a tensor product of basis functions in one - dimension @xcite . [ sec : numersoln ] the discrete variational problem in equation involves the computation of the following fields square - root electron - density , electrostatic potential and kernel potentials . two solution procedures , suggested in prior efforts @xcite , for solving this discrete variational problem include : ( i ) a simultaneous solution of all the discrete fields in the problem ; ( ii ) a nested solution procedure , where for every trial square - root electron - density the discrete electrostatic and kernel potential fields are computed . given the non - linear nature of the problem , the simultaneous approach is very sensitive to the starting guess and often suffers from lack of robust convergence , especially for large - scale problems . the nested solution approach , on the other hand , while constituting a robust solution procedure , is computationally inefficient due to the huge computational costs incurred in computing the kernel potentials which involves the solution of a series of helmholtz equations ( cf . equation ) . thus , in the present work , we will recast the local variational problem in equation as the following fixed point iteration problem : [ eq : fixedpoint ] @xmath197 @xmath198 we solve this fixed point iteration problem using a mixing scheme , and , in particular , we employ the anderson mixing scheme @xcite with full history in this work . our numerical investigations suggest that the fixed point iteration converges , typically , in less than ten self - consistent iterations even for large - scale problems , thus , providing a numerically efficient and robust solution procedure for the solution of the local variational orbital - free dft problem . we note that this idea of fixed point iteration has independently and simultaneously been investigated by another group in the context of finite difference discretization @xcite , and have resulted in similar findings . in the fixed point iteration problem , we employ a simultaneous solution procedure to solve the non - linear saddle point variational problem in @xmath190 and @xmath191 ( equation ) . we employ an inexact newton solver provided by the petsc package @xcite with field split preconditioning and generalized - minimal residual method ( gmres ) @xcite as the linear solver . the discrete helmholtz equations in equation are solved by employing block jacobi preconditioning and using gmres as the linear solver . an efficient and scalable parallel implementation of the solution procedure has been developed to take advantage of the parallel computing resources for conducting the large - scale simulations reported in this work . in this section , we discuss the numerical studies on al , mg and al - mg intermetallics to investigate the accuracy and transferability of the real - space formulation of orbital - free dft ( rs - ofdft ) proposed in section [ sec : rs - ofdft ] . wherever applicable , we benchmark the real - space orbital - free dft calculations with plane - wave based orbital - free dft calculations conducted using profess @xcite , and compare with kohn - sham dft ( ks - dft ) calculations conducted using the plane - wave based abinit code @xcite . further , we demonstrate the usefulness of the proposed real - space formulation in studying the electronic structure of isolated defects . [ sec : calc ] , scaledwidth=46.0% ] , scaledwidth=46.0% ] in all the real - space orbital - free dft calculations reported in this section , we use the local reformulation of the density - dependent wgc @xcite kinetic energy functional proposed in section [ sec : rs - formulation ] , the local density approximation ( lda ) @xcite for the exchange - correlation energy , and bulk derived local pseudopotentials ( blps ) @xcite for al and mg . cell stresses and ionic forces are calculated using the unified variational formulation of configurational forces developed in section [ sec : configurationalforces ] . in the second order taylor expansion of the density - dependent wgc functional about the bulk electron density ( cf . section [ sec : ofdft ] ) , we only retain the @xmath46 term for the computation of bulk properties as the contributions from @xmath46 dominate those of @xmath47 for bulk materials systems . however , in the calculations involving mono - vacancies , where significant spatial perturbations in the electronic structure are present , we use the full second order taylor expansion of the density dependent wgc functional . we recall from section [ sec : rs - formulation ] that in order to obtain a local real - space reformulation of the extended interactions in the kinetic energy functionals , the kernels ( @xmath199 , @xmath110 , @xmath47 , @xmath46 ) are approximated using a sum of @xmath200 partial fractions where the coefficients of the partial fractions are computed using a best fit approximation ( cf . equation ) . these best fit approximations for @xmath201 that are employed in the present work are given in the supplemental material . it has been shown in recent studies that @xmath202 suffices for al @xcite . however , we find that @xmath203 is required to obtain the desired accuracy in the bulk properties of mg , and table [ tab : bulktrf2 ] shows the comparison between the kernel approximation with @xmath203 and plane - wave based orbital - free dft calculations conducted using profess @xcite for mg . thus , we use the best fit approximation of the kernels with @xmath202 for al , and employ the approximation with @xmath203 for mg and al - mg intermetallics . henceforth , we will refer by rs - ofdft - fe the real - space orbital - dft calculations conducted by employing the local formulation and finite element discretization proposed in section [ sec : rs - ofdft ] . the ks - dft calculations used to assess the accuracy and transferability of the proposed real - space orbital - free dft formalism are performed using the lda exchange correlation functional @xcite . the ks - dft calculations are conducted using both local blps as well as the non - local troullier - martins pseudopotential ( tm - nlps ) @xcite in order to assess the accuracy and transferability of both the model kinetic energy functionals in orbital - free dft as well as the local pseudopotentials to which the orbital - free dft formalism is restricted to . the tm - nlps for al and mg are generated using the fhi98pp code @xcite . within the fhi98pp code , we use the following inputs : @xmath204 angular momentum channel as the local pseudopotential component for both al and mg , default core cutoff radii for the @xmath205 , @xmath206 , and @xmath204 angular momentum channels , which are @xmath207 bohr and @xmath208 bohr for al and mg respectively , and the lda @xcite exchange - correlation . for brevity , henceforth , we refer to the ks - dft calculations with blps and tm - nlps as ks - blps and ks - nlps , respectively . in all the rs - ofdft - fe calculations reported in this work , the finite - element discretization , order of the finite - elements , numerical quadrature rules and stopping tolerances are chosen such that we obtain 1 mev/ atom accuracy in energies , @xmath209 accuracy in cell stresses and @xmath210 accuracy in ionic forces . similar accuracies in energies , stresses and ionic forces are achieved for ks - dft calculations by choosing the appropriate k - point mesh , plane - wave energy cutoff , and stopping tolerances within abinit s framework . all calculations involving geometry optimization are conducted until cell stresses and ionic forces are below threshold values of @xmath211 and @xmath212 , respectively . [ sec : convergence ] we now study the convergence of energy and stresses with respect to the finite - element discretization of the proposed real - space orbital - free dft formulation . in a prior study on the computational efficiency afforded by higher - order finite - element discretization in orbital - free dft @xcite , it was shown that second and third - order finite - elements offer an optimal choice between accuracy and computational efficiency . thus , in the present study , we limit our convergence studies to hex27 and hex64 finite - elements , which correspond to second- and third - order finite - elements . as a benchmark system , we consider a stressed fcc al unit cell with a lattice constant @xmath213 bohr . we first construct a coarse finite - element mesh and subsequently perform a uniform subdivision to obtain a sequence of increasingly refined meshes . we denote by @xmath188 the measure of the size of the finite - element . for these sequence of meshes , we hold the cell geometry fixed and compute the discrete ground - state energy , @xmath214 , and hydrostatic stress , @xmath215 . the extrapolation procedure proposed in motamarri et . al @xcite allows us to estimate the ground - state energy and hydrostatic stress in the limit as @xmath216 , denoted by @xmath217 and @xmath218 . to this end , the energy and hydrostatic stress computed from the sequence of meshes using hex64 finite - elements are fitted to expressions of the form @xmath219 to determine @xmath220 . in the above expression , @xmath221 denotes the number of elements in a finite - element mesh . we subsequently use @xmath217 and @xmath218 as the exact values of the ground - state energy and hydrostatic stress , respectively , for the benchmark system . figures [ fig : energyconv ] and [ fig : stressconv ] show the relative errors in energy and hydrostatic stress plotted against @xmath222 , which represents a measure of @xmath188 . we note that the slopes of these curves provide the rates of convergence of the finite - element approximation for energy and stresses . these results show that we obtain close to optimal rates of convergence in energy of @xmath223 , where @xmath224 is polynomial interpolation order ( @xmath225 for hex27 and @xmath226 for hex64 ) . further , we obtain close to @xmath227 convergence in the stresses , which represents optimal convergence for stresses . the results also suggest that higher accuracies in energy and stress are obtained with hex64 in comparison to hex27 . thus , we will employ hex64 finite - elements for the remainder of our study . .[tab : bulktrf1 ] [ cols="^,^,^,^,^,^",options="header " , ] in order to understand this boundary condition dependence of the cell - size effects , we compute the perturbations in the electronic fields due to the presence of the mono - vacancy by subtracting from the electronic fields corresponding to the mono - vacancy the electronic fields of a perfect crystal . to this end , we define the normalized perturbations in the electronic fields computed on the finite - element mesh to be @xmath228 in the above , @xmath229 and @xmath230 denote the electronic fields in the computational domain with the vacancy and those without the vacancy ( perfect crystal ) , respectively . @xmath231 denotes the volume average of an electronic field over the computational cell . as a representative metric , in the definition of @xmath232 and @xmath233 we only consider the kernel potentials corresponding to @xmath83 . figures [ fig : monovaccf1 ] and [ fig : monovaccf2 ] shows the normalized corrector fields for the mono - vacancy , computed using periodic boundary conditions , along the face - diagonal of the periodic boundary . it is interesting to note from these results that the perturbations in the electronic structure due to the vacancy are significant up to @xmath234 computational cells . thus , although the vacancy formation energy appears converged by @xmath235 computational cell while using periodic boundary conditions , the electronic fields are not converged till a cell - size of @xmath234 computational cell . on the other hand , the cell - size convergence in mono - vacancy formation energy suggested by the bulk dirichlet boundary conditions is inline with the convergence of electronic fields . these results unambiguously demonstrate that the cell - size effects in the electronic structure of defects are larger than those suggested by a cell - size study of defect formation energies employing periodic boundary conditions . using bulk dirichlet boundary conditions for the cell - size study of defect formation energies provides a more accurate estimate of the cell - size effects in the electronic structure of defects , and the extent of electronic structure perturbations due to a defect . further , while periodic boundary conditions are limited to the study of point defects , bulk dirichlet boundary conditions can be used to also study defects like isolated dislocations @xcite , whose geometry does not admit periodic boundary conditions . we have developed a local real - space formulation of orbital - free dft with wgc kinetic energy functionals by reformulating the extended interactions in electrostatic and kinetic energy functionals as local variational problems in auxiliary potentials . the proposed real - space formulation readily extends to all - electron orbital - free dft calculations that are commonly employed in warm dense matter calculations . building on the proposed real - space formulation we have developed a unified variational framework for computing configurational forces associated with both ionic and cell relaxations . further , we also proposed a numerically efficient approach for the solution of ground - state orbital - free dft problem , by recasting the local saddle point problem in the electronic fields electron density and auxiliary potential fields as a fixed point iteration problem and employing a self - consistent iteration procedure . we have employed a finite - element basis for the numerical discretization of the proposed real - space formulation of orbital - free dft . our numerical convergence studies indicate that we obtain close to optimal rates of convergence in both ground - state energy and configurational forces with respect to the finite - element discretization . we subsequently investigated the accuracy and transferability of the proposed real - space formulation of orbital - free dft for al - mg materials system . to this end , we conducted a wide range of studies on al , mg and al - mg intermetallics , including computation of bulk properties for these systems , formation energies of al - mg intermetallics , and ionic forces in bulk and in the presence of point defects . our studies indicate that orbital - free dft and the proposed real - space formulation is in good agreement with kohn - sham dft calculations using both local pseudopotentials as well as non - local pseudpotentials , thus providing an alternate linear - scaling approach for electronic structure studies in al - mg materials system . we finally investigated the cell - size effects in the electronic structure of a mono - vacancy in al , and demonstrated that the cell - size convergence in the vacancy formation energy computed by employing periodic boundary conditions is not commensurate with the convergence of the electronic fields . on the other hand , the true cell - size effects in the electronic structure are revealed by employing the bulk dirichlet boundary conditions , where the perturbations in the electronic fields due to the defect vanish on the boundary of the computational domain . our studies indicate that the true cell - size effects are much larger than those suggested by periodic calculations even for simple defects like point defects . we note that the proposed real - space formulation and the finite - element basis are crucial to employing the bulk dirichlet boundary conditions that are otherwise inaccessible using fourier based formulations . the proposed formulation , besides being amenable to complex geometries , boundary conditions , and providing excellent scalability on parallel computing platforms , also enables coarse - graining techniques like the quasi - continuum reduction @xcite to conduct large - scale electronic structure calculations on the energetics of extended defects in al - mg materials system , and is an important direction for future studies . we gratefully acknowledge the support from the u.s . department of energy , office of basic energy sciences , division of materials science and engineering under award no . de - sc0008637 that funds the predictive integrated structural materials science ( prisms ) center at university of michigan , under the auspices of which this work was performed . v.g . also gratefully acknowledges the hospitality of the division of engineering and applied sciences at the california institute of technology while completing this work . we also acknowledge advanced research computing at university of michigan for providing the computing resources through the flux computing platform . 999 r. m. martin , _ electronic structure : basic theory and practical methods _ ( cambridge university press , cambridge , 2011 ) . p. hohenberg and w. kohn , phys . rev . * 136 * , b864 ( 1964 ) . w. kohn and l. j. sham , phys . rev . * 140 * , a1133 ( 1965 ) . s. goedecker , rev . 71 * , 1085 ( 1999 ) . d. r. bowler and t. miyazaki , rep . prog . phys . * 75 * , 036503 ( 2012 ) . p. motamarri and v. gavini , phys . b * 90 * , 115127 ( 2014 ) . r. parr and w. yang , _ density - functional theory of atoms and molecules _ ( oxford university press , 1989 ) . wang and m. p. teter , phys . b * 45 * , 13196 ( 1992 ) . e. smargiassi and p. a. madden , phys . b * 49 * , 5220 ( 1994 ) . y. a. wang , n. govind , and e. a. carter , phys . b * 58 * , 13465 ( 1998 ) . y. a. wang , n. govind , and e. a. carter , phys . b * 60 * , 16350 ( 1999 ) . karasiev , travis sjostrom , and s.b . trickey , phys . b * 86 * , 115101 ( 2012 ) . y. ke , f. libisch , j. xia , l .- w . wang , and e. a. carter , phys . lett . , * 111 * , 066402 ( 2013 ) . karasiev , d.chakraborty , and s.b . trickey , in _ many - electron approaches in physics , chemistry , and mathematics _ , l. delle site and v. bach eds . ( springer , heidelberg , 2014 ) , 113 - 134 . y. ke , f. libisch , j. xia , and e. a. carter , phys . b , * 89 * , 155112 ( 2014 ) . f. lambert , j. clrouin , and g. zrah , phys . e * 73 * , 016403 ( 2006 ) . f. lambert , j. clrouin , j .- f . danel , l. kazandjian , and g. zrah , phys . e * 77 * , 026402 ( 2008 ) . d. a. horner , f. lambert , j. d. kress , and l. a. collins , phys . b * 80 * , 024305 ( 2009 ) . l. burakovsky , c. ticknor , j. d. kress , l. a. collins , and f. lambert , phys . e * 87 * , 023104 ( 2013 ) . d. sheppard , j. d. kress , s. crockett , l. a. collins , and m. p. desjarlais , phys . e * 90 * , 063314 ( 2014 ) . v. gavini , k. bhattacharya , and m. ortiz , phys . b * 76 * , 180101 ( 2007 ) . g. ho , m. t. ong , k. j. caspersen , and e. a. carter , phys . * 9 * , 4951 ( 2007 ) . q. peng , x. zhang , l. hung , e. a. carter , and g. lu , phys . b * 78 * , 054118 ( 2008 ) . i. shin , a. ramasubramaniam , c. huang , l. hung , and e. a. carter , philos . mag . * 89 * , 3195 ( 2009 ) . i. shin and e. a. carter , phys . b * 88 * , 064106 ( 2013 ) . m. iyer , b. radhakrishnan , and v. gavini , j. mech . solids * 76 * , 260 ( 2015 ) . l. hung , c. huang , i.shin , g. ho , v. l. ligneres , and e. a. carter , comput . comm . , * 181 * , 2208 ( 2010 ) . m. chen , j. xia , c. huang , j. m. dieterich , l. hung , i. shin , and e. a. carter , comp . * 190 * , 228 ( 2015 ) . v. gavini , j. knap , k. bhattacharya , and m. ortiz , j. mech . solids * 55 * , 669 ( 2007 ) . b. g. radhakrishnan and v. gavini , phys . b * 82 * , 094117 ( 2010 ) . p. motamarri , m. iyer , j. knap , and v. gavini , j. comput . phys . * 231 * , 6596 ( 2012 ) . garcia - cervera , comm . comp . phys . * 2 * , 334 ( 2007 ) . p. suryanarayana and d. phanish , j. comp . phys . * 275 * , 524 ( 2014 ) . s. ghosh and p. suryanarayana , arxiv preprint arxiv:1412.8250 ( 2014 ) . v. gavini , l. liu , j. mech . solids * 59 * , 1536 ( 2011 ) . d. m. ceperley and b. j. alder , phys . rev . lett . * 45 * 566 ( 1980 ) . j. p. perdew and a. zunger , phys . b * 23 * 5048 ( 1981 ) . d. c. langreth and m. j. mehl , phys . b * 28 * , 1809 ( 1983 ) . j. p. perdew , j. a. chevary , s. h. vosko , k. a. jackson , m. r. pederson , d. j. singh , and c. fiolhais , phys . b * 46 * , 6671 ( 1992 ) . c. huang and e. a. carter , phys . chem . phys.*10 * , 7109 ( 2008 ) . j. xia and e. a. carter , phys . b * 86 * , 235109 ( 2012 ) . c. huang and e. a. carter , phys . rev . b * 85 * , 045126 ( 2012 ) . m. finnis , _ interatomic forces in condensed matter _ ( oxford university press , 2003 ) . n. choly and e. kaxiras , solid state communications * 121 * , 281 ( 2002 ) . v. gavini , k. bhattacharya , and m. ortiz , j. mech . solids * 55 * , 697 ( 2007 ) . brenner and l.r . scott , _ the mathematical theory of finite - element methods _ ( springer , new york , 2002 ) . p. motamarri , m. nowak , k. leiter , j. knap , and v. gavini , j. comput . phys . * 253 * , 308 ( 2013 ) . d. g. anderson , j. assoc . * 12 * , 547 ( 1965 ) . s. balay , j. brown , k. buschelman , v. eijkhout , w. d. gropp , d. kaushik , m. g. knepley , l. c. mcinnes , b. f. smith , h. zhang , _ petsc 3.4 users manual _ , argonne national laboratory , 2013 . y. saad and m. h. schultz , siam j. sci . * 7 * , 856 ( 1986 ) . x. gonze , j .- m . beuken , r. caracas , f. detraux , m. fuchs , g .- rignanese , l. sindic , m. verstraete , g. zerah , f. jollet , m. torrent , a. roy , m. mikami , p. ghosez , j .- y . raty and d. c. allan , comput . sci . * 25 * , 478 ( 2002 ) . x. gonze , b. amadon , p.m. anglade , j .- beuken , f. bottin , p. boulanger , f. bruneval , d. caliste , r. caracas , m. cote , t. deutsch , l. genovese , ph . ghosez , m. giantomassi , s. goedecker , d. hamann , p. hermet , f. jollet , g. jomard , s. leroux , m. mancini , s. mazevet , m.j.t . oliveira , g. onida , y. pouillon , t. rangel , g .- rignanese , d. sangalli , r. shaltaf , m. torrent , m.j . verstraete , g. zrah , j.w . zwanziger , comput . . commun . * 180 * , 2582 ( 2009 ) . n. troullier and j. l. martins , phys . rev . b * 43 * , 1993 , ( 1991 ) . m. fuchs and m. scheffler , comput . * 119 * , 67 ( 1999 ) . m. feuerbacher , c. thomas , j.p.a . makongo , s. hoffmann , w. carrillo - cabrera , r. cardoso , y. grin , g. kreiner , j .- joubert , t. schenk , j. gastaldi , h. nguyen - thi , n. mangelinck - noel , b. billia , p. donnadieu , a. czyrska - filemonowicz , a. zielinska - lipiec , b. dubiel , t. weber , p. schaub , g.r krauss , v. gramlich , j. christensen , s. lidin , d. fredrickson , m. mihalkovic , w. sikora , j. malinowski , s. brhne , t. proffen , w. assmus , m. de boissieu , f. bley , j .- l . chemin , j. schreuer , z. kristallographie * 222 * , 259 ( 2007 ) . n. chetty , m. weinert , t. s. rahman and j. w. davenport , phys . b * 52 * , 6313 ( 1995 ) . m. j. gillan , j. phys . : condens . matter * 1 * , 689 ( 1989 ) . iyer , m. , gavini , v. , j. mech . solids * 59 * , 1506 ( 2011 ) .

we propose a local real - space formulation for orbital - free dft with density dependent kinetic energy functionals and a unified variational framework for computing the configurational forces associated with geometry optimization of both internal atomic positions as well as the cell geometry . the proposed real - space formulation , which involves a reformulation of the extended interactions in electrostatic and kinetic energy functionals as local variational problems in auxiliary potential fields , also readily extends to all - electron orbital - free dft calculations that are employed in warm dense matter calculations . we use the local real - space formulation in conjunction with higher - order finite - element discretization to demonstrate the accuracy of orbital - free dft and the proposed formalism for the al - mg materials system , where we obtain good agreement with kohn - sham dft calculations on a wide range of properties and benchmark calculations . finally , we investigate the cell - size effects in the electronic structure of point defects , in particular a mono - vacancy in al . we unambiguously demonstrate that the cell - size effects observed from vacancy formation energies computed using periodic boundary conditions underestimate the extent of the electronic structure perturbations created by the defect . on the contrary , the bulk dirichlet boundary conditions , accessible only through the proposed real - space formulation , which correspond to an isolated defect embedded in the bulk , show cell - size effects in the defect formation energy that are commensurate with the perturbations in the electronic structure . our studies suggest that even for a simple defect like a vacancy in al , we require cell - sizes of @xmath0 atoms for convergence in the electronic structure .

thermal dilepton production in the mass region below 1 gev / c@xmath1 is largely mediated by the light vector mesons @xmath0 , @xmath2 and @xmath3 . among these , the @xmath0(770 mev / c@xmath1 ) is the most important , due to its strong coupling to the @xmath4 channel and its short lifetime of only 1.3 fm / c . changes both in width and in mass were originally suggested as precursor signatures of the chiral transition @xcite and subsequent models have tied these changes directly @xcite or indirectly @xcite to chiral symmetry restoration . the first na60 results focused on the space - time averaged spectral function of the @xmath0 @xcite ; more details were added recently @xcite . the present paper concentrates mainly on new developments , in particular first results on acceptance - corrected @xmath5 and @xmath6 spectra for different mass windows details of the na60 apparatus can be found in @xcite , while the different analysis steps ( including the critical assessment of the combinatorial background from @xmath7 and @xmath8 decays through event mixing ) are described in @xcite . the results reported here were obtained from the analysis of data taken in 2003 for in - in at 158 agev . the left part of fig . [ fig1 ] shows the opposite - sign , background and signal dimuon mass spectra , integrated over all collision centralities . after subtracting the combinatorial background and the signal fake matches , the resulting net spectrum contains about 360000 muon pairs in the mass range 0 - 2 gev / c@xmath9 , roughly 50% of the total available statistics . the associated average charged - particle multiplicity density measured by the vertex tracker is @xmath10 = 120 . vector mesons @xmath2 and @xmath3 are completely resolved with a mass resolution at the @xmath2 of 20 mev / c@xmath1 . most of the analysis is done in four classes of collision centrality ( defined through the charged - particle multiplicity density ) : peripheral ( 4@xmath1130 ) , semiperipheral ( 30@xmath11110 ) , semicentral ( 110@xmath11170 ) and central ( 170@xmath11240 ) . the peripheral data can essentially be described by the expected electromagnetic decays of the neutral mesons @xcite . this is not the case in the more central bins , due to the presence of a strong excess . to isolate this excess without any fits , a novel procedure has been devised @xcite . the resulting spectrum for all centralities and all @xmath12 shows a a peaked structure residing on a broad continuum . the same feature can be seen for each centrality bin , with a yield strongly increasing with centrality , but remaining essentially centered around the position of the nominal @xmath0 pole . a more quantitative analysis of the shape of the excess mass spectra _ vs. _ centrality has been performed , using a finer subdivision of the data into 12 centrality bins @xcite . on the basis of this analysis it is possible to rule out that the excess shape can be accounted for by the cocktail @xmath0 residing on a broad continuum , independent of centrality . for different mass windows . right : rapidity distribution of the excess data for the mass window @xmath13 gev/@xmath14 and for three selected @xmath5 bins . the measured data ( full markers ) are reflected around midrapidity ( open markers).,title="fig:",scaledwidth=40.0% ] for different mass windows . right : rapidity distribution of the excess data for the mass window @xmath13 gev/@xmath14 and for three selected @xmath5 bins . the measured data ( full markers ) are reflected around midrapidity ( open markers).,title="fig:",scaledwidth=40.0% ] a differential analysis in @xmath12 was originaly performed and mass spectra associated with three different @xmath12 windows , without acceptance correction , were obtained @xcite . the na60 acceptance relative to @xmath15 as a function of @xmath12 as shown in fig . [ fig3 ] ( left ) , implies that one should in principle perform the acceptance correction using a 3-dimensional grid in ( @xmath16 ) space . this can lead , however , to large errors once the correction is applied . instead , the correction is performed in 2-dimensional ( @xmath17 ) space , using the measured rapidity distribution as an input . the latter was determined with an acceptance correction found , in an iterative way , from monte carlo simulations matched to the data in @xmath18 and @xmath12 . on the basis of this rapidity distribution , 0.1 gev/@xmath19 bins in m and 0.2 gev/@xmath20 bins in @xmath12 were used to determine the remaining 2-dimensional correction . after correction the results were integrated over the three extended mass windows @xmath21 , @xmath13 and @xmath22 gev / c@xmath1 . in fig . [ fig3 ] ( right ) the rapidity distribution of the central mass window is shown for three different @xmath12 windows , exhibiting a close resemblance to the distribution of inclusive pion production , as measured by na49 for pb - pb and na60 for in - in . the results for the acceptance - corrected @xmath5 spectra are summarized in fig . the errors shown are purely statistical . systematic errors arise from the acceptance corrections including the rapidity distribution used , the subtraction of the cocktail , and the subtraction of the combinatorial background plus fake matches . for @xmath23 gev / c , the combinatorial background contributes most , ranging from 10 to 25% for semiperipheral up to central . for @xmath24 gev / c , the statistical errors dominate . the data show a significant dependence on mass , but hardly any on centrality . to bear out the differences in mass more clearly , the data were summed over the three more central bins and plotted in fig . [ fig5 ] ( left ) as a function of @xmath25 . the inverse slope parameter @xmath26 as determined from differential fits of the @xmath6 spectra with @xmath27 , using a sliding window in @xmath5 , is plotted on the right . instead of flattening at very low @xmath6 as should be expected from radial flow , strikingly all spectra _ steepen _ , which is equivalent to very small values of @xmath26 . for comparison , the @xmath3 resonance , placed just in between the upper two mass windows , does flatten as expected . moreover , depending on the fit region , @xmath26 covers an unusually large dynamic range . finally , the largest masses have the steepest @xmath6 spectrum , _ i.e. _ the smallest value of @xmath26 everywhere , again contrary to radial flow and to what is usually observed for hadrons . all this suggests that different mass regions are coupled to basically different emission sources . summarizing , the previously measured excess mass spectra and the new acceptance - corrected @xmath5 and @xmath6 spectra present an unexpected behaviour . beyond the @xmath0 spectral function this may lead to a better understanding of the continuum part of the spectra for @xmath28 gev / c@xmath9 , possibly disentangling parton - hadron duality . 00 r. d. pisarski , phys . * 110b * , 155 ( 1982 ) g. e. brown , m. rho , phys . rept . * 363 * , 85 ( 2002 ) r. rapp and j. wambach , adv . . phys . * 25 * , 1 ( 2000 ) r. arnaldi _ et al . _ ( na60 collaboration ) , phys . * 96 * ( 2006 ) 162302 s. damjanovic _ et al . _ ( na60 collaboration ) , nucl - ex/0609026 ; s. damjanovic _ et al . _ ( na60 collaboration ) , nucl - ex/0701015 g. usai _ et al . _ ( na60 collaboration ) , eur . j. * c43 * , 415 ( 2005 ) ; g. usai _ et al . _ ( na60 collaboration ) , these proceedings . m. keil _ et al . _ , nucl . instrum . meth . * a539 * , 137 ( 2005 ) and * a546 * , 448 ( 2005 ) r. shahoyan _ et al . _ ( na60 collaboration ) , nucl . * a774 * , 677 ( 2006 ) r. shahoyan _ et al . _ ( na60 collaboration ) , these proceedings

the na60 experiment at the cern sps has studied low - mass muon pairs in 158 agev in - in collisions . a strong excess of pairs is observed above the yield expected from neutral meson decays . the unprecedented sample size of close to 400k events and the good mass resolution of about 2% have made it possible to isolate the excess by subtraction of the decay sources ( keeping the @xmath0 ) . the shape of the resulting mass spectrum exhibits considerable broadening , but essentially no shift in mass . the acceptance - corrected transverse - momentum spectra have a shape atypical for radial flow and show a significant mass dependence , pointing to different sources in different mass regions .

galaxy cluster mergers are ideal probes of gravitational collapse and the hierarchical structure formation in the universe . observations of the evolving cluster mass function provide a sensitive cosmological test that is both independent of , and complementary to , other methods ( e.g. , bao , sn , cmb ) @xcite . the use of galaxy clusters as cosmological probes relies on the accuracy of scaling relations between the total mass and observable quantities . galaxy cluster mergers will disrupt the intracluster gas and cause departures from these scaling relations ( e.g. , * ? ? ? * ; * ? ? ? given that these mass scaling relations are a necessary ingredient for the interpretation of on - going cosmological surveys , a detailed understanding of the intracluster medium ( icm ) gas physics in mergers has become increasingly important . llccclcc + satellite & pointing & obsid & r.a . & dec & date obs & exposure & pi + & & & & & & xis0/xis1/xis3 & + & & & & & & ( ks ) + + _ suzaku _ & north & 806096010 & 13 31 15.53 & - 01 39 13.3 & 2011 jul 2 & 74.7/74.7/74.7 & s. randall + _ suzaku _ & center & 806095010 & 13 30 46.63 & - 01 53 14.3 & 2011 jul 24 & 38.0/38.0/38.0 & s. randall + _ suzaku _ & south & 806097010 & 13 30 13.15 & - 02 06 22.7 & 2011 jul 9 & 70.2/70.2/70.2 & s. randall + _ suzaku _ & southeast & 806098010 & 13 31 27.19 & - 02 04 19.9 & 2011 jul 6 & 55.9/55.9/56.0 & s. randall + _ suzaku _ & southeast & 806098020 & 13 31 28.58 & - 02 02 29.4 & 2011 dec 23 & 11.3/11.3/11.3 & s. randall + _ chandra _ & north & 11878 & 13 31 10.83 & - 01 43 21.0 & 2010 may 11 & 19.4@xmath3 & s. murray + _ chandra _ & center & 11879 & 13 30 50.30 & - 01 52 28.0 & 2010 may 9 & 19.7@xmath3 & s. murray + _ chandra _ & south & 12914 & 13 30 15.80 & - 02 02 28.7 & 2011 mar 16 & 36.8@xmath3 & s. murray + + + [ table : obs ] the properties of the icm in the cores of merging clusters have been studied in detail , since the high density and surface brightness of the gas in this region is well - suited to high angular resolution observations with _ chandra _ and _ xmm - newton _ ( see * ? ? ? * for a review ) . with the launch of the low particle - background _ suzaku _ mission , it has become possible to probe the low gas density and faint surface brightness regions at the virial radii of nearby galaxy clusters ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? observational studies at these radii have mostly focused on relatively relaxed , massive , cool - core systems . due to the limited number of observations , the dynamical evolution of the icm in strong merger events out to the viral radius is not clearly understood . strongly merging , bimodal clusters are where we expect to find the large - scale filaments and accretion shocks . comparing results from observations of mergers and relaxed clusters at the virial radius will provide an important confirmation of our current picture of large - scale structure formation . the double clusters identified from _ einstein _ observations ( a1750 , a98 , a115 , a3395 ) are ideal targets for studying the virial radii of strongly merging clusters @xcite . these canonical binary galaxy clusters have two separated peaks of x - ray emission , and distortions in their x - ray surface brightness distributions suggest ongoing merger events ( e.g. , * ? ? ? most of these systems are in fact triple clusters , with all sub - clusters lying roughly along the same line , suggesting the presence of large - scale structure filaments . a1750 is a triple merger system at a redshift of 0.085 , with an average temperature of 4.5 kev @xcite . it contains three main sub - clusters with x - ray centroids : a1750n ( j2000 , ra : 202.79@xmath4 , dec : @xmath51.73@xmath4 ) , a1750c ( j2000 , ra : 202.71@xmath4 , dec : @xmath51.86@xmath4 ) , and a1750s ( j2000 , ra : 202.54@xmath4 , dec : @xmath52.105@xmath4 ) . mmt data provided redshifts for the brightest cluster galaxies of 0.0836 , 0.0878 , and 0.0865 ( see section [ sec : resultsopt ] for details ) . a1750 was identified as a strongly merging double " cluster due to the presence of two bright x - ray subcluster peaks , which are clearly visible in the _ einstein _ image @xcite . the centers of a1750n and a1750c are separated by 9.7@xmath6 ( 930 kpc ; see figure [ fig : image ] ) . and observations indicate possible shock heated gas with an elevated temperature of 5.5 kev between these sub - clusters , suggesting that they are in an early stage merger @xcite . more recent _ xmm - newton _ observations confirm this region of elevated temperature , and also indicate that a1750c may itself be undergoing a merger @xcite . a1750s was identified with observations . its center is located 17.5@xmath6 ( 1.68 mpc ) to the southwest of a1750c , along the same line connecting a1750c and a1750n , presumably tracing a large - scale filament . the 0.2 @xmath5 10 kev luminosities of the two brighter sub - clusters are 1.3 @xmath7 ergs s@xmath8 for a1750n and 2.2 @xmath7 ergs s@xmath8 for a1750c @xcite . the x - ray luminosity of the fainter , southern sub - cluster a1750s is 6.4 @xmath9 ergs s@xmath8 , estimated from pspc observations . here , we present results from mosaic _ suzaku _ observations of a1750 out to the virial radius . these new observations , together with archival _ chandra _ and _ xmm - newton _ observations , probe the icm properties from the subcluster cores out to their viral radii . previous studies of other ( non - merging ) systems have found entropy profiles that flatten at large radii , in contradiction with theoretical predictions , possibly due to the presence of unresolved cool gas clumps @xcite . this behavior shows some variation with azimuth , suggesting a connection with large - scale structure and gas accretion @xcite . we use our observations , which extend both along and perpendicular to the putative large - scale structure filament , to look for correlations between the icm properties , the surrounding large - scale environment , and to examine the merger dynamics . this paper is organized as follows : in section [ sec : obs ] , we describe the _ suzaku _ , _ chandra _ , and _ mmt _ data used in our analysis . in section [ sec : analysis ] , the analysis of the x - ray and optical observations is described in detail . in section [ sec : syst ] , we discuss systematic errors that are relevant to the _ suzaku _ x - ray measurements at large radii . in sections [ sec : resultsopt],[sec : resultsxray ] , and [ sec : thermo ] we discuss our results and present our conclusions in section [ sec : conc ] . throughout the paper , a standard @xmath10cdm cosmology with h@xmath11 = 70 km s@xmath8 mpc@xmath8 , @xmath12 = 0.7 , and @xmath13 = 0.3 is assumed . in this cosmology , 1@xmath14 at the redshift of the cluster corresponds to @xmath15 96.9 kpc . unless otherwise stated , reported errors correspond to 90% confidence intervals . r vs r ) plot of galaxies included in the spectroscopic catalog , with selected passive cluster members plotted in red . * lower panel : * color - color ( g @xmath5 r vs r @xmath5 i ) plot of galaxies included in the spectroscopic catalog , with selected passive cluster members plotted in red.[fig : colormag ] , width=340 ] the majority of the galaxy spectroscopic redshifts used in this analysis are new observations obtained using the hectospec instrument @xcite at the _ mmt _ observatory 6.5 m telescope on mt . hopkins , az . a single hectoscpec configuration places up to 300 fibers in a region of the sky approximately one degree in diameter . we use data from two such configurations , which resulted in 517 individual spectroscopic redshift measurements . to supplement our hectospec spectroscopy , we include data from the literature , when available . specifically , we use 12 spectroscopic redshift measurements from @xcite , 68 from @xcite , seven from @xcite , 19 from the 6df galaxy survey @xcite , and 200 from the sloan digital sky survey ( sdss ; * ? ? ? the sdss selection includes all objects within a 0.5 degree radius of the centroids of the x - ray emission of a1750n , a1750c , and a1750s , and with a spectroscopic redshift falling in the interval 0.03 @xmath16 0.15 , which easily captures the range of recessional velocities of galaxies associated with a1750 . we then check for duplicate entries across the different input redshift catalogs , resulting in 24 removals and a final data set of 799 spectroscopic redshifts . in addition to optical spectroscopy , we also use optical photometry from the sdss catalogs . we perform a query of all objects classified as galaxies within a 0.5 degree radius of the centroid of the x - ray surface brightness of each subcluster and download all of the available optical photometry in the _ ugriz _ bands for those sources . [ fig : image ] a1750 was observed with _ suzaku _ with five pointings during july 2011 and december 2011 ( see table [ table : obs ] ) . we process the unfiltered _ suzaku _ data with _ heasoft _ version 6.13 , and the latest calibration database caldb as of may 2014 . the raw event files are filtered using the ftool _ aepipeline_. in addition to the standard filtering performed by _ aepipeline _ , we require an earth elevation angle @xmath17 5@xmath18 , a geomagnetic cut - off rigidity of @xmath17 6 gv / c , and exclude data collected during passages through the south atlantic anomaly as described in @xcite . the data taken with 3@xmath193 and 5@xmath195 clocking modes are merged and the corners of the chips illuminated by the fe - calibration sources are excluded from further analysis . we carefully examine each light curve after the initial screening to ensure that the data are free from background flaring events . due to the increase in charge injection in data taken with xis1 after 2011 june 1 , the two rows adjacent to the standard charge - injected rows are removed . the region lost due to a putative micrometeorite hit on xis0 is also excluded from our analysis . the net exposure times of each xis0 , xis1 , and xis3 pointing after filtering are given in table [ table : obs ] . due to our strict filtering , 30 ks of the total exposure time was lost . the total filtered _ suzaku _ xis0/xis1/xis3 exposure time is 250.1/250.1/250.2 ks . the _ chandra _ observations that were used in the analysis are summarized in table [ table : obs ] . for each observation , the aimpoint was on the front - side illuminated acis - i ccd . all data were reprocessed from the level 1 event files using ciao 4.6 and caldb 4.4.7 . cti and time - dependent gain corrections were applied . lc_clean was used to check for periods of background flares . the mean event rate was calculated from a source free region using time bins within 3@xmath20 of the overall mean , and bins outside a factor of 1.2 of this mean were discarded . there were no periods of strong background flares . to model the background we used the caldb blank sky background files appropriate for this observation , normalized to match the 10 - 12 kev count rate in our observations to account for variations in the particle background . the total filtered acis - i exposure time is 75.9 ks . sdss _ ugriz _ photometry samples the full optical spectral energy distribution ( sed ) for galaxies in a1750 , including the 4000 break that is located in the @xmath21-band at the redshift of a1750 . the 4000 break is a strong feature , characteristic of the passive red sequence galaxies that dominate the galaxy populations of evolved galaxy clusters @xcite . we identify candidate cluster member galaxies of a1750 using the red sequence in the _ gri _ bands , which span the break . the red sequence selection involves two steps . the initial selection is made in color - magnitude space ( @xmath22 vs. @xmath23 ; figure [ fig : colormag ] top panel ) with a manual identification of the over - density of galaxies with approximately the same @xmath22 color . we then perform a linear fit in color - magnitude space to define the red sequence in a1750 , and flag all galaxies within @xmath240.125 in @xmath22 magnitudes as candidate red sequence galaxies . the second step occurs in color - color space ( @xmath22 vs @xmath25 ; figure [ fig : colormag ] bottom panel ) , where we identify an over - density of candidate red sequence galaxies with similar @xmath25 colors . galaxies that satisfy the initial color - magnitude selection while also falling within @xmath240.125 magnitudes of the mean @xmath25 color of the over density in color - color space are flagged as red sequence galaxies . the range of color values that we use accounts for both the observed intrinsic scatter in the red sequence of massive galaxy clusters ( @xmath260.05 - 0.1 mags ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and the typical sdss photometric uncertainties of @xmath150.025 magnitudes . we extract an image of a1750 in the 0.5 @xmath5 7 kev energy band and mosaic the pointings in sky coordinates . the non - x - ray background ( nxb ) images are generated using the ` night - earth ' data ( nte ) using the ftool _ xisnxbgen _ @xcite . the nxb images are then subtracted from the mosaicked image prior to exposure correction . to generate the exposure maps , we first simulate a monochromatic photon list assuming a 20@xmath6 uniform extended source for each observation with the _ xrt _ ray - tracing simulator _ xissim _ @xcite . these vignetting - corrected photon lists are then used with _ xisexpmapgen _ to generate exposure maps of each pointing , as described in detail in @xcite . regions with @xmath27% of the maximum exposure time are removed . the resulting exposure maps for each pointing are merged . the particle background subtracted , vignetting - corrected image is shown in figure [ fig : image ] ( left panel ) . 1.11.5 lccccccccccc + region & n1 & n2 & n3 & n4 & se1 & se2 & se3 & s1 & s2 + + + n1 & 40.88 & 15.64 & 1.41 & 0.45 & - & - & - & - & - + n2 & 7.82 & 56.23 & 15.95 & 1.11 & - & - & - & - & - + n3 & 1.18 & 15.95 & 58.37 & 13.58 & - & - & - & - & - + n4 & 0.07 & 0.57 & 8.54 & 55.84 & - & - & - & - & - + se1 & - & - & - & - & 56.64 & 7.86 & 0.05 & - & - + se2 & - & - & - & - & 10.81 & 61.54 & 2.20 & - & - + se3 & - & - & - & - & 0.45 & 4.65 & 57.16 & - & - + s1 & - & - & - & - & - & - & - & 60.78 & 14.35 + s2 & - & - & - & - & - & - & - & 9.05 & 69.38 + + + + [ table : psf ] to detect x - ray point sources unresolved by _ , we use the three _ chandra _ pointings of the cluster , which overlap with the northern , central , and southern _ suzaku _ pointings . the locations of the point sources in the field - of - view ( fov ) are detected using ciao s _ wavdetect _ tool and are shown in the right panel of figure [ fig : image ] . since the point spread function ( psf ) sizes of _ suzaku _ and _ chandra _ are different , the extents of the point sources reported by _ wavdetect _ can not be used directly to exclude point source in the _ suzaku _ fov . we use the following procedure to determine a reliable and conservative radius for point source exclusion . the brightest point source within both the _ chandra _ and _ suzaku _ fov ( j2000 ; ra : 202.603@xmath4 , dec : @xmath51.808@xmath4 ) is selected as a test case ( shown with green circle in figure [ fig : image ] left panel ) . the source is located in a fairly faint region ( 9@xmath14 away from the center of a1750 to northeast ) . spectrum of the point - source is extracted using ciao s _ specextract _ tool and is fitted with an absorbed power - law model with an index fixed to 1.4 ( the slope associated with the x - ray background spectrum at 0.5@xmath58 kev ; e.g. @xcite ) , while the normalization is left free . based on the best - fit power - law index and normalization ( 5.22 @xmath28 photons kev@xmath8 @xmath29 s@xmath8 ) obtained from the _ chandra _ fits , a 120 ks long _ suzaku _ xis observation is simulated using the _ xissim _ tool . to assess the impact of the source flux on the measured parameters of the diffuse emission , we add the simulated source spectrum to a typical diffuse emission spectrum with 1000 net counts . we then incrementally increase the source exclusion radius ( thereby decreasing the contribution of the point source to the total emission ) and examine the effect on the best - fitting parameters to the total ( source plus diffuse emission ) spectrum . we find that for all exclusion radii @xmath30 the best - fitting parameters ( kt , abundance , and normalization ) are not significantly affected by the point source contribution . since this estimate is based on the analysis of the brightest brightest point source in a faint region , the exclusion radius for fainter point sources would be smaller . we note that since all our spectral extraction regions include at least 2000 total counts , this radius represents a conservative estimate . we therefore exclude regions with radii of 35@xmath31 around point sources detected by _ chandra _ from our _ suzaku _ analysis . the southeast _ suzaku _ pointing does not have an overlapping _ chandra _ observation . therefore , the point sources in this region are detected from the suzaku data using ciao s _ wavdetect _ tool . the detection is performed using _ suzaku _ s half - power radius of 1@xmath6 as the wavelet radius , as done in @xcite . the point sources detected with _ suzaku _ are shown as green regions in the left panel in figure [ fig : image ] . spectra are extracted from the filtered event files in _ xselect_. corresponding detector redistribution function ( rmf ) files are constructed using the _ xisrmfgen _ tool , while the ancillary response function ( arf ) files are constructed using the _ xisarfgen _ tool assuming a uniform surface brightness in a 20@xmath6 radius . cutoff - rigidity - weighted particle - induced background spectra are extracted from the nte data for each detector using the _ xisnxbgen _ tool . the particle induced background spectrum is subtracted from each source spectrum prior to fitting . spectral fitting is performed in the 0.5 @xmath5 7 kev energy band where the _ suzaku _ xis is the most sensitive . the cluster emission is modeled with an absorbed single temperature thermal plasma model with atomdb version 2.0.2 @xcite . _ xspec _ v12.8.2 is used to perform the spectral fits @xcite with the extended c - statistic as an estimator of the goodness of fits . we co - add front illuminated ( fi ) xis0 and xis3 data to increase the signal - to - noise , while the back illuminated ( bi ) xis1 data are modeled simultaneously with the front illuminated observations due to the difference in energy responses . we adopt the solar abundance table from @xcite . the galactic column density is frozen at the leiden / argentine / bonn ( lab ) galactic hi survey value @xcite of 2.37 @xmath19 10@xmath32 @xmath29 in our fits . we examine the local x - ray background emission using the rosat all sky survey ( _ rass _ ) data extracted from a 1@xmath52 degree annulus surrounding the central sub - cluster s centroid . a region 19@xmath6 @xmath5 21@xmath6 away from the central sub - cluster a1750c in the southeast pointing is used to extract the local background ( see figure [ fig : image ] ) . the rass spectrum is simultaneously fit with the local background xis fi and bi spectra using two gaussian models for solar wind charge exchange at 0.56 and 0.65 kev , an unabsorbed _ apec _ model for local hot bubble ( lhb ) emission , and an absorbed _ apec _ model for galactic halo ( gh ) emission @xcite . the abundances of these models are set to solar , while the redshifts are fixed at zero . an absorbed power - law component with a photon index of 1.4 is added to the model to include emission from unresolved extragalactic sources ( primarily agn ) . we note that statistical uncertainties in the observed local background parameters given in this section are 1@xmath20 . the best - fit temperature of the lhb component is 0.14@xmath33 kev , with a normalization of 3.22@xmath34 @xmath35 arcmin@xmath36 . the best - fit temperature and normalization of the gh component is 0.69@xmath37 kev and 1.79@xmath38 @xmath39 @xmath35 arcmin@xmath36 . the normalization for the cxb power - law component is 5.84@xmath40 photons kev@xmath8 @xmath29 s@xmath8 arcmin@xmath41 at 1 kev , corresponding to a cxb flux of ( 1.15 @xmath42 ergs s@xmath8 @xmath29 deg@xmath41 . the flux ( 6.22 @xmath24 0.16 ) @xmath43 ergs s@xmath8 @xmath29 deg@xmath41 in the 0.5@xmath52 kev band is in agreement with the value ( 7.7@xmath240.4 @xmath43 ergs s@xmath8 @xmath29 deg@xmath41 ) reported by @xcite . in studies of low surface brightness emission , it is crucial to estimate the contribution of various systematic uncertainties , particularly those related to background modeling , to the total error budget . we consider the following potential sources of systematic error in our analysis ; i ) uncertainties due to stray light contamination and the large size of the psf of _ suzaku _ s mirrors ; ii ) uncertainties due to intrinsic spatial variations in the local soft background ; iii ) systematics associated with the nxb ; vi ) uncertainties due to the intrinsic spatial variation of unresolved point sources . due to _ suzaku s relatively large psf , some x - ray photons that originate from one particular region on the sky may be detected elsewhere on the detector . the psf spreading in each direction is calculated by generating simulated event files using the ray - tracing simulator _ xissim _ @xcite . _ chandra _ x - ray images of each annular sector ( shown in figure [ fig : image ] left panel ) and the best - fit spectral models obtained from the _ suzaku _ observations are used to simulate event files with 1@xmath44 photons . the fraction of photons that are spread into the surrounding annuli is calculated for each xis detector and annulus sector . relative contributions are weighted by the effective area at 1.5 kev of each detector to calculate the overall percentage contribution ( given in table [ table : psf ] ) . we find that the majority of photons originating in an annulus on the sky are detected in the same annulus ( except region n1 ) on the detector . up to 15% of the photons may be detected in surrounding annuli . however , the percentage fraction of photons that scatter into the outermost annuli at @xmath2 is defined as the radius within which the average matter density of the cluster is 200 times the critical density of the universe at the cluster redshift . ] to the north and southeast from the bright cores is small ( @xmath11% ) . these results are consistent with the photon fractions reported in @xcite and @xcite . considering the shallow temperature distribution of a1750 measured by _ observations , the psf is expected to have a minimal effect on the measurements of temperature in the outermost regions . to estimate the effect of psf spreading on our temperature and normalization measurements , we jointly fit the spectra of sectors with models , with the normalizations scaled according to the fractions listed in table [ table : psf ] . in all cases , the change in best - fit parameter values due to scattered flux from other annuli is significantly less than the statistical errors on the measured observables ( see table [ table:1tfits ] ) . 1.11.7 lcccc pointing & region 1 & region 2 & region 3 & region 4 + north & 1 & 4 & 8 & 19 + southeast & 0.7 & 3.4 & 25 & - + south & 4 & 20 & - & - + + + + [ table : bkgfluc ] to model the soft x - ray foreground and cosmic x - ray background , we jointly fit rass data with local xis background spectra , as described in detail in section [ sec : suzanalysis ] . we find that the local x - ray background is consistent with the rass data . however , spatial variations in the background level can introduce additional systematic uncertainties on x - ray observables . to estimate the effect of these uncertainties , we perform 10,000 monte carlo realizations of the background model . the model parameters are allowed to vary simultaneously within their 1@xmath20 uncertainty ranges obtained from the joint rass - local background fit . a variation of up to @xmath153.6% of the nxb level is also taken into account @xcite . the percent systematic uncertainty contributions due to the variance in cosmic , local , and particle background on the temperature estimates are given in table [ table : bkgfluc ] . we find that the effect on temperature and normalization is negligible ( @xmath15 1% ) and smaller than the statistical uncertainties in the inner regions ( shown in table [ table:1tfits ] ) , while it can be as large as 25% in the outskirts near @xmath2 . these uncertainties are included in the total error budget in our analysis by adding them in quadrature . 1.11.7 .estimated 1@xmath20 fluctuations in the cxb level due to unresolved point sources in the _ suzaku _ fov in units of @xmath45 ergs @xmath29 s@xmath8 deg@xmath41 . [ cols="<,^,^,^,^",options="header " , ] [ table : cxbfluc ] the intrinsic variations in the unresolved cxb component can be an important source of uncertainty in the analyses of cluster outskirts with _ suzaku_. to estimate the magnitude of this component , we follow a similar approach to that described in @xcite . the _ suzaku _ data alone allow us to detect point sources to a limiting flux of 1.3 @xmath46 ergs @xmath29 s@xmath8 deg@xmath41 in our observations . the contribution of unresolved point sources to the total flux in ergs @xmath29 s@xmath8 deg@xmath41 can be estimated as @xcite : @xmath47 the source flux distribution in the 2@xmath510 kev band is described by the analytical function @xmath48 \rm{erg\ cm^{-2}\ s^{-1 } } , \label{eqn : psflux}\ ] ] where @xmath49 , @xmath50 , @xmath51 , and @xmath52 . we then integrate equation [ eqn : cxbflux ] from a lower limit of @xmath53 ergs @xmath29 s@xmath8 ( the flux of the faintest source in our fov ) up to the upper limit of @xmath54 ergs @xmath29 s@xmath8 @xcite . the integration gives an unresolved 2 @xmath5 10 kev flux of ( 1.20 @xmath55 ergs @xmath29 s@xmath8 deg@xmath41 . the expected deviation in the cxb level due to unresolved point sources is @xmath56 where @xmath57 is the solid angle @xcite . using the power - law relation ( given in equation [ eqn : psflux ] ) in equation [ eqn : fluc ] , we calculate the 1@xmath58 rms fluctuations in the cxb ( given in table [ table : cxbfluc ] ) . we find that the variation is 4.3 @xmath43 erg @xmath29 s@xmath8 deg@xmath41 in the faintest outermost se3 region , which extends to @xmath2 . these estimates are consistent with the values reported by @xcite and @xcite . the 1@xmath20 uncertainty on the measured cxb ( from joint rass and local background fits ) is comparable to the expectation value of fluctuations on the cxb brightness calculated here . we include this variation in our monte - carlo markov realizations of the x - ray background to account for the cxb variation ( as described in detail in section [ sec : systcxb ] ) . the final systematic errors on the observed quantities were added in quadrature . [ fig : rsmap ] in this section , we describe our search for substructure in the optical data and use them to further constrain the dynamical state and merger history of a1750 and determine whether the subclusters are bound to each other . in figure [ fig : rsmap ] we overplot the spatial density of red sequence ( rs ) galaxies on the sky on the _ suzaku _ x - ray image . this map is created from a cloud - in - cell interpolation of the spatial distribution of red sequence selected galaxies on the sky , where galaxies are weighted by their r - band magnitudes ( brighter galaxies weighted more heavily ) . the resulting map of the surface density of red sequence light traces the collisionless galaxy component of the system . we then applied a broad gaussian smoothing kernel of 54@xmath31 to generate the contours shown in the image . the peaks observed in the red light distribution roughly align with the peaks in the x - ray emission . we note that we do not find strong evidence for extended filaments along the axis of the aligned clumps in rs light , thus no large - size groups are detected along the filament direction to the north . the lack of evidence indicates that the filaments do not contain significant large group - like structures with a detectable red sequence population . the large sample of spectroscopic redshifts available in the a1750 field provides an opportunity to investigate the dynamical state of cluster member galaxies . we first characterize the cluster member dynamics for the entire system by making an initial selection of cluster members that are within a projected physical radius of 1 mpc of the centroid of the x - ray emission of each subcluster ( this is approximately equal to the region covered by our hectospec observations ) , and which have redshifts in the interval 0.07 @xmath59 0.1 . we then use the bi - weight location and scale estimators @xcite as the starting guess for the median and dispersion of cluster member velocities . we then iterate this process , rejecting galaxies with redshifts more than 3-@xmath20 away from the median until the redshift sample converges . this results in an estimate of the velocity dispersion for the entire system of @xmath60 km s@xmath8 , and a median redshift of @xmath61 ( a recession velocity , @xmath62 km s@xmath8 ) based on 243 cluster member redshifts . . , width=302 ] [ fig : vdist ] 1.11.5 lcccc + region & @xmath63 & @xmath64 & @xmath65 & @xmath66 + & & & ( km s@xmath8 ) & ( km s@xmath8 ) + total system & 243 & 0.0861@xmath240.0028 & 780@xmath2430 & 0 + a1750n & 25 & 0.0832@xmath240.0006 & 750@xmath24160 & -810 + a1750c & 40 & 0.0864@xmath240.0004 & 835@xmath24120 & 90 + a1750s & 33 & 0.0868@xmath240.0019 & 532@xmath2460 & 200 + + + + [ table : memberdynamics ] given that we have hundreds of spectroscopic cluster members , we can also test for line - of - sight velocity differences between the three individual x - ray sub - clusters . we define subsets of spectra that originate from galaxies located in three non - overlapping 3@xmath14 radius circular regions on the sky that are centered on the x - ray peaks of each of the three distinct sub - clusters . for each of these regions , we use all of the cluster member galaxies that satisfy the @xmath243-@xmath20 velocity range for the total cluster system from above , and compute the bi - weight location and scale estimates of the median and dispersion in the galaxy velocities . these regions extend out radially @xmath15300 kpc from each x - ray centroid , and therefore only include a relatively small fraction of the full sample of 243 cluster member spectra ( between 25@xmath540 cluster members per subregion ) . the resulting kinematics estimates are given in table [ table : memberdynamics ] ; the central and southern x - ray clumps have redshifts that are similar to the median for the total system , but the northern clump is blue - shifted , with a peculiar velocity of @xmath67 km s@xmath8 ( see figure [ fig : vdist ] ) . the observed peculiar velocities imply that any relative motion between the central and southern clumps is in the plane of the sky , while the northern clump is moving at least partly along a vector that is normal to the sky . the velocity dispersion of galaxies within the total structure is not larger than the velocity dispersion of the individual clumps , indicating that the system is unrelaxed . the individual subclusters havent begun to virialize into the final larger cluster , i.e. the total mass of all three clumps is @xmath68 , and 840 km s@xmath8 is well below the velocity dispersion of a virialized structure of that mass . the shape of the velocity distributions within the different subcluster regions ( plotted in figure [ fig : vdist ] ) , suggest that northern and central subclusters are less well - structured ( with asymmetric velocity dispersion profiles ) than that of the southern subcluster . this could be due to some degree of interaction between the central and northern subclusters , while the southern subcluster may still be infalling ( i.e. , has not started tidally interacting with the other systems ) . we further calculated the implied virial masses of individual subclusters based on the velocity dispersions . using the @xcite scaling relations , the virial masses of a1750n , a1750c , and a1750s are 4.6@xmath69 m@xmath70 , 6.4@xmath71 m@xmath70 , and 1.7@xmath72 m@xmath70 , respectively . these masses are consistent with total masses of each subcluster obtained from x - ray observations ( see section [ sec : mass ] for detailed calculations ) . 1.11.5 cccccc + @xmath73&@xmath74 & @xmath75 & @xmath76 & @xmath77 & @xmath78 + ( rad ) & ( degrees ) & ( kpc ) & ( kpc ) & ( @xmath79 ) & ( @xmath80 ) + 4.542 & 71.73 & 2966.7 & 5072.2 & 931.2 & 22 + 5.319 & 23.86 & 1016.9 & 4727.4 & 2186.2 & 78 + + + + we apply a dynamical model introduced by @xcite and @xcite to evaluate the dynamical state of the subclusters a1750n and a1750c . this model allows us to estimate the most likely angle between the merger axis and the plane of the sky . the equations of motion take two different forms , depending on whether the subclusters are gravitationally bound or not . for the case where they are gravitationally bound , we parameterize the equations of motion in the following form : @xmath81 where @xmath76 is the subclusters separation at the moment of maximum expansion , @xmath82 is the system s total mass , and @xmath73 is the variable used to parametrize friedmann s equation , also know as development angle . for the case of not gravitationally bound subclusters , the equations are parametrized as : @xmath83 where @xmath84 is the velocity of expansion at the asymptotic limit . @xmath85 , the radial velocity difference , and @xmath86 , the projected distance , are related to the parameters of the equations by @xmath87 where @xmath74 is the projection angle of the system with respect to the plane of the sky . the virial mass of this subclusters is @xmath88 ( sum of the masses of both subclusters within @xmath2 uncertainties are quoted here as 68% confidence intervals ; see section [ sec : mass ] for detailed calculations ) derived from @xcite icm models . we assume that the subclusters velocities are the median velocities of their galaxies . the projected distance on the plane of the sky between the x - ray center of each subcluster is @xmath89 = 0.93 mpc . the difference of the median redshifts of these subclusters yields a radial velocity difference of @xmath90 . by setting @xmath91 = 12.4 gyr , the age of the universe at the mean redshift of these subclusters ( @xmath92 ) , we close the system of equation . the parametric equations are then solved via an iterative procedure , which computes the radial velocity difference @xmath93 for each projection angle @xmath74 . using simple energy considerations , we determine the limits of the bound solutions : @xmath94 figure [ fig : alpha_vr ] presents the projection angle ( @xmath74 ) as a function of the radial velocity difference ( @xmath93 ) between the subclusters . the uncertainties in the measured radial velocity and mass of the subclusters lead to a range in the solutions for the projection angles ( @xmath95 and @xmath96 ) . we compute the relative probabilities of these solutions by : @xmath97 where each solution is represented by the index @xmath98 . we then normalize the probabilities by @xmath99 . ) as a function of the radial velocity difference ( @xmath93 ) between the subclusters . uo , bi , and bo stand for unbound outgoing , bound incoming , and bound outgoing solutions . solid red and blue lines correspond to unbound and bound solutions , respectively . the vertical solid line corresponds to the radial velocity difference between the median velocities of the galaxies in each subcluster . dashed lines correspond to 68@xmath80 confidence ranges . [ fig : alpha_vr ] ] 1.11.5 cccccc + @xmath73&@xmath74 & @xmath75 & @xmath76 & @xmath77 & @xmath78 + ( rad ) & ( degrees ) & ( kpc ) & ( kpc ) & ( @xmath79 ) & ( @xmath80 ) + 1.692 & 86.45 & 15025.5 & 886.0 & 610.4 & 0.02 + + + + 1.11.5 ccccccc + @xmath73&@xmath74 & @xmath75 & @xmath76 & @xmath77 & @xmath78 & relative + ( rad ) & ( degrees ) & ( kpc ) & ( kpc ) & ( @xmath79 ) & ( @xmath80 ) & motion + 2.871 & 77.68 & 7871.0 & 8017.1 & 113.0 & 10 & outgoing + 3.385 & 74.76 & 6390.7 & 6486.5 & 114.4 & 15 & incoming + 4.987 & 4.39 & 1684.9 & 4620.9 & 1442.4 & 75 & incoming + + + + solving the parametric equations we obtain two bound solutions and one unbound solution . for the case of the bound solutions , the subclusters are either approaching each other at 931 km s@xmath8 ( 22@xmath80 probability ) or at 2186 km s@xmath8 ( 78@xmath80 probability ) . the former solution corresponds to a collision in less than 3.1 gyr , given their separation of @xmath15 2.97 mpc . the latter corresponds to a collision in less than 460 myr , given their separation of @xmath15 1017 kpc . the unbound solution ( 0.02@xmath80 probability ) corresponds to a separation of @xmath15 15 mpc . the parameters of these solutions are presented in tables [ tab : bound ] and [ tab : unbound ] . given its very low probability , the unbound solution can be neglected , while the bound solution in which the separation between the clusters is @xmath15 1017 kpc is highly favored ( @xmath100 probability ) . as mentioned in @xcite , the method to determine the dynamical state of a system of clusters from @xcite assumes a purely radial infall . also , the way the probabilities are computed , by integrating over the angles determined by the uncertainties on the mass of the system , favors small angle solutions . therefore , the probabilities for the solutions should be treated with caution , as we have no information about the angular momentum of this subclusters . now , we apply the same procedure to determine the dynamical state of the pair a1750c - a1750s . using the virial mass estimated from the velocity dispersion of the galaxies in the southern subcluster , the total mass of this system is @xmath101 ( uncertainties are quoted here at the 68% confidence level ) . the difference between the median redshifts of these subclusters yields a radial velocity difference of @xmath102 . solving the system of parametric equations ( equations ( [ eq : r_bound ] ) ( [ eq : v_r_alpha ] ) ) yields the results presented in table [ tab : bound2 ] , with a1750c - a1750s being bound in all solutions . the most likely solution ( 75% probability ) indicates that the merger is happening very close to the plane of the sky ( @xmath103 degrees ) , also supporting the scenario in which all three subclusters are merging along a cosmic filament . we extract spectra in concentric annular sectors along the north ( filament ) , south ( filament ) , and southeast ( off - filament ) directions from the regions shown in figure [ fig : image ] . each spectral extraction region is selected to include at least 2000 net source counts . the lhb+cxb+gh components are fixed to the values determined from fits to local background and rass data as described in section [ sec : suzanalysis ] . we stress that the systematic errors are included as explained in section [ sec : syst ] . the _ suzaku _ spectra are fitted using an absorbed single temperature ( 1 t ) model with free temperature , abundance , and normalization . we first examine the _ suzaku _ spectra extracted along the north direction starting from the center of a1750n . the spectra are extracted from four consecutive annular sectors ; 0@xmath6@xmath52.5@xmath6 , 2.5@xmath6@xmath55@xmath6 , 5.0@xmath6@xmath57.5@xmath6 , and 7.5@xmath6@xmath512.5@xmath6 . the total source counts in the co - added fi observations in regions n1 , n2 , n3 , and n4 are 3300 , 4400 , 2700 , and 3300 , respectively . the bi spectra in the same regions have total source counts of 2400 , 3300 , 2100 , and 2600 . both fi and bi spectra of the outermost 7.5@xmath6 @xmath5 12.5@xmath6 region are dominated by the nxb background at @xmath176 kev , thus this band is excluded from further analysis . 1.11.4 lcccc + region & @xmath104 & abund & @xmath105 & c - stat + & ( kev ) & ( a@xmath106 ) & ( @xmath107 @xmath35 ) & ( dof ) + + + n@xmath108 & 3.33 @xmath109 & 0.28 @xmath24 0.5 & 110.18 @xmath24 2.95 & 178.10 ( 177 ) + n@xmath110 & 2.80 @xmath111 & 0.15 @xmath24 0.4 & 48.87 @xmath24 1.70 & 183.31 ( 244 ) + n@xmath112 & 1.98 @xmath24 0.18 & 0.2@xmath3 & 17.55 @xmath24 0.77 & 272.86 ( 165 ) + n@xmath113 & 1.61 @xmath24 0.30 & 0.2@xmath3 & 6.08 @xmath24 1.09 & 322.35 ( 241 ) + + s@xmath108 & 2.61 @xmath24 0.21 & 0.19 @xmath24 0.08 & 22.28 @xmath24 1.79 & 148.68 ( 144 ) + s@xmath110 & 2.04 @xmath114 & 0.20 @xmath115 & 2.76 @xmath24 1.15 & 201.25 ( 187 ) + + se@xmath108 & 4.72 @xmath116 & 0.31 @xmath24 0.01 & 154.79 @xmath24 5.33 & 712.29 ( 746 ) + se@xmath110 & 4.83 @xmath24 0.40 & 0.2@xmath3 & 31.93 @xmath24 1.44 & 776.43 ( 747 ) + se@xmath112 & 2.47 @xmath117 & 0.2@xmath3 & 4.26 @xmath24 1.10 & 495.63 ( 473 ) + + + + + [ table:1tfits ] to investigate the nature of the gas along the filament , we first fit the fi and bi spectra simultaneously with a 1 t model . the parameters of the fi and bi spectral models are tied to each other . the abundances are only constrained by the observations in regions n1 and n2 . the best - fit temperatures are 3.33@xmath109 kev and 2.80@xmath111 kev , respectively . a 1 t model produces an acceptable fit to the spectra of the innermost two regions . adding an additional model does not significantly improve the fits for these regions . the model parameters are given in table [ table:1tfits ] . 1.11.7 lccccc + region & @xmath118 & abund & @xmath119 & @xmath120 & @xmath121 + & ( kev ) & a@xmath106 & ( @xmath107 @xmath35 ) & ( kev ) & ( @xmath107 @xmath35 ) + + + n@xmath112 & 3.24@xmath122 & 0.1@xmath123 & 10.32@xmath242.05 & 1.01@xmath124 & 9.19@xmath125 + n@xmath112 & 2.93@xmath126 & 0.2@xmath123 & 11.76@xmath241.66 & 0.99@xmath240.07 & 4.87@xmath127 + n@xmath112 & 2.93@xmath128 & 0.3@xmath123 & 11.75@xmath241.53 & 0.95@xmath240.08 & 3.29@xmath129 + n@xmath113 & 1.95@xmath130 & 0.1@xmath123 & 4.81@xmath131 & 0.79@xmath132 & 2.98@xmath133 + n@xmath113 & 2.12@xmath134 & 0.2@xmath123 & 4.53@xmath135 & 0.81@xmath240.12 & 1.74@xmath136 + n@xmath113 & 2.29@xmath137 & 0.3@xmath123 & 4.21@xmath24 1.33 & 0.80@xmath138 & 1.28@xmath24 0.25 + + [ table:2tfits ] error bars of _ xmm - newton _ and _ suzaku _ temperatures include systematic and statistical uncertainties . temperatures reported by three satellites are in a good agreement . we are able to extend the gas temperature measurements out to 0.9 @xmath2 of a1750n ( r@xmath139 ) and @xmath2 of a1750c ( r@xmath140 ) clusters.,width=325 ] a 1 t model produces best - fit temperatures of 1.98@xmath240.18 kev and 1.61@xmath240.30 kev in regions n3 and n4 , respectively . abundances are not constrained ; we therefore assume an abundance of 0.2a@xmath106 , as observed in the outskirts of low mass clusters @xcite . the projected temperature profile to the north is shown in figure [ fig : projtprof ] . we compare the _ suzaku _ results with those from _ chandra _ ( this work ) and _ xmm - newton _ @xcite . we note that the _ chandra _ results shown in figure [ fig : projtprof ] do not include the systematic uncertainties , and are shown here for a rough check on the _ suzaku _ temperature estimates . we find good agreement between measurements from each satellite . we note that both _ suzaku _ and _ chandra _ observations cover the radial range out to 0.9 @xmath2 ( @xmath15 14@xmath14 , see section [ sec : mass ] ) , and the best - fit temperatures measured by _ suzaku _ and _ chandra _ are in agreement at the 1@xmath20 confidence level . however , since _ suzaku _ has a lower background at large radii ( @xmath15@xmath2 ) and more precise temperature measurements ( i.e. , smaller systematic+statistical uncertainties ) , we will use _ suzaku _ temperature and density measurements hereafter . the residuals in the spectrum after a model fit in the softer 0.5 @xmath5 2.0 kev band ( shown in the left panel of figure [ fig : nn3 ] ) suggest the possible presence of a second , cooler thermal component in the regions n3 and n4 . to investigate this , we add another absorbed component to the model ( 2 t ) and re - do the fit . both the temperature and the normalization of the second component are left free , while the abundances are tied to each other between the two models . the best - fit parameters of the 2 t model and the improvement in the fits are given in table [ table:2tfits ] . figure [ fig : nn3 ] ( right panel ) shows the improvements in the fits of both region n3 and n4 . the temperature of the primary component increases from 1.98 @xmath24 0.09 kev to 2.93@xmath126 kev , while the temperature of the secondary component is estimated to be 0.99 @xmath24 0.07 kev in region n3 . the change in the goodness of the fit statistics is significant , with a @xmath141c - statistic of 64.5 for an additional two d.o.f . the c - statistic value does not provide a statistical test to quantify the significance of the improvement in the fit from adding the second component , thus we calculate the corresponding @xmath142 values before and after addition of the secondary model . we find that adding two d.o.f . ( additional temperature and its normalization ) improves the @xmath142 by 28.5 . in region n4 , the best - fit temperature of the primary _ apec _ becomes 2.12@xmath134 kev in the 2 t fits , while the temperature of the secondary component is 0.81 @xmath24 0.12 kev . the @xmath143 value of 24.4 with an additional two d.o.f . , corresponding to a null hypothesis probability of @xmath144 , suggests that the detection is significant . the best - fit parameters of these 2 t models are summarized in table [ table:2tfits ] . the derived _ xspec _ normalizations , i.e. emission measures , and temperatures depend on the assumed metallicity . we provide the measurements of these observables for various solar abundance fractions . we note that the assumed metallically does not have a significant impact on temperature or emission measure of the hotter component in our fits . the discussion of the nature of this gas is provided in section [ sec : fila ] . considering that the calibration of xis below 0.7 kev is uncertain , we re - perform the 1 t and 2 t model fits in the n3 and n4 regions to investigate the effect of this uncertainty on the temperature and normalization ( i.e. density ) . fixing the abundance at 0.2 @xmath145 , we find that the temperatures and normalizations of both models are consistent with results from the 0.57 kev band fits within the total ( statistical plus systematic ) uncertainties . the results from the 1 t and 2 t model fits in the 0.77 kev band are given in table [ table : fits-0p7 ] . we conclude that the detection of the cooler @xmath151 kev gas is not significantly affected by the effective area uncertainties below 0.7 kev . 1.11.7 lccccc + region & @xmath118 & @xmath119 & @xmath120 & @xmath121 + & ( kev ) & ( @xmath107 @xmath35 ) & ( kev ) & ( @xmath107 @xmath146 ) + + + n@xmath112 & 1.96@xmath240.18 & 16.45@xmath147 & @xmath5 & @xmath5 + n@xmath112 & 2.90@xmath148 & 10.82@xmath242.19 & 0.99@xmath149 & 4.75 @xmath24 1.21 + n@xmath112 & 1.59@xmath240.29 & 5.73@xmath241.12 & @xmath5 & @xmath5 + n@xmath113 & 2.09@xmath150 & 4.22@xmath151 & 0.79@xmath152 & 1.73@xmath153 + + [ table : fits-0p7 ] to investigate the x - ray emission along the filament to the south , we extract spectra from two annular sectors ( regions s1 and s2 ) extending south from the center of a1750s . these regions are shown in figure [ fig : image ] . region s1 extends from the cluster core to 4@xmath6 , and region s2 extends from 4@xmath6 to 9.7@xmath6 . the source counts in the combined fi and bi observations are 2600 and 1700 in region s1 , and 2200 and 2000 in region s2 . we first fit the spectra with a 1 t model . the best - fit temperatures of 2.61 @xmath24 0.21 kev and 2.04@xmath154 kev , and abundances of 0.19 @xmath24 0.08 a@xmath106 and 0.20@xmath115 a@xmath106 are measured in regions s1 and s2 , respectively . the results are shown in table [ table:1tfits ] with the goodness of the fits . abundance measurements of 0.2a@xmath106 are consistent with the abundances measured in low mass systems @xcite . the possible presence of the cool @xmath15 1 kev gas is tested by performing 2 t fits . the additional secondary model does not significantly improve the fits . unlike the detection in the north , we find no evidence for such a component in the south . the x - ray emission to the southeast , perpendicular to the putative large - scale filament , is examined using spectra extracted in annular sectors ( se1 , se2 , and se3 shown in figure [ fig : image ] ) with radii of 0@xmath6@xmath54@xmath6 , 4.0@xmath6@xmath58.0@xmath6 , and 8@xmath14 extending out to @xmath2 ( @xmath1516@xmath6 ) of the central sub - cluster . the total source counts in the fi and bi observations are 7800 and 5000 in region se1 , and 4000 and 2600 in region se2 , and 3000 and 2500 in region se3 . to study the nature of the gas along the off - filament direction we followed a similar approach to that outlined in section [ sec : fila ] . the fi and bi spectra of each region are first fit with a 1 t model . the best - fit parameters and the goodness of these fits are given in table [ table:1tfits ] . the temperature and abundance in the innermost region are 4.72@xmath116 kev and 0.31 @xmath24 0.01 a@xmath106 . the best - fit temperature of the se2 region is 4.83 @xmath24 0.40 kev . unlike in region se1 , we are not able to constrain the abundance in region se2 , thus the abundance parameter is fixed at 0.2a@xmath106 . to test if the best - fit temperature is sensitive to the assumed metallicity , we perform the fit with abundances of 0.1a@xmath106 and 0.3a@xmath106.the best - fit temperature declines to 4.73 @xmath24 0.39 kev for an assumed abundance of 0.1a@xmath106 , while it increases to 4.95 @xmath24 0.38 kev for an abundance of 0.3a@xmath106 . however , the change in the measured temperature is not statistically significant . the spectrum from region se3 are dominated by the nxb above 5 kev . therefore , we perform our fits in the 0.5@xmath55 kev energy band in this region . the best - fit temperature is 2.47@xmath117 kev for an assumed abundance of 0.2a@xmath106 . the temperature is 2.56@xmath155 kev and 2.85@xmath156 kev for fixed abundances of 0.1a@xmath106 and 0.3a@xmath106 , respectively . the temperatures for our assumed abundances are all consistent within the 1@xmath20 level . in all cases , we observe a significant sharp decline in the projected temperature at @xmath157 ( 10.6@xmath14 ; see section [ sec : mass ] ) to the southeast . taking a similar approach as in section [ sec : fila ] , we fit the spectra of the outermost regions se2 and se3 with a 2 t model . the temperature of the secondary component is not constrained , and this addition does not improve the fit significantly . thus , we find no evidence for a softer thermal component in the off - filament direction . to further test if the @xmath151 kev gas detected along the filament to the north is observable along the off - filament southeast direction , we scale the normalization of the softer component detected in region n4 ( see table [ table:2tfits ] ) by the ratio of the area of regions se3 and n4 . freezing the normalization to the scaled value of 1.5@xmath158 @xmath35 and the observed temperature to 0.99 kev , we refit the fi and bi spectra of the se3 region . the temperature and normalizations of the primary component are unconstrained after the fit is performed . the sharp decline in the goodness - of the fit ( c - statistics value of 4931.15 for 471 d.o.f . ) suggests that if the @xmath151 kev gas detected along the filament direction existed in this region with the same surface brightness , it would be detected . thus , this component is clearly absent in the off - filament direction . we investigate the distribution of the gas temperature between a1750n @xmath5 a1750c and between a1750c @xmath5 a1750s along the merger axis . we define rectangular regions along the line connecting the centroids of the three sub - clusters ( figure [ fig : bridge ] , right ) , which are marked with dashed lines in figure [ fig : bridge ] , left . we fit the spectra of the selected regions using a 1 t model . figure [ fig : bridge ] ( left panel ) displays the projected temperature as a function of distance . we find that , starting from the northernmost region , the temperature keeps rising towards the center of a1750n , and reaches a peak temperature of 3.37 @xmath24 0.10 kev . due to the large psf of _ suzaku _ , we can not rule out or confirm the suggestion that a1750n is a cool core cluster @xcite . continuing past a1750n , the temperature rises up to 5.49 @xmath24 0.59 kev with a sharp increase at @xmath156@xmath6 ( @xmath15 0.5 mpc ) . this increase in the temperature is significant at a level of 2.7@xmath20 . hot , presumably shock - heated gas between a1750n and a1750c , coinciding with the location where we detect hot gas with _ suzaku _ , has previously been observed in _ chandra _ and _ xmm - newton _ data @xcite . the presence of hot gas in this region is an indication of an interaction between the a1750n and a1750c sub - clusters . a1750c shows a relatively uniform temperature around the centroid , with a peak temperature of 4.25 @xmath24 0.16 kev . we detected another temperature peak located 7@xmath6 away from a1750c , in the southwest direction , with a temperature of 4.74 @xmath24 0.70 kev . southwest of this peak , the temperature declines to 3.19 @xmath240.42 kev . this sharp decrease is significant at a 4@xmath20 level , suggesting an interaction between the sub - clusters a1750s and a1750c . a hot region , where the peak detected by _ suzaku _ observations , was previously detected in the vicinity of a1750c @xcite . due to large error bars on the temperature ( 5.7@xmath159 kev ) , the authors were unable to determine the true nature of the structure and claimed that it could due to a point source . similarly a hot region was observed in _ data @xcite coinciding with the reported location of the peak . here we confirm the extended nature of the emission and suggest a potential interaction between a1750c and a1750s . although , we note that the optical data do nt show any evidence of interaction between these clusters ( see section [ sec : optspec ] for discussion ) . the projected temperature continues to decline towards the center of the southern sub - cluster a1750s . the central temperature of a1750s is 2.93 @xmath24 0.21 kev . the radial temperature profile shows that the temperature decreases smoothly moving across the center of a1750s towards the southwest . to examine the radial profiles of cluster masses and thermodynamical quantities such as entropy and pressure , we determine the deprojected density and temperature . the electron density is obtained from the best - fit normalization @xmath160 of the model in _ xspec _ using the relation , @xmath161 where @xmath162 is the angular size distance to the source in units of cm , and @xmath163 and @xmath164 are the electron and hydrogen number densities in units of @xmath165 . we note that the arfs generated by _ assume a uniform source occupying an area of 400@xmath166 square arcminutes . we therefore apply a correction factor to each region and normalization prior to deprojection . an ` onion - peeling ' method is used to deproject the temperature and density profiles @xcite . the resulting deprojected density and temperature profiles to the north , southeast , and south directions are shown in figure [ fig : deprojprof ] . we extend the temperature and density profiles out to 0.9@xmath2 for a1750n to the north and @xmath2 for a1750c to the southeast with the new _ suzaku _ observations ( see in figure [ fig : image ] ) . the temperature profiles to the north and southeast decline with radius and reach half of the peak value at @xmath2 . similar temperature declines have been reported for other clusters ( e.g. , * ? ? ? * ; * ? ? ? we observe a rather gradual decline in temperature to the north and south . however , the profile to the southeast indicates a uniform temperature within 8@xmath6 and falls relatively rapidly beyond @xmath167 . based on the average deprojected density and temperature , we estimated the mass of each sub - cluster within @xmath168 using the @xmath169 scaling relation ( * ? ? ? * v09 hereafter ) . the spectra between 0.15 @xmath5 1@xmath168 are extracted to determine the global properties for each cluster . to avoid flux contamination , adjacent sub - clusters were excluded . a1750n has a best - fit global temperature of 3.14@xmath170 kev , and an abundance of 0.15 @xmath24 0.03 a@xmath106 . our measurement is consistent with the temperature of 3.17 @xmath24 0.1 kev reported in @xcite . the scaling relation predicts a total mass of 1.98 @xmath171 at @xmath168 ( 9.3@xmath14 ) . the best - fit temperature of a1750c is 4.15@xmath172 kev , with an abundance of 0.21@xmath173 a@xmath106 . the global temperature reported in @xcite is slightly lower ( @xmath174 kev ) . their extraction region excludes the hotter plasma between a1750n and a1750c , which may account for the difference observed in temperature . the v09 scaling relation predicts a total mass of 3.03 @xmath171 enclosed within @xmath175 ) . the spectral fit to a1750s gives a best - fit temperature of 3.59@xmath176 kev and an abundance of 0.20@xmath177 . the estimated total mass within @xmath168 ( 9.9@xmath14 ) is 2.43 @xmath171 . lcccccc + & north & southeast + + @xmath178 ( @xmath179 @xmath165 ) & 1.78 @xmath180 & 2.19 @xmath181 + @xmath182 ( kev ) & 3.89 @xmath24 0.22 & 5.49 @xmath24 0.27 + @xmath183 & 4.49 @xmath184 & 6.01 @xmath185 + @xmath186 ( arcmin ) & 300@xmath3 & 480@xmath3 + @xmath187 & 2.0@xmath3 & 2.0@xmath3 + @xmath142 ( dof ) & 5.35 ( 5 ) & 3.21 ( 3 ) + + [ table : b10params ] [ fig : fits ] to investigate the radial behavior of the gas mass , the total mass , and the gas mass fraction , we employ a physically motivated icm model described in ( * ? ? ? * ; * ? ? ? * b10 , hereafter ) . the b10 model is based on the assumption that the icm is a polytropic gas in hydrostatic equilibrium in the cluster s gravitational potential . the deprojected density and temperature profiles are fit simultaneously using the b10 model . the fitting was performed using a markov chain monte - carlo ( mcmc ) approach , with metropolis - hastings sampling , to determine posterior distributions for the best - fit model parameters . the temperature profile is @xmath188 , \label{eqn_polytropic_temperature}\ ] ] where the normalization constant @xmath189 is @xmath190 using the relation between temperature and gas density provided by the polytropic relation , the gas density is @xmath191^{n},\ ] ] where @xmath187 + 1 is the slope of the total density distribution , @xmath183 is the polytropic index , @xmath186 is the scale radius , and @xmath189 and @xmath178 are the central temperature and density of the polytropic function . this model has sufficient fitting flexibility to describe x - ray data , while making simple physical assumptions @xcite . we note that the core taper function in the b10 model is omitted in the fits performed in this work , since observations are not able to resolve the cluster cores . figure [ fig : fits ] shows the best - fit models to the density ( left panel ) and temperature ( right panel ) in the off - filament and filament directions . @lcccccccc cluster & @xmath168&@xmath192 & @xmath193 & @xmath194 & @xmath2 & @xmath195 & @xmath196 & @xmath197 + & ( arcmin ) & ( @xmath198 ) & ( @xmath199 ) & & ( arcmin ) & ( @xmath198 ) & ( @xmath199 ) + + a1750n & 9.3@xmath14 & 1.86 @xmath24 0.38 & 1.54 @xmath200 & 0.12 @xmath201 & 14.1@xmath14 & 3.41 @xmath202 & 2.32 @xmath203 & 0.15 @xmath204 + + a1750c & 10.6@xmath14 & 3.15 @xmath205 & 3.04 @xmath206 & 0.10 @xmath201 & 16.2@xmath14 & 5.46 @xmath24 0.16 & 4.85 @xmath207 & 0.11 @xmath208 + + [ table : mass ] due to the limited number of data points compared to the number of free model parameters of the b10 model ( five in this case ) , we were not able to constrain all of the free parameters of the model . the @xmath187 parameter is fixed to the slope of the navarro - frenk - white profile @xcite , while the scale radius r@xmath209 ( fixed in our fits ) , the radius beyond which the temperature starts declining , is estimated from the temperature profiles ( see figure [ fig : fits ] ) . the rest of the model parameters ( @xmath183 , @xmath178 , and @xmath182 ) are allowed to vary independently . the best - fit parameters of the model are given in table [ table : b10params ] , along with the goodness of the fits . the best - fit models for the density and temperature profiles are displayed in figure [ fig : fits ] , with 90% confidence intervals . given the limited number of data points , the profiles to the south are not constrained . the total mass enclosed within radius @xmath23 is @xmath210 . \end{aligned } \label{eqn : totalmass}\ ] ] the normalization factor for the total matter density is @xmath211 $ ] . the gas mass @xmath212 is computed by integrating the gas density profile within the volume , @xmath213 where @xmath214 and @xmath215 are the mean molecular weight per electron and the proton mass . the gas mass fraction is @xmath216 the gas mass , total mass , and @xmath217 are measured at @xmath168 , determined using the v09 scaling relations , and are given in table [ table : mass ] . following @xcite , we assume @xmath168 = 0.659@xmath2 . the total mass , gas mass , and gas mass fraction profiles are plotted in figure [ fig : mass ] . we find that the total masses enclosed within @xmath168 are well within agreement with the total masses estimated using the v09 scaling relations . the gas mass fractions of a1750n and a1750c are consistent with the gas mass fraction expected for clusters in this mass range based on the v09 scaling relations ( @xmath218 0.11 ) at @xmath168 . the b10 model was then used to calculate the masses and mass fraction at @xmath2 . we found that the gas mass fraction of a1750c and a1750n at @xmath2 is 0.11@xmath208 and 0.15@xmath204 . the virial masses of the a1750n and a1750c subclusters are in agreement with the mass estimates from the optical observations at a 2.7@xmath20 level ( see section [ sec : optspec ] ) . however , we note that , the cluster mass inferred from x - ray analysis depends on the geometry of the merger , hydrostatic equilibrium , and other model parameters ( e.g. scale radius ) of the merging clusters . the gas fractions derived in the filament and off - filament directions are consistent with the cosmic baryon fraction derived from wmap seven - year data of 0.166 @xcite . similarly , gas mass fractions consistent with the cosmic value , were observed in rx j1159 + 5531 @xcite , a1689 @xcite , and a1246 @xcite . however , we note that the total mass estimates are based on a few assumptions on the distribution of the gas properties . spherical symmetry and isotropy are assumed when calculating these masses . such assumptions may bias our results , particularly in a merger system at large radii . [ fig : mass ] the entropy ( @xmath219 ) and pressure ( @xmath220 ) profiles are calculated using the electron density ( @xmath163 ) and deprojected temperature ( @xmath104 ) . the profiles along the filament and off - filament directions are shown in figure [ fig : deprojentropress ] . in the absence of non - gravitational processes , such as radiative cooling and feedback , cluster entropy profiles are expected to follow the simple power - law relation @xmath221 where we assume a cosmic baryon fraction of @xmath222 = 0.15 , with a characteristic entropy of @xmath223 @xcite . we used an @xmath224 ( the total mass within @xmath168 of a1750c ) of 3 @xmath171 , as determined in [ sec : mass ] . the resulting expected self - similar entropy profile for a1750c is shown as the dashed lines in figure [ fig : deprojentropress ] ( left ) . we find that the entropy along the filament directions ( to the north and south ) and off - filament direction derived from _ suzaku _ data alone are in good agreement with each other within @xmath225 . profiles obtained from _ xmm - newton _ observations are consistent with those from _ suzaku _ data within @xmath226 . the observed entropy exceeds the self - similar model prediction within @xmath227 , which we attribute to the influence of non - gravitational processes ( e.g. , agn feedback , infalling substructures due to violent merging events ) in the subcluster cores . such an influence on the entropy profiles of a sample of low redshift clusters ( @xmath228 0.25 ) was reported by @xcite . the entropy profiles follow a flatter profile beyond a radius of @xmath229 , and become consistent with the self - similar model , both along the northern filament and off - filament directions . we find that the entropy profile towards the northern filament reaches the self - similar level at smaller radii ( @xmath150.4r@xmath230 ) as compared with the off - filament direction . this may be due to the lower temperature gas ( @xmath151 kev ) observed to the north , which biases the average temperature low , and depresses the measured value of the entropy . the entropy profile along the off - filament direction stays above the self - similar expectation to @xmath231 . beyond this radius it remains consistent with the self - similar prediction . if the entropy contribution from the cool gas detected to the north is removed , the entropy rises to 1245.6 @xmath24 486.5 kev @xmath232 ( shown in figure [ fig : deprojentropress ] with the dashed data point in red ) and becomes more consistent with the entropy to the southeast . this provides evidence that the cool gas does indeed lead to a slight decrease in the entropy , although not at the level seen in other systems where it is likely arises from gas clumping @xcite . [ fig : deprojentropress ] unlike the rising , self - similar entropy observed in a1750 , a flattening of entropy profiles near @xmath2 appears to be a common feature in other relaxed and disturbed clusters ( for a review , see * ? ? ? * ) . a few detailed studies of nearby bright merging systems have probed the physical properties of the icm at large radii , e.g. the coma cluster @xcite and the virgo cluster @xcite . _ xmm - newton _ observations of the dynamically young cool virgo cluster revealed a suppressed entropy profile beyond 450 kpc by a factor of 22.5 below the expectation from pure gravitational collapse models . authors attributed this flattening to gas clumping at large radii . in the merging coma cluster , @xcite find no evidence for entropy flattening along the relatively relaxed directions , although due to large uncertainties they are unable to exclude entropy flattening at the level of what is observed in some relaxed clusters . there has been great effort in the literature to explain the seeming ubiquity of flattened entropy profiles at large radii . in the hierarchical model of structure formation , clusters form by accreting material from their surrounding large - scale structure . accretion of infalling subhalos can cause gas motions and clumpiness " around @xmath2 . these subhalos tend to have lower temperature and higher density than the surrounding icm , leading to a bias towards lower temperatures and higher densities in the emission measure - weighted spectra , if the subhalos are unresolved . the level of gas inhomogeneities is characterized through the clumping factor ( @xmath233 ) . as a result of overestimation of density , the gas mass , and subsequently the gas mass fraction , are biased high ( i.e. above the cosmic baryon fraction ) . the observed excess in the gas mass fraction ( @xmath234 ) in the _ suzaku _ observations of the perseus cluster was explained with a very large clumping factor of 3 @xmath5 4 around @xmath2 @xcite . @xcite reported that the expected clumpiness factor at @xmath2 can be as large as 2 and confirmed the flattened entropy profiles beyond @xmath235 in their non - radiative and cooling+star - formation simulations . however , @xcite examined entropy profiles for a sample of relaxed clusters at @xmath236 0.25 out to @xmath2 and concluded that the gas clumping calculated in the numerical simulations is insufficient to reproduce the observed flattening of the entropy . an alternative explanation to the flattening was proposed by @xcite and @xcite , and is based on the electron - ion non - equilibrium in the cluster outskirts . if the energy is not transferred to the electrons through electron - ion collisions sufficiently rapidly , the electron temperature remains low compared to that of ions , leading to an apparent entropy suppression at @xmath2 . @xcite and @xcite proposed that the flattening in the entropy is a result of a weakened accretion shock as it expands . the bulk energy carried along with the shock increases the turbulence and non - thermal pressure support in the outskirts , but the shock is not energetic enough to raise the intra - cluster entropy . the decreasing thermalization in low - density regions results in a tapered entropy around @xmath15@xmath2 . this claim supports the observed azimuthal variations in entropy in cool - core clusters @xcite and in the non - cool core coma cluster @xcite . other proposed explanations of entropy flattening include a rapid radial fall of the gas temperature caused by non - gravitational effects @xcite and cosmic - rays consuming as a significant sink for the kinetic energy in the outskirts @xcite . on the other hand , @xcite have performed a joint planck sz and rosat x - ray analysis of 18 galaxy clusters and concluded that entropy profiles are consistent with a self - similar power - law increase expected from pure gravitational infall . the discrepancy between the @xcite and the @xcite results is due to the differing dependence on sz and x - ray signals to the electron pressure used to derive entropy profiles @xcite . self - similar entropy profiles at @xmath2 have been previously observed in low mass relaxed fossil groups , e.g. rx j1159 + 5531 @xcite . on the other hand , the entropy of morphologically relaxed groups has been found to be significantly higher than self - similar at @xmath237 @xcite . however , massive mergers ( @xmath238 ) are expected to have a higher level of gas clumping , since they have a larger fraction of lower - temperature gas that is not detectable in the x - ray band @xcite . although a1750 is a dynamically young , massive system , we do not find evidence for gas clumping in this merger system . entropy profile measurements along the off - filament and filament directions are in agreement with each other and with the universal expectation with a power - law relation @xmath239 . remarkably in a1750 , the entropy profiles within @xmath2 do not seem to have been influenced by the apparent filamentary structure of the system . our results suggest that gravitational collapse is the main driver of the temperature and density profiles in the outskirts . we also examine the pressure profiles along the off - filament and filament directions . pressure profiles are calculated assuming an ideal gas law with @xmath240 , and compared to the universal pressure profiles of @xcite ( a10 , hereafter ) and @xcite ( planck13 , hereafter ) for clusters with mean redshifts of 0.11 and 0.17 , respectively . the a10 universal pressure profile is @xmath241^{2/3+\kappa}\\ & \times \mathbb{p}(r / r_{500})\ h_{70}^{2 } \ \ \rm{kev}\ { cm^{-3 } } , \end{aligned } \label{eqn : arnaud}\ ] ] where the scaled pressure profile is characterized based on the generalized navarro - frenk - white profile @xcite @xmath242^{(\beta-\gamma)/\alpha}},\ ] ] and @xmath243 with best - fit parameters of @xmath244 = 8.403@xmath245 , @xmath246 = 1.177 , @xmath247 = 0.3081 , @xmath74 = 1.0510 , and @xmath187 = 5.4905 . the first term in equation [ eqn : kappa ] , @xmath248 , is an approximation which depends on the departures from the standard scaling relations , while the second term , @xmath249 , represents a break from self - similarity . since non - gravitational processes become less dominant at large radii , the latter term is negligible at @xmath250 . the a10 universal pressure profile primarily samples the inner regions , while the planck13 profile samples the cluster outskirts . the pressure profiles derived from the planck observations for a sample of 62 galaxy clusters found slightly higher pressure than that predicted by a10 in the outskirts of clusters . these profiles were obtained by averaging pressure profiles from all azimuths for a large sample of clusters with different dynamical states . the dispersion over the universal profiles can be as large as 100 % at @xmath15@xmath168 ( see figure 8 in a10 ) . we compare the pressure profiles of a1750 with the universal profiles of a10 and planck13 in the right panel of figure [ fig : deprojentropress ] . while the pressure profile along the filament direction to the north agrees with the universal profile , the profile along the off - filament direction is higher and the profile to the south is lower than the expectation within @xmath1 0.2@xmath2 . on the other hand , the profile in the filament direction to the south and to the north is consistent with the a10 and planck13 universal profiles at large radii ( @xmath15@xmath2 ) at the 2.7@xmath20 level . the pressure to the southeast exceeds the universal models at all radii . pressure excesses at large radii have been previously reported in other relaxed clusters , e.g. pks 0745 - 191 @xcite , the centaurus cluster @xcite , and the fossil group rx j1159 + 5531 @xcite , and were attributed to gas clumping . figure [ fig : deprojprof ] indicates that the excess in the pressure along the southeast direction compared to the north or south directions in a1750 is due to high temperature ( not high density ) . on the contrary , clumping ( if it existed in this system ) would bias the density measurements high , leading to an excess in pressure and a decrement in entropy in the outskirts . therefore , the deviation from the universal profile in a1750 is unlikely to be due to clumpy gas , since other evidence for clumping , e.g. entropy flattening and an excess in gas mass fraction ( see section [ sec : mass ] ) , are not observed in this system . we note that @xcite reported the detection of a weak @xmath251 , shock resulting from a merger event intrinsic to a1750c along the southeast direction . this merger event may elevate the temperature and cause deviations from the universal profile . in any event , given the large dispersion among pressure profiles of clusters in the a10 and planck13 samples , we do not expect the pressure profiles derived in a1750 in perfect agreement with their results . the cool @xmath15 1 kev gas detected in regions n@xmath112 and n@xmath113 ( see figure [ fig : nn3 ] ) may be 1 ) the hot dense whim connecting a1750n to the large - scale filament , 2 ) stripped icm gas formed as a result of infalling groups , or 3 ) gas stripped from a1750n itself , as it interacts with filament gas or with a1750c . the feature is relatively extended with an observed radial range of @xmath17 0.62 mpc . assuming a geometry for the merger system , the mass of the feature can be calculated ( see section [ sec : dynamics ] for the detailed calculation ) . assuming that the density of the feature is constant within each region ( 5.56 @xmath252 @xmath165 ) , and can be described as a cylinder that extends to 1.2 mpc with a line - of - sight depth of the structure ( _ l _ ) , we obtain a gas mass of 4.13 @xmath253 . the observed flux of the feature ( 1 @xmath254 erg @xmath29 s@xmath8 arcmin@xmath41 ) , density , and temperature ( @xmath150.8 kev ) are consistent with the expected surface brightness and temperature of the dense portion of the whim , where the large - scale structure interacts with the cluster s icm @xcite . such a filamentary feature also may be due to an additional small subcluster infalling into a1750n which is being disrupted as it interacts with the main cluster . the bulk of the halo gas lags behind the infalling groups , and is stripped by the ram pressure of the ambient icm . such halos are expected to have an average temperature of @xmath151 kev with a typical halo mass of 3@xmath255 @xcite . bright , large - scale ( @xmath15 700 kpc ) stripped tails have been observed in the outskirts of galaxy clusters , e.g. the virgo cluster @xcite , a85 @xcite , and a2142 @xcite . the stripped gas from infalling halos may seed gas inhomogeneities ( i.e. clumping ) , which suppress the average entropy inferred at large radii . in such systems , a flattening of the entropy profile , as well as an excess in the gas mass fraction as compared with the cosmic value , have often been observed in cluster outskirts . in the case of a1750 , the entropy profile remains consistent with the self - similar prediction out to @xmath2 , and the gas mass fraction is consistent with the cosmic value ( see section [ sec : mass ] ) , implying that the observed cool gas could indeed be the densest and hottest parts of the warm - hot intergalactic medium . in addition , dense and cool clumps in the outer cluster regions are expected to lead to more entropy flattening since they will lower the average temperature , and , more importantly raise the average density @xcite . along the north direction of a1750n , there is sufficient cool gas to be detected , but it does not cause the dramatic entropy flattening seen in some other clusters , suggesting that its density can not be too high . completely ruling out the ram pressure - stripping scenario for this cool gas requires deeper _ chandra _ observations with good angular resolution . the whim interpretation of this feature can not be firmly established based on the _ suzaku _ data . we present an analysis of the strongly merging cluster a1750 using _ suzaku _ and _ chandra _ x - ray observations , and _ mmt _ optical observations out to the cluster s virial radius . the deep _ suzaku _ observations allow us to constrain the entropy , pressure , and mass profiles at the outskirts , both along and perpendicular to the large - scale filament . we use optical observations to constrain the dynamical state of the cluster . our major results are : 1 . a1750n and a1750c have a 78% chance of being bound . there is an apparent hot region with a temperature of 5.49 @xmath24 0.59 kev in between these subclusters implying an interaction . the red galaxy distribution and the velocity dispersion data prefer a pre - merger scenario . in an early pre - merger scenario , one expects the outer icm atmospheres of the subclusters to interact subsonically , driving shocks , and ultimately creating a heated icm region between the subclusters , e.g. , n7619 and n7626 @xcite . we find overall a good agreement between the measured entropy profiles and the self - similar expectation predicted by gravitational collapse near @xmath2 both along and perpendicular to the putative large - scale structure filament . unlike some other clusters , the entropy profiles at large radii , both perpendicular and along the filamentary directions , are consistent with each other . agreement of the entropy with the self - similar expectation at @xmath2 in this massive and dynamically young system suggests that a1750 exhibits little gas clumping at large radii . the gas mass fractions in both the filament and off - filament directions are consistent with the cosmic baryon fraction at @xmath2 . this may indicate that gas clumping may be less common in such smaller , lower temperature ( @xmath256 4 kev ) systems ( with a few exceptions , e.g. the virgo cluster , * ? ? ? cluster mass may therefore play a more important role in gas clumping than dynamical state . 4 . an extended gas ( @xmath17 0.62 mpc ) is observed to the north of the a1750n subcluster along the large scale structure , where one would expect to detect the densest part of the whim in a filament , near a massive cluster . the measured temperature ( 0.8 @xmath51 kev ) , density , and radial extent of this cool gas is consistent with the whim emission . the thermodynamical state of the gas at that radius ( i.e. self - similar like entropy profile , and gas mass fraction consistent with the cosmic value ) favors the whim emission interpretation . however , a deeper observation with _ resolution is required to distinguish this diffuse filamentary gas from an infalling substructure , or gas from ram pressure - stripping . we thank gabriel pratt for kindly providing temperature , density , and entropy profiles from _ xmm - newton _ data . we also thank mike mcdonald and john zuhone for useful comments and suggestions . eb was supported in part by nasa grants nnx13ae83 g and nnx10ar29 g . swr was supported by the _ chandra _ x - ray center through nasa contract nas8 - 03060 and by the smithsonian institution . mbb acknowledges support from the nsf through grant ast-1009012 . elb and rpm was partially supported by the national science foundation through grant ast-1309032 . cls was funded in part by _ chandra _ grants go4 - 15123x and go5 - 16131x and nasa xmm grant nnx15ag26 g . aedm acknowledges partial support by _ chandra _ grants go2 - 13152x and go3 - 14132x . authors thank prof . nihal e. ercan for providing the support for ce .

we present results from recent _ suzaku _ and _ chandra _ x - ray , and _ mmt _ optical observations of the strongly merging double cluster " a1750 out to its virial radius , both along and perpendicular to a putative large - scale structure filament . some previous studies of individual clusters have found evidence for icm entropy profiles that flatten at large cluster radii , as compared with the self - similar prediction based on purely gravitational models of hierarchical cluster formation , and gas fractions that rise above the mean cosmic value . weakening accretion shocks and the presence of unresolved cool gas clumps , both of which are expected to correlate with large scale structure filaments , have been invoked to explain these results . in the outskirts of a1750 , we find entropy profiles that are consistent with self - similar expectations , and gas fractions that are consistent with the mean cosmic value , both along and perpendicular to the putative large scale filament . thus , we find no evidence for gas clumping in the outskirts of a1750 , in either direction . this may indicate that gas clumping is less common in lower temperature ( @xmath0 kev ) , less massive systems , consistent with some ( but not all ) previous studies of low mass clusters and groups . cluster mass may therefore play a more important role in gas clumping than dynamical state . finally , we find evidence for diffuse , cool ( @xmath1 1 kev ) gas at large cluster radii ( @xmath2 ) along the filament , which is consistent with the expected properties of the denser , hotter phase of the whim .

@xmath2li is destroyed in stellar interiors where temperatures exceed @xmath3 k , and li - depleted material can in principle reach the stellar surfaces where it can be observed . thus , if one is to infer pre - stellar @xmath2li abundances from current - epoch observations , it is important to understand the stellar processing of this species . it has widely , though not universally , been supposed that warm ( @xmath4 k ) , metal - poor ( [ fe / h ] @xmath5 ) stars retain their pre - stellar abundances ( spite & spite 1982 ; bonifacio & molaro et al . 1997 ; but see also deliyannis 1995 ; ryan et al . 1996 ) . although claims had been made of an intrinsic spread in the li abundances by 0.04 0.1 dex ( deliyannis , pinsonneault , & duncan 1993 ; thorburn 1994 ) , ryan , norris & beers ( 1999 ) attributed these to an embedded @xmath6(li ) vs [ fe / h ] dependence , and underestimated errors , respectively . ryan et al . ( 1999 ) set tight limits on the intrinsic spread of @xmath2li in metal - poor field stars as essentially zero , stated conservatively as @xmath7 dex . however , the subset of ultra - li - deficient stars identified by spite , maillard , & spite ( 1984 ) , hobbs & mathieu ( 1991 ) , hobbs , welty & thorburn ( 1991 ) , thorburn ( 1992 ) , and spite et al . ( 1993 ) stands out as a particular exceptional counter - example to the general result . these stars have only upper limits on their @xmath2li abundances , typically 0.5 dex or more below otherwise similar stars of the same @xmath8 and metallicity . detailed studies of other elements in these objects have revealed some chemical anomalies , but none common to all , or which might explain _ why _ their li abundances differ so clearly from those of otherwise similar stars ( norris et al . 1997a ; ryan , norris & beers 1998 ) . in contrast to the situation for population ii stars , a wider range of li behaviors is seen in population i. in addition to a stronger increase with metallicity , thought to be due to the greater production of li in later phases of galactic chemical evolution ( ryan et al . 2001 ) , there is also substantial evidence of li depletion in certain temperature ranges . open cluster observations , for example , show steep dependences on temperature for @xmath9 k ( e.g. , hobbs & pilachowski 1988 ) and in the region of the f - star li gap ( 6400 k @xmath10 7000 k ; boesgaard & tripicco 1986 ) . more problematic , for the young cluster @xmath11 per ( age 50 myr ) and the pleiades ( age 100 myr ) , is the presence of a large apparent li spread even at a given mass . various explanations have been proposed involving mixing in addition to that due to convection . extra mixing processes include rotationally - induced mixing ( e.g. , chaboyer , demarque & pinsonneault 1995 ) , structural changes associated with rapid rotation ( martn & claret 1996 ) , and different degrees of suppression of mixing by dynamo - induced magnetic fields ( ventura et al . gravity waves have been proposed as yet another different mixing mechanism ( schatzman 1993 ; montalbn & schatzman 1996 ) . consensus has not yet emerged concerning the range of possible mechanisms , or the relative importance of each . jeffries ( 1999 ) even questions the reality of a li abundance spread in low mass pleiades stars , due to a similar spread being seen in the resonance line . amongst older open clusters , the spread at a given effective temperature is generally much less , though m67 ( jones , fischer , & soderblom 1999 ) is an exception . a class of stars with higher lithium abundances than otherwise similar stars is short - period tidally - locked binaries ( deliyannis et al . 1994 ; ryan & deliyannis 1995 ) which give credence to the view that physics related to stellar rotation can and does influence the evolution of li in approximately solar - mass stars . the fraction of warm , metal - poor stars that fall in the ultra - li - deficient category has previously been estimated at approximately 5% ( thorburn 1994 ) . however , recent measurements of li in a sample of 18 warm ( @xmath12 k ) , metal - poor ( @xmath13[fe / h ] @xmath14 ) stars yielded four ultra - li - deficient objects , i.e. more than 20% of the sample ( ryan et al . the poisson probability of a 5% population yielding 4 or more objects in a sample of this size is just 0.013 . clearly , the selection criteria for this sample have opened up a regime rich in ultra - li - poor stars . we now examine those criteria , and discuss the implications for the origin of such systems and for our understanding of li - poor and li - normal stars . we note some similarities between li - deficient halo stars and blue stragglers . although these two groups have traditionally been separated due to the different circumstances of their _ discovery _ , we question whether there is a reliable _ astrophysical _ basis for this separation . one must ask whether the process(es ) that gives rise to blue stragglers is capable only of producing stars whose mass is greater than that of the main sequence turnoff of a @xmath1513 gyr old population . if , as we think is reasonable , the answer is `` no '' , then one may ask what the sub - turnoff mass products of this process(es ) would be . our proposal is that they would be li - deficient , but otherwise difficult to distinguish from the general population , and in this regard very similar to the ultra - li - deficient halo stars . the ultra - li - poor halo stars we consider were identified serendipitously in a study of predominantly high proper - motion halo stars having @xmath16 k and @xmath17 [ fe / h ] @xmath18 , and are listed in table 1(a ) . details of the sample selection and abundance analysis are given by ryan et al . ( 2001 ) ; the key points are that high resolving power ( @xmath19 ) chelle spectra were obtained , equivalent widths were measured , and abundances were computed using a model atmosphere spectrum - synthesis approach . two of the li - poor stars were subsequently found to have previous li measurements ; wolf 550 was identified as g66 - 30 , and g202 - 65 had been observed by hobbs & mathieu ( 1991 ) in a study targeted at blue stragglers . the new spectra of the four stars , plus one with normal li for comparison , are shown in figure 1 . the comparison star , cd@xmath20305 , has @xmath21 k , [ fe / h ] = @xmath22 , and @xmath6(li ) = 2.24 ( ryan et al . 2001 ) . for convenience , previously known li - depleted halo stars are listed in table 1(b ) . the full sample of ryan et al . ( 2001 ) is plotted in figure 2 , along with additional stars from the literature . it is immediately apparent that three of the four ultra - li - deficient stars are amongst the hottest in our sample , though not _ the _ hottest in the figure . it seems likely that high temperature is one biasing characteristic of these objects . the stars with @xmath23 k and _ normal _ li abundances are listed in table 1(c ) . these have had comparatively high values of de - reddening applied , and it is possible that they are in reality cooler than table 1 shows . an indication that high temperature is not the _ only _ biasing characteristic of ultra - li - poor stars is that the ryan et al . ( 1999 ) study of 23 very metal - poor ( @xmath24 [ fe / h ] @xmath25 ) stars in the same temperature range included only one ultra - li - deficient star , g186 - 26 . this rate , 1 in 23 , is consistent with previous estimates for population ii stars as a whole . however , very few relatively metal - rich ( @xmath26 [ fe / h ] @xmath18 ) halo stars in this temperature range had been studied previously , so earlier works may have been biased against discovering ultra - li - poor objects . it appears , then , that the fraction of ultra - li - deficient stars is higher as metallicity increases . this may explain why our study , which targeted stars in the higher metallicity range _ and _ with @xmath27 k , was so successful at yielding ultra - li - deficient stars . figure 3 shows the distribution of objects in the @xmath28 , [ fe / h ] plane . whereas previously no region of parameter space stood out as `` preferred '' by li - deficient stars , the objects are now more conspicuous as a result of their high temperatures and relatively high metallicities . also shown in figure 3 are the @xmath28 of the main - sequence turnoff as a function of metallicity , for 14 and 18 gyr isochrones . the isochrones shown are the oxygen - enhanced curves of bergbusch & vandenberg ( 1992 ; solid curves ; y=0.235 ) , and , for comparison , the revised yale isochrones of green , demarque & king ( 1987 ; dotted curves ; y=0.24 ) . clearly there is disagreement of @xmath294 gyr between the two sets as to the ages that would be assigned to these stars , and there are uncertainties in the color-@xmath28 transformations that have been applied to the observed data . however , these difficulties are not the issue here . rather , we use the isochrones to indicate the _ shape _ of the turnoff locus in the @xmath28 vs [ fe / h ] plane , and on that point the four isochrones are in overall agreement . they emphasise that even though hd 97916 is cooler than five other li - depleted stars in the study , it is nevertheless close to the turnoff . that is , a star with @xmath28 = 6100 k would appear below the turnoff if [ fe / h ] = @xmath30 , but will be close to the turnoff if [ fe / h ] = @xmath31 . even excluding the definite blue straggler bd+25@xmath321981 , there are four li - depleted stars amongst the eight whose symbols lie above or touch the 14 gyr revised yale isochrone . clearly , all of these are very close to the turnoff once their metallicities are taken into account . besides these li - depleted stars close to the turnoff , four are 100200 k cooler than the turnoff . we discuss later in this paper whether the these two groupings might have different origins . blue stragglers are recognised observationally as stars that are considerably bluer than the main - sequence turnoff of the population to which they belong , but having a luminosity consistent with main - sequence membership . such objects were originally identified in globular clusters ( e.g. , m3 ; sandage 1953 ) , but are also known in the field ( e.g. , carney & peterson 1981 ) , and in population i as well as population ii ( e.g. , leonard 1989 ; stryker 1993 ) . their origin is not known with certainty , and it is possible that more than one mechanism is responsible for their presence . a range of explanations was examined by leonard ( 1989 ) , but the discovery of li destruction in blue stragglers in the halo field and the open cluster m67 led hobbs & mathieu ( 1991 ) and pritchet & glaspey ( 1991 ) to conclude that `` virtually all mechanisms for the production of blue stragglers _ other than _ mixing , binary mass transfer , or binary coalescence appear to be ruled out ... . '' as hobbs & mathieu emphasized , internal mixing alone is also ruled out ; mixing out to the surface is required . recent advances in high - resolution imaging have verified that the blue straggler fractions in at least some globular clusters are higher in their cores , strongly supporting the view that some blue stragglers are formed through stellar collisions , probably involving the coalescence of binary stars formed and/or hardened through exchanges , in these dense stellar environments ( e.g. , ferraro et al . however , it is neither established nor required that a single mechanism will explain all blue stragglers , and it is unclear how the field examples and those in the tenuous dwarf galaxy ursa minor ( feltzing 2000 , priv . comm . ) relate to those in the dense cores of globular clusters . probably even the halo field and dwarf spheroidal stars formed in clusters of some description ( since the formation of stars in isolation is unlikely ) , but one should not be too quick to link the properties of surviving globular clusters to diffuse populations . this view is supported by preston & sneden s ( 2000 ) conclusion that at more than half ( 62% 100% ) of their field blue metal - poor binaries are blue stragglers formed by mass transfer rather than mergers , due to the long orbital periods and low eccentricities of the field systems they observed . their conclusion is entirely consistent with the views of ferraro et al . ( 1995 ) , who ascribed blue straggler formation to interactions _ between _ systems in high - density environments , but _ within _ systems ( primordial binaries ) in lower - density clusters . in contrast to but not contradicting preston & sneden s result for field systems , mateo et al . ( 1990 ) argue that all of the blue stragglers in the globular cluster ngc 5466 are the result of close binary mergers . the mechanism for li destruction in field blue stragglers is not known . it is unclear what degree of mixing will occur as a result of coalescence . early work by webbink ( 1976 ) suggested substantial mixing would occur , whereas more recent simulations of head - on collisions by sills et al . ( 1997 ) , and grazing collisions and binary mergers by sandquist , bolte , & hernquist ( 1997 ) , have suggested otherwise . sills , bailyn & demarque ( 1995 ) argue , however , that to account for the blue stragglers observed in ngc 6397 , mixing is nevertheless required ( unless the collision products have more than twice the turnoff mass ) , and may occur after the initial coalescence . this is perhaps consistent with the result of lombardi , rasio , & shapiro ( 1996 ) that some mixing could occur as a merger remnant re - contracts to the main sequence . due to the fragility of li , if some mixing of surface material does occur during the coalescence it will at least dilute , and possibly also destroy , any lithium remaining in the stars thin convective surface zones up to that time . one might suppose that mass transfer in a detached system also destroys li , though one could also imagine gentle mass - transfer processes where the rate is slow enough that the original envelope is not subjected to additional mixing , and where the transferred matter itself does not undergo additional li - destruction . of course , mass transfer via roche lobe overflow in a detached system , or wind accretion from a more distant companion , involve mass from an evolved star which may have _ already _ depleted its surface li due to single - star evolutionary processes . consequently , the mass transferred may be already devoid of li , as in the scenario quantified by norris et al . ( 1997a ) . we also note the possibility that the accretor in a mass - transfer system , or the progenitors of a coalescence , was ( were ) devoid of li prior to that event . li is ( normally ) preserved in halo stars only over the temperature range from the turnoff ( @xmath1 k ) to about @xmath33 k , corresponding to a mass range from 0.80 down to 0.70 @xmath34 . therefore it is likely that any mass accretor , and certain that any merger remnant , now seen in this mass range began life as one ( or two ) stars with initial mass(es ) less than 0.70 @xmath34 and had already destroyed li normally , as lower - mass stars are known to do , prior to mass exchange . in such a scenario , it is not _ necessary _ for any li to have been destroyed as a result of the blue - straggler formation process itself , though this could occur as well . in view of the distributions of the ultra - li - deficient stars in the @xmath28 , [ fe / h ] plane , with four at the turnoff and four 100200 k cooler , we consider whether all represent the same phenomenon , or the possibility that two distinct processes have been in operation . it is not a trivial matter to answer this question , because we do not know with certainty what mechanism(s ) has affected any of the stars . however , we explore a number of possibilities in the discussion that follows . ignoring again the obvious blue straggler bd+25@xmath321981 , of the 111 stars shown in figure 3 , 8 are ultra - li - deficient . if all ultra - li - poor stars have the same origin , then we should begin by restating the frequency of such li - weak objects as @xmath297% of plateau stars rather than @xmath295% as estimated previously when the parameter space was incompletely sampled , and with strong metallicity and temperature dependences in that fraction . historically , blue stragglers and ultra - li - deficient stars have been regarded as separate phenomena . however , we have been driven to consider whether there is any astrophysical basis for this separation . one must ask whether the process(es ) that gives rise to blue stragglers is capable only of producing stars whose mass is greater than that of the main sequence turnoff of a @xmath1513 gyr old population . if , as we think is reasonable , the answer is `` no '' , then one may ask what the sub - turnoff mass products of this process(es ) would be . our proposal is that they would be li - deficient , but otherwise difficult to distinguish from the general population . for ultra - li - poor stars redder than the main sequence turnoff , hipparcos parallaxes have established that g186 - 26 is on the main sequence rather than on the subgiant branch . of those _ at _ the turnoff , wolf 550 , g202 - 65 , and bd+51@xmath321817 also have hipparcos parallaxes ; two are almost certainly dwarfs , while g202 - 65 is subject to larger uncertainties and may be more evolved ( see ryan et al . 2001 , table 2 ) . the argument that the evolutionary rate of subgiants is too rapid to explain the high frequency of observed li - deficient objects , which persuaded norris et al . ( 1997a ) to reject the proposition that they might be the _ redward_-evolving ( post - turnoff ) progeny of blue - stragglers , is therefore redundant . however , the detection of several li - weak stars at the bluest edge of the colour distribution has prompted us to re - examine their possible association with blue stragglers . we would describe g202 - 65 as `` at '' the turnoff rather than classify it as a blue straggler in the conventional sense , as it is only marginally hotter ( bluer ) than the main sequence turnoff for its metallicity ( see figure 3 ) . hobbs & mathieu , on the other hand , classified it as a blue straggler , based presumably on the photometry of laird , carney & latham ( 1988 ) which they referenced . ( indeed , carney et al ( 1994 ) declare it as a `` blue straggler candidate '' , and carney et al . ( 2000 ) treat it as one , though acknowledging at the same time that some normal stars may be included in this classification . ) our purpose is _ not _ to debate how this star should be classified , but rather to underline the main suggestion of our work , that the blue straggler and halo ultra - li - deficient stars may have a common origin . although blue stragglers have historically been recognised because they are bluer than the main - sequence turnoff , it is essential to remember that stars that have accreted mass from a companion , or that result from a coalescence can have a mass less than the current turnoff . such stars would be expected to share many of the properties of blue stragglers , but would not _ yet _ appear bluer than the turnoff . however , at some future time , once the main - sequence turnoff reaches lower masses , these non - standard objects would lag the evolution of normal stars and hence appear bluer , showing canonical straggling behaviour . therefore , such stars might , for the present , be regarded as `` blue - stragglers - to - be , '' and our speculation is that the ultra - li - deficient halo stars in are in fact members of such a population . note that this proposition is distinct from that of _ redward_-evolving systems considered and rejected by norris et al . . if ultra - li - deficient stars and blue stragglers are manifestations of the same process , then li deficiency may be the only way of distinguishing sub - turnoff - mass blue - stragglers - to - be from normal main - sequence stars , prior to their becoming classical blue stragglers . mass transfer during their formation may also help clarify some of the unusual element abundances found by norris et al . ( 1997a ; see also ryan et al . 1998 ) . whereas an appeal to extra mixing ( in a single - star framework ) to explain the li depletion would not necessarily affect other elements , mass transfer in a binary with an agb donor may be capable of altering s - process abundances as well . in this regard , we recall that two of the ultra - li - deficient stars studied by norris et al . ( 1997a ; also ryan et al . 1998 ) had non - standard sr and ba abundances . mass transfer from an rgb donor would presumably leave a different chemical signature . @xmath36 , as many as 25% have c overabundances ( e.g. norris , ryan , & beers 1997b ) . at least some but not all of these ( norris , ryan , & beers 1997c ) have s - process anomalies . detailed studies have yet to be completed , so it is unclear what fraction of stars are formed from anomalous material and what fraction became modified later in their life . whilst we can not presently rule out the possibility that the s - process anomalies seen in some ultra - li - deficient stars were inherited at birth , our expectation is that mass transfer from a companion star will be a more common mechanism . ] some constraints on the progenitors of the li - deficient stars may be obtained from their rotation rates and radial velocity variations . webbink s ( 1976 ) calculations of a coalesced star ( @xmath37 ) show that a high rotation rate is maintained at least until it reaches the giant branch . in contrast , previously known blue stragglers appear not to have uncommonly high rotation rates ( e.g. , carney & peterson 1981 ; pritchet & glaspey 1991 ) . this tends to argue against the blue stragglers as having originated from coalesced main - sequence contact binaries , and points towards one of the other binary mass - transfer scenarios , unless mass loss ( e.g. , via webbink s excretion disk ) and magnetic breaking can dissipate envelope angular momentum during the main sequence lifetime of a coalesced star . to spin down , stars must have a way of losing surface angular momentum . in single stars , most of this is believed to occur during the pre- and early - main - sequence phase when magnetic coupling of the stellar surface to surrounding dust creates a decelerating torque on the star . it is not clear that two mature stars which merge will still have this coupling , because of the much lower mass loss rates beyond the early stages of evolution ( unless they produce an excretion disk ) and lower magnetic field strengths . ( see also discussion by sills et al . 1997 , 5.5 . ) leonard & livio ( 1995 ) have proposed that the merger product acquires the distended form of a pre - main - sequence - like star which then spins down as it again approaches the main sequence , losing angular momentum in much the same way as conventional pre - main - sequence stars . for the four stars observed in this work , three had previous radial velocity measurements accurate to @xmath29 1 km s@xmath38 ( carney et al . 1994 ) . the new measurements ( ryan et al . 2001 ; table 2 ) showed residuals of + 1.0 ( bd+51@xmath321817 ) , @xmath393.3 ( g202 - 65 ) , and @xmath396.9 km s@xmath38 ( wolf 550 ) ; compared with the expected radial velocity accuracy of @xmath40 = 0.30.7 km s@xmath38 , these are consistent with significant motion . carney et al . ( 2000 ) indicate periods of 168 to 694 days for these systems , and low eccentricities , except for wolf 550 ( @xmath41 = 0.3 ) . similarly , the metal - poor field blue straggler cs 22966 - 043 has an orbital period of 319 days ( preston & landolt 1999 ) . if the brighter component has a mass of 0.8 m@xmath42 and its companion has a mass between 0.4 and 1.4 m@xmath42 ( appropriate to a white dwarf ) then the _ current _ semi - major axis of the system will be in the range @xmath43 = 200260 r@xmath42 ( from kepler s third law ) . companions having a canonical white - dwarf mass . ] their second system , cs 29499 - 057 , may have an even longer period of 2750 days , implying @xmath43 = 9001100 r@xmath42 . the periods of these and carney et al s systems , and hence their large current separations , are more compatible with mass loss from an evolved companion rather than being short - period systems in contact on the main sequence . the evidence presented to date has argued against internal mixing alone as an adequate explanation for the ultra - li - deficient stars whose neutron - capture elements show abundance anomalies . note , though , that certainly not all ultra - li - deficient stars and blue stragglers exhibit neutron - capture element anomalies ( carney & peterson 1981 ; norris et al . 1997a ; ryan et al . if mass transfer has occurred , systems in which s - process elements are abnormal would presumably indicate material originating with an agb companion , whereas s - process - normal remnants would indicate mass transfer during an earlier stage of evolution ( rgb ) or from a pre - thermal - pulsing agb mass donor . ( we have no data on the n abundance , and the ch band in these stars is too weak to hope to measure the @xmath44c/@xmath45c ratio . ) likewise , the rotation rates of both blue stragglers and ultra - li - deficient stars are apparently normal , arguing against coalescences having already occurred on the main sequence . of the three mechanisms found to be viable by pritchet & glaspey ( 1991 ) and hobbs & mathieu ( 1991 ) , this leaves mass transfer from a companion as the only one remaining , _ if _ we are correct in speculating that the ultra - li - deficient and blue straggler phenomena are manifestations of the same process . in the absence of an adequate theory for why eight otherwise - normal halo stars ( excluding the traditional blue straggler bd+21@xmath321981 ) should have low ( zero ? ) li abundances , it may be useful to consider the hot subsample ( 6200 k @xmath46 @xmath28 @xmath46 6300 k ) as a distinct group . several possibilities then arise that might account for the observed li deficiency , including diffusion ( the sinking of li to below the photosphere ) , the f - star li dip , and an unknown process that may be responsible for depletion in some ( but not all ) disk stars . we consider each of these in turn . we note that the three li - deficient stars with @xmath1 k are confirmed binaries , whereas most cooler ones show no evidence of binary motion . the binary / single distinction between warmer / cooler li - depleted stars is pronounced ; see table 1 , where the binary status ( carney et al . 1994 , 2000 ; latham 2000 , priv.comm . ) is given in the final column . if such a dichotomy is maintained as more li - poor systems are discovered , it may indicate a genuine difference in the origin of the turnoff and sub - turnoff systems . deliyannis , demarque & kawaler ( 1990 ) and proffitt & michaud ( 1991 ) have computed the predicted effects of diffusion on the surface li abundances of warm halo stars . diffusion is more significant in hotter stars because their surface convective zone is thinner . the degree of depletion expected at @xmath0 k is a function of effective temperature , changing by @xmath29 0.2 dex per 100 k in the former ( for @xmath11 = 1.1 ) , and @xmath29 0.2 and @xmath47 dex per 100 k in the latter ( for @xmath11 = 1.7 and 1.5 respectively ) . this does not match the behavior observed ( see figure 2 ) . for comparison , our ultra - li - poor stars are depleted by @xmath48 dex . this alone appears to rule out diffusion as the explanation , except possibly for the lower-@xmath11 model of proffitt & michaud . however , li diffusion appears to have been inhibited in all other metal - poor samples ( e.g. , ryan et al . 1996 ) , so it would be unusual to see it suddenly present and with such effect only in isolated stars in our new sample . boesgaard & tripicco ( 1986 ) and hobbs & pilachowski ( 1988 ) showed that li is severely depleted in population i open cluster stars over the interval 6400 k @xmath49 k. various explanations have been proposed , including mass loss ( e.g. , schramm , steigman , & dearborn 1990 ) , diffusion ( e.g. , turcotte , richer & michaud 1998 ) , and slow mixing of various forms ( e.g. , deliyannis & pinsonneault 1997 ) , but none has been convincingly established as responsible , and several mechanisms may be acting in concert ( e.g. , turcotte et al . ) . whatever the correct explanation(s ) , is it possible that the hottest ultra - li - deficient stars are encroaching on this regime and are affected by this phenomenon ? although this can not be ruled out completely for the hot subset , especially since we have questioned the reliability of the @xmath50 ( and hence @xmath28 ) values of the hottest li - preserving stars in figure 2 , the onset of destruction in the f - star dip seems too gradual with @xmath28 to explain the new data . the hyades observations ( boesgaard & tripicco 1986 ) show a decrease of only 0.3 dex from 6200 to 6400 k , substantially less than the @xmath51 dex deficit in the ultra - metal - poor objects around 6300 k. as noted above , hipparcos parallaxes are available for five of the eight known ultra - li - deficient stars and , with the possible exception of g202 - 65 , rule out the possibility that these stars are redward - evolving _ descendants _ of the li - dip . lambert , heath & edvardsson ( 1991 ) found that , in almost all cases , the low li abundances in their population i sample could be ascribed to their being evolved descendants of li - dip stars , or else being dwarfs exhibiting the li - depletion that increases towards _ lower _ temperature , as is normally associated with pre - main - sequence and/or main - sequence burning . anomalously high li depletions were found in only 13 cases out of some 26 old - disk stars , and for a similar fraction of young - disk stars . based on this fraction , lambert et al . proposed that a new class of highly li - depleted stars , comprising less than about 10% of the population , might exist . it is interesting to note that this proposal pre - dated the discovery of ultra - li - deficient halo dwarfs . the uncertain number of cases stated above arises because lambert et al . recognised that uncertainties in the stellar luminosities , and hence mass , could drive stars into or out of the region of importance . we now have the benefit of accurate hipparcos parallaxes . these indicate that two of the seven stars highlighted by their study , hd 219476 and hr 4285 , are indeed considerably more massive than reported in lambert et al.s tables and hence are probably descendants of the li gap , thus reducing the number of _ genuine _ cases to 2 out of 26 old - disk stars , and 3 out of a similar number of young - disk stars . that is , the fraction of anomalously li - depleted stars appears to be around 8 - 10% , albeit sensitive to small - number statistics . ultra - li - depleted population i stars are also seen in young open clusters . they can be recognised , for example , in fig . 1 of ryan & deliyannis ( 1995 ) , where @xmath296% of the hyades stars cooler than the f - star dip appear to be ultra - li - deficient . is it possible that the li - depleted halo stars are of the same type ? the lack of examples in the two pop i and pop ii classes to compare with precludes a detailed analysis , but we note that we see li deficiency in about 7% of halo objects , which is comparable to the ratio for the pop i objects . that is , the pop i and pop ii examples could arise due to the same process , even though it remains unclear what that process is . we note , for completeness , that ryan et al . ( 2001 ) showed that the kinematics of the new ultra - li - depleted stars are clearly those of halo objects , and thus they genuinely belong to the halo population , despite their metallicities being close to those of the most metal - poor thick - disk stars . the stars remaining on lambert et al s list of unusually li - deficient objects are : hr 3648 , hr 4657 , hr 5968 , hr 6541 , and hd 30649 . upon searching the literature for evidence of binarity or abundance anomalies in these systems , we found that not only was hr 4657 a 850 day period binary , but fuhrmann & bernkopf ( 1999 ) had also been driven to consider this star as a blue straggler . it has an unexpectedly high rotational velocity ( in contrast to the blue stragglers studied by carney & peterson 1981 ) . there is no evidence of s - process anomalies , but other unusual characteristics of the system include an observable soft x - ray flux and the very likely association of this object with grb 930131 . hr 3648 (= 16 uma = hd 79028 ) is a 16.2 day period chromospherically - active single - lined spectroscopic binary ( basri , laurent , & walter 1985 ) . hd 30649 (= g81 - 38 ) and hr 6541 ( = hd 159332 ) , in contrast , show no significant evidence of binarity ( carney et al . hr 5968 (= @xmath52 crb ) does not appear to have a stellar companion , though it has a planetary companion ( noyes et al . 1997 ) , but ryan ( 2000 ) argues that li in this star is _ not _ anomalous . hr 3648 and hr 4657 have ba abundance measurements from the study by chen et al . the latter also has been observed by fuhrmann & bernkopf ( 1999 ) , but neither star appears abnormal in this element . ryan et al . ( 1999 ) have argued that the ultra - li - deficient halo stars are distinct from the majority of halo stars that occupy the spite plateau , and , in particular , that they do _ not _ merely represent the most extreme examples of a _ continuum _ of li depletion . if the association with blue stragglers ( or , for that matter , any distinct evolutionary phenomenon ) is correct , then the mechanism for their unusual abundances will at last be understood and they will be able to be neglected with certainty from future discussion of the spite plateau . in the present work , we have proposed and discussed the possibility that ultra - li - depleted halo stars and blue stragglers are manifestations of the same phenomenon , and described the former as `` blue - stragglers - to - be . '' we proposed that their li was destroyed either during the formation process of blue stragglers or during the _ normal _ single - star evolutionary processes of their precursors , namely during pre - main - sequence and/or main - sequence phases of low - mass stars , or during post - main - sequence evolution of mass donors , as in the scenario quantified by norris et al . ( 1997a ) . we note that in a study carried out separately but over the same time period as ours , carney et al . ( 2000 ) have examined the orbital characteristics of blue stragglers , and have been driven towards similar considerations as we have . there are clearly still details to be clarified , but our two groups appear to be converging on a view unifying blue stragglers and ultra - li - deficient systems . because there are numerous observational and theoretical issues surrounding this unified view , we seek to clarify the main arguments and possibilities using an itemised summary . observations : @xmath53 in a study of 18 halo stars with @xmath17 [ fe / h ] @xmath18 and 6000 k @xmath54 6400 k , we have found four ultra - li - deficient objects , i.e. a 22% detection rate . @xmath53 the fraction of ultra - li - deficient stars is very much higher amongst the hottest and most metal - rich halo main - sequence stars ( @xmath2920% ) than amongst cooler and more metal - poor ones ( @xmath295% ) . @xmath53 ultra li - deficient stars exist both at the turnoff , and cooler than the turnoff , and with well - determined main - sequence luminosities from hipparcos . @xmath53 all of the turnoff ultra - li - deficient halo stars , but none of the sub - turnoff ultra - li - deficient halo stars , appear to be binaries . this may indicate that two different mechanisms are causing the halo ultra - li - deficient phenomenon . theoretical framework : @xmath53 blue stragglers may form from _ several _ mechanisms , but seem to require at least one of either complete mixing , binary mass transfer , or coalescence ( hobbs & mathieu 1991 ; pritchet & glaspey 1991 ) . origins : @xmath53 we speculate that ultra - li - deficient stars and blue stragglers are manifestations of the same process , and that sub - turnoff - mass ultra - li - deficient stars may be regarded as `` blue - stragglers - to - be . '' @xmath53 li could be destroyed at several stages : ( i ) in a mass - transfer event which induces extensive mixing ; ( ii ) by single - star evolutionary processes ( convective mixing ) in a post - main - sequence mass donor ; ( iii ) by single - star evolutionary processes ( mixing ) in pre - main - sequence ( or possibly main - sequence ) low - mass stars prior to their gaining mass . @xmath53 mass - transfer scenarios from an agb star seem better able to explain the unusual neutron - capture element ratios _ sometimes _ seen in ultra - li - depleted stars ( norris et al . 1997a ) than internal mixing , since @xmath29 0.8 m@xmath42 core - hydrogen - burning stars are not expected to process neutron - capture elements . this argues against internal mixing as the sole explanation for the existence of ultra - li - depleted stars with unusual neutron - capture abundances . ( mass transfer from pre - agb ( most likely rgb ) donors would produce the stars with normal neutron - capture abundances . ) @xmath53 coalesced binaries are expected to maintain high rotation rates until they reach the giant branch , but neither blue stragglers nor ultra - li - depleted halo stars have high rotation rates . this argues against coalescence of a binary as the explanation for these objects unless they have spun down . @xmath53 the orbital periods of metal - poor field blue stragglers ( preston & landolt 1999 ; carney et al . 2000 ) suggest current semi - major axes in the range 2001100 r@xmath42 , arguing against these being coalescing stars ( unless they began their lives as triple systems ) . @xmath53 the arguments against solely internal mixing , and against coalescence of main - sequence contact binaries , leaves mass transfer as the most viable mechanism for field binaries . this is _ not _ to say that li was destroyed during the transfer ; it may have been destroyed by single - star mechanisms already . @xmath53 the observed d@xmath6(li)/d@xmath28 is too steep compared with models of diffusion to be due to that process . @xmath53 the observed d@xmath6(li)/d@xmath28 is too steep compared with the hyades data to be due to the f - star li dip . @xmath53 the halo ultra - li - deficient stars could be related to the pop i anomalously - li - depleted stars identified in the field by lambert et al . ( 1991 ) and also seen in open clusters . @xmath53 hipparcos parallaxes rule out the possibility that the ultra - li - deficient stars are redward - evolving post - turnoff stars . they have not descended from the f - star li dip . implications : @xmath53 severe li depletion may be the ( only ? ) signature of sub - turnoff - mass blue stragglers . the halo population fraction comprising ultra - li - poor stars is 7% . @xmath53 understanding the ultra - li - depleted stars as resulting from a distinct process ( not normally affecting single stars ) would eliminate the need to include them in discussions of processes affecting the evolution of normal spite plateau stars , and would explain why they appear so radically different from the vast majority of halo stars ( ryan et al . the authors gratefully acknowledge the support for this project given by the australian time assignment committee ( atac ) and panel for the allocation of telescope time ( patt ) of the aat and wht respectively , and for practical support given by the staff of these facilities . they also express gratitude to d. a. latham and b. w. carney for conveying the results of their program in advance of publication , and to an anonymous referee for his / her comments that helped us clarify our arguments . s.g.r . sends a special thanks to colleagues at the university of victoria : to c. j. pritchet for a most memorable snow - shoeing expedition on 1991 february 10 during which li deficiency in blue stragglers was discussed , to d. a. vandenberg for discussing and supplying isochrones , and to f. d. a. hartwick for once asking whether there were blue stragglers in the halo field . t.c.b acknowledges partial support from grant ast 95 - 29454 from the national science foundation . basri , g. , laurent , r. , & walter , f. m. 1985 , apj , 298 , 761 bergbusch , p. a. , & vandenberg , d. a. 1992 , apjs , 81 , 163 boesgaard , a. m. & tripicco , m. j. 1986 , apj , 302 , l49 bonifacio , p. & molaro , p. 1997 , mnras , 285 , 847 carney , b. w. , latham , d. w. , laird , j. b. , grant , c. e , & morse , j. a. 2000 , preprint carney , b. w. , latham , d. w. , laird , j. b. , & aguilar , l. a. 1994 , aj , 107 , 2240 carney , b. w. , & peterson , r. c. 1981 , apj , 251 , 190 chaboyer , b. 2000 , `` the galactic halo ; from globular clusters to field stars '' a. noels , p. magain , d. caro , e. jehin , g. parmentier , & a. thoul ( eds ) ( u. lige , lige ) in press chaboyer , b. , demarque , p. , & pinsonneault , m. h. 1995 , apj , 441 , 876 chen , y. q. , nissen , p. e. , zhao , g. , zhang , h.w . , & benoni , t. 2000 , a&as , 141 , 491 deliyannis , c. p. 1995 , the light element abundances , ed . p. crane , ( berlin : springer - verlag ) , 395 deliyannis , c. p. , demarque , p. , & kawaler , s. d. 1990 , apjs , 73 , 21 deliyannis , c. p. , king , j. r. , boesgaard , a. m. , & ryan , s. g. 1994 , apj , 434 , l71 deliyannis , c. p. & pinsonneault , m. h. 1997 , apj , 488 , 836 deliyannis , c. p. , pinsonneault , m. h. & duncan , d. k. 1993 , apj , 414 , 740 ferraro , f. r. , fusi pecci , f. , & bellazini , m. 1995 , a&a , 294 , 80 ferraro , f. r. , palterinieri , b. , rood , r. t. , & dorman , b. 1999 , apj , 522 , 983 fuhrmann , k. , & bernkopf , j. 1999 , a&a , 247 , 897 to field stars , ed . a noels & p. magain , 35@xmath55 lige int . astroph . , in press green , e. m. , demarque , p. , & king , c. r. 1987 , the revised yale isochrones and luminosity functions ( new haven : yale univ . press ) hobbs , l. m. , & mathieu , r. d. 1991 , pasp , 103 , 431 hobbs , l. m. & pilachowski , c. 1988 , apj , 334 , 734 hobbs , l. m. , welty , d. e. , & thorburn , j. a. 1991 , apj , 373 , l47 jeffries , r. d. 1999 , mnras , 309 , 189 jones , b. f. , fischer , d. , & soderblom , d. r. 1999 , aj , 117 , 330 laird , j. b. , carney , b. w. , & latham , d. w. 1988 , aj , 95 , 1843 lambert , d. l. , heath , j. e. , and edvardsson , b. 1991 , mnras , 253 , 610 leonard , p. j. t. 1989 , aj , 98 , 217 leonard , p. j. t. , & livio , m. 1995 , apj , 447 , l121 lombardi , j. c. jr , rasio , f. a. , & shapiro , s. l. 1996 , apj , 468 , 797 martn , e. l. , & claret , a. 1996 , a&a , 306 , 408 mateo , m. , harris , h. c. , nemec , j. , & olszewski , e. w. 1990 , aj , 100 , 469 montalbn , j. , & schatzman , e. 1996 , a&a , 305 , 513 norris , j. e. , ryan , s. g. , & beers , t. c. 1997b , apj , 488 , 350 norris , j. e. , ryan , s. g. , & beers , t. c. 1997c , apj , 489 , l169 norris , j. e. , ryan , s. g. , beers , t. c. , & deliyannis , c. p. 1997a , apj , 485 , 370 noyes , r. w. , jha , s. , korzennik , s. g. , krockenberger , m. , nisenson , p. , brown , t. m. , kennelly , e. j. , & horner , s. d. 1997 , apj , 483 , l111 piotto , g. 2000 , `` the galactic halo ; from globular clusters to field stars '' a. noels , p. magain , d. caro , e. jehin , g. parmentier , & a. thoul ( eds ) ( u. lige , lige ) in press portegies zwart , s. 2000 , proc . . the influence of binaries on stellar population studies , brussels 2000 , d. vanbeveren ( ed ) , in press preston , g. w. , & landolt , a. u. 1999 , aj , 118 , 3006 preston , g. p. & sneden , c. 2000 , in press pritchet , c. j. , & glaspey , j. w. 1991 , apj , 373 , 105 proffitt , c. r. , & michaud , g. 1991 , apj , 371 , 584 rebolo , r. , molaro , p. & beckman , j. e. 1988 , a&a , 192 , 192 ryan , s. g. 2000 , mnras , 316 , l35 ryan , s. g. , beers , t. c. , deliyannis , c. p. , & thorburn , j. a. 1996 , apj , 458 , 543 ryan , s. g. & deliyannis , c. p. 1995 , apj , 453 , 819 ryan , s. g. , kajino , t. , beers , t. c. , suzuki , t. , romano , d. , matteucci , f. , & rosolankova , k. 2001 , apj , 548 ( 20 feb ) ryan , s. g. , norris , j. e. , & beers , t. c. 1998 , apj , 506 , 892 ryan , s. g. , norris , j. e. , & beers , t. c. 1999 , apj , 523 , 654 sandage , a. r. 1953 , aj , 58 , 61 sandquist , e. l. , bolte , m. , & hernquist , l. 1997 , apj , 477 , 335 schatzman , e. , 1993 , a&a , 279 , 431 schramm , d. n. , steigman , g. , and dearborn , d. s. p. 1990 , apj , 359 , l55 sills , a. p. , bailyn , c. d. , & demarque , p. 1995 , apj , 455 , l163 sills , a. , lombardi , j. c. jr , bailyn , c. d. , demarque , p. , rasio , f. a. , & shapiro , s. l. 1997 , apj , 487 , 290 spite , f. & spite , m. 1982 , a&a , 115 , 357 spite , m. , maillard , j. p. , & spite , f. 1984 , a&a , 141 , 56 spite , m. , molaro , p. , franois , p. , & spite , f. 1993 , a&a , 271 , l1 spite , m. , spite , f. , cayrel , r. , hill , v. , depagne , e. , nordstrom , b. , beers , t. , & nissen , p. e. 2000 , `` the evolution of the light elements : iau symp . 198 '' l. da silva , r. de medeiros , & m. spite ( eds ) , ( asp conf ser . ) , in press stryker , l. l. 1993 , pasp , 105 , 1081 thorburn , j. a. 1992 , apj , 399 , l83 thorburn , j. a. 1994 , apj , 421 , 318 thorburn , j. a. & beers , t. c. 1993 , apj , 404 , l13 turcotte , s. , richer , j. , & michaud , g. 1998 , apj , 504 , 559 ventura , p. , zeppieri , mazzitelli , i. , & dantona , f. 1998 , a&a , 331 , 1011 webbink , r. f. 1976 , apj , 209 , 829

we present data for four ultra - li - deficient , warm , halo stars . the li deficiency of two of these is a new discovery . three of the four stars have effective temperatures @xmath0 k , in contrast to previously known li - deficient halo stars which spanned the temperature range of the spite plateau . in this paper we propose that these , and previously known ultra - li - deficient halo stars , may have had their surface lithium abundances reduced by the same mechanism as produces halo field blue stragglers . even though these stars have yet to reveal themselves as blue stragglers , they might be regarded as `` blue - stragglers - to - be . '' in our proposed scenario , the surface abundance of li in these stars could be destroyed ( a ) during the normal pre - main - sequence single star evolution of their low mass precursors , ( b ) during the post - main - sequence evolution of a evolved mass donor , and/or ( c ) via mixing during a mass - transfer event or stellar merger . the warmest li - deficient stars at the turnoff would be regarded as emerging `` canonical '' blue stragglers , whereas cooler ones represent sub - turnoff - mass `` blue - stragglers - to - be . '' the latter are presently hidden on the main sequence , li depletion being possibly the clearest signature of their past history and future significance . eventually , the main sequence turnoff will reach down to their mass , exposing those li - depleted stars as canonical blue stragglers when normal stars of that mass evolve away . arguing _ against _ this unified view is the observation that the three li - depleted stars at @xmath1 k are _ all _ binaries , whereas very few of the cooler systems show evidence for binarity ; it is thus possible that two separate mechanisms are responsible for the production of li - deficient main - sequence halo stars .

expanding maps of the unit interval have been widely studied in the last decades and the associated transfer operators have proven to be of vital importance in solving problems concerning the statistical behaviour of the underlying interval maps @xcite . in recent years an increasing amount of interest has developed in maps which are expanding everywhere except on an unstable fixed point ( that is , an indifferent fixed point ) at which trajectories are considerably slowed down . this leads to an interplay of chaotic and regular dynamics , a characteristic of intermittent systems @xcite . from an ergodic theory viewpoint , this phenomenon leads to an absolutely continuous invariant measure having infinite mass . therefore , standard methods of ergodic theory can not be applied in this setting ; indeed it is wellknown that birkhoff s ergodic theorem does not hold under these circumstances , see for instance @xcite . in this paper we will be concerned with @xmath0-farey maps , see figure [ fig1 ] . these maps are of great interest since they are piecewise linear and expanding everywhere except for at the indifferent fixed point where they have ( right ) derivative one . this makes the @xmath0-farey maps a simple model for studying the physical phenomenon of intermittency @xcite . moreover , an induced version of the @xmath0-farey maps are given by the @xmath0-lroth maps introduced in @xcite , which have significant meaning in number theory , see for instance @xcite . thaler @xcite was the first to discern the asymptotics of the transfer operator of a class of interval maps preserving an infinite measure . this class of maps , to which the @xmath0-farey maps do not belong , have become to be known as thaler maps . in an effort to generalise this work , by combining renewal theoretical arguments and functional analytic techniques , a new approach to estimate the decay of correlation of a dynamical system was achieved by sarig @xcite . subsequently , gouzel @xcite generalised these methods . using these ideas and employing the methods of garsia and lamperti @xcite , erickson @xcite and doney @xcite , recently melbourne and terhesiu ( * * theorem 2.1 to 2.3 ) proved a landmark result on the asymptotic rate of convergence of iterates of the induced transfer operator and showed that these result can be applied to gibbs - markov maps , thaler maps , afn maps , and pomeau - manneville maps . thus , the question which naturally arises is , whether this asymptotic rate can be related to the asymptotic rate of convergence of iterates of the actual transfer operator . the results of this paper give some positive answers to this question for @xmath1-expansive @xmath0-farey maps . as mentioned above , in this paper , we will consider the @xmath0-farey map @xmath2 \to [ 0 , 1]$ ] , which is given for a countable infinite partition @xmath3 of @xmath4 by non - empty intervals @xmath5 . it is assumed throughout that the atoms of @xmath0 are ordered from right to left , starting with @xmath6 , and that these atoms only accumulate at zero . further , we assume that @xmath5 is right - open and left - closed , for all natural numbers @xmath7 . we define the @xmath0-_farey map _ \to [ 0 , 1]$ ] by @xmath8 where @xmath9 is equal to the lebesgue measure @xmath10 of the atom @xmath11 and @xmath12 denotes the lebesgue measure of the @xmath7-th tail @xmath13 of @xmath0 , see figure [ fig1 ] . throughout , we will assume that the partition @xmath0 satisfies the condition that the sequence @xmath14 is not summable . for @xmath15 $ ] , an @xmath0-farey map @xmath16 is said to be _ @xmath1-expansive _ if the sequence @xmath17 is regularly varying of order @xmath18 , that is , if there exists a slowly varying function @xmath19 such that @xmath20 , for all @xmath21 . ( recall that @xmath22 is called a _ slowly varying function _ , if it is measurable , locally riemann integrable and @xmath23 , for each @xmath24 and for some @xmath25 , see @xcite for further details . ) in this situation , ( * ? ? ? * theorem 1.5.10 ) implies that @xmath26 therefore , the lebesgue measure of the @xmath7-th tail of @xmath0 is asymptotic to a regularly varying function of order @xmath27 . thus , @xmath1-expansive implies expansive of order @xmath1 in the sense of @xcite . however , an expansive @xmath0-farey map of order @xmath1 is not necessarily @xmath1-expansive . throughout , let @xmath28 denote the @xmath16-invariant measure which is determined by @xmath29 and let @xmath30 denote the borel @xmath31-algebra of @xmath32 $ ] . here and in the sequel , for a given borel set @xmath33 , we let @xmath34 denote the indicator function on @xmath35 . it is verified in @xcite that , since the sequence @xmath17 is regularly varying of order @xmath36 , the map @xmath16 is conservative , ergodic and measure preserving on the infinite and @xmath31-finite measure space @xmath37 , \mathscr{b } , \mu_{\alpha})$ ] . the dynamical system @xmath38 , \mathscr{b } , \mu_{\alpha } , f_{\alpha})$ ] will be referred to as a _ @xmath0-farey system_. following the definitions and notations of @xcite , throughout , we let @xmath39)$ ] ( respectively @xmath40)$ ] ) denote the class of measurable functions @xmath41 with domain @xmath32 $ ] for which @xmath42 is @xmath28-integrable ( respectively @xmath43-integrable ) , and for @xmath44)$ ] ( respectively @xmath40)$ ] ) , define @xmath45 ( respectively @xmath46 ) by @xmath47 further , for a given measurable function @xmath48 , we set @xmath49 } \rvert w(x ) \lvert$ ] . the _ @xmath0-farey transfer operator _ @xmath50 ) \to \mathcal{l}^{1}_{\mu_{\alpha}}([0,1])$ ] is the positive linear operator given by @xmath51 where @xmath52})^{-1}$ ] and @xmath53})^{-1}$ ] refer to the inverse branches of @xmath16 . in particular , for all @xmath54)$ ] and all measurable functions @xmath48 with @xmath55 , @xmath56 ( the above equality is a direct consequence of ( * ? ? ? * lemma 2.5 ) . ) note that the equality given in is the usual defining relation for the _ transfer operator _ of @xmath16 . however , the relation in only determines values of the transfer operator of @xmath16 applied to an observable @xmath28-almost everywhere . thus the @xmath0-farey transfer operator is a version of the transfer operator of @xmath16 . in order to state our main theorems , we will also require the following function spaces . we let @xmath57 denote the _ first return time _ , given by @xmath58 , and we write @xmath59 . let @xmath60 denote the countable - infinite partition @xmath61 of @xmath6 and let @xmath62 denote the set of functions with domain @xmath32 $ ] that are supported on a subset of @xmath63 and which have finite @xmath64-norm , where @xmath65 and where @xmath66 in particular , if @xmath67 , then @xmath41 is lipschitz continuous on each atom of @xmath60 , zero outside of @xmath63 and bounded ( everywhere ) . we then define @xmath68 ) : \lvert v \rvert_{\infty } < \infty\ ; \text{and } \ ; \widehat{f}^{n-1}_{\alpha}(v \cdot \mathds{1}_{a_{n } } ) \in \mathcal{b}_{\alpha } \ ; \text{for all } \ ; n \in \mathbb{n } \right\}. \ ] ] for examples of observables belonging to @xmath69 , we refer the reader to example [ ex : ex1 ] and the discussion succeeding our main results , theorems [ thm : main2 ] and [ thm : main1 ] . let us also recall from @xcite that the _ wandering rate _ of @xmath16 is given by @xmath70 further , as we will see in , if @xmath71 and if the given @xmath0-farey system is @xmath1-expansive , then the wandering rate is regularly varying of order @xmath72 . also , in the case that @xmath73 , if @xmath74 is a bounded sequence , then we say that the wandering rate @xmath75 is _ moderately increasing_. here and in the sequel for @xmath76 we let @xmath77 denote the smallest integer greater than or equal to @xmath78 . with the above preparations , we are now in a position to state the main results , theorems [ thm : main2 ] and [ thm : main1 ] . theorem [ thm : main2 ] provides mild conditions under which the asymptotic behavior of the iterates of an @xmath0-farey transfer operator _ restricted to _ @xmath6 can be extended to all of @xmath79 $ ] and is used in our proof of theorem [ thm : main1 ] . ( note that , by , any @xmath1-expansive @xmath0-farey system satisfies the requirements of theorem [ thm : main2 ] . ) one of the facets of theorem [ thm : main1 ] is that it gives sufficient conditions on observables which guarantee that iterates of an @xmath0-farey transfer operator applied to such an observable is asymptotic to a constant times the wandering rate . these results complement ( * ? ? ? * theorem 10.5 ) and show that additional assumptions are required in ( * ? ? ? * theorem 10.4 ) . namely , in the case that @xmath73 , we show that the statement of ( * ? ? ? * theorem 10.4 ) holds true , with the additional assumption that the wandering rate is moderately increasing ( theorem [ thm : main1][i ] ) ; for @xmath80 , we provide an example which demonstrates that additional requirements are necessary for the expected convergence ( theorem [ thm : main1][iii ] ) and provide sufficient conditions ( theorem [ thm : main1][ii ] ) . [ thm : main2 ] for an @xmath0-farey system @xmath37 , \mathscr{b } , \mu_{\alpha } , f_{\alpha})$ ] for which the wandering rate satisfies the condition @xmath81 , we have that , if @xmath82)$ ] satisfies @xmath83 uniformly on @xmath63 , then the same holds on any compact subset of @xmath79 $ ] . the same statement holds when replacing uniform convergence by almost everywhere uniform convergence . for @xmath71 , a @xmath1-expansive @xmath0-farey system has wandering rate satisfying @xmath81 . [ thm : main1 ] let @xmath37 , \mathscr{b } , \mu_{\alpha } , f_{\alpha})$ ] be a @xmath1-expansive @xmath0-farey system . 1 . [ i ] let @xmath84 and assume that the wandering rate is moderately increasing.if@xmath85andif @xmath86 then uniformly on compact subsets of @xmath79 $ ] , @xmath87 2 . [ ii ] for @xmath88 $ ] , if @xmath89)$ ] with @xmath90 bounded and if 1 . the sequence @xmath91 is bounded and 2 . there exist constants @xmath92 and @xmath93 with @xmath94 , for all @xmath21 . + then uniformly on compact subsets of @xmath79 $ ] , @xmath95 here , @xmath96 and @xmath97 denotes the gamma function . [ iii ] for @xmath80 , there exists a positive , locally constant , riemann integrable function @xmath98 of bounded variation satisfying the inequality in , such that , for all @xmath99 , @xmath100 it is immediate that if @xmath101 , then @xmath102 and @xmath103 , and that these parameters give rise to an example of an @xmath0-farey system which satisfies the conditions of theorem [ thm : main1][i ] . indeed there exist many examples of @xmath0-farey systems for which the conditions of theorem [ thm : main1][i ] are satisfied , but where the wandering rate behaves very differently to the function @xmath104 . letting @xmath73 , as we will see in lemma [ lem : powerslowly][sv(vi ) ] , the sequence @xmath105 is slowly varying and @xmath106 . we also have that @xmath107 using this we deduce the following . 1 . if @xmath108 , then @xmath109 and @xmath110 . if @xmath111 , then @xmath112 and @xmath113 , + where @xmath114 . indeed the above two sets of parameters give rise to examples of @xmath115-expansive @xmath0-farey systems whose wandering rate is moderately increasing . moreover , @xmath116 demonstrating that two moderately increasing wandering rates , although they are all slowly varying , do not have to be asymptotic to each other nor to the function @xmath104 . in the case that @xmath16 is a @xmath115-expansive @xmath0-farey map , we have that the wandering rate @xmath75 is a slowly varying function . we remark here that it is not the case that every slowly varying function is moderately increasing , namely , it is not the case that if @xmath117 \to \mathbb{r}$ ] is a slowly varying function , then the sequence @xmath118 is bounded . for instance consider the following . let @xmath119 be a decreasing sequence of positive real numbers which converge to zero and , for @xmath120 , set @xmath121 where @xmath122 . we define @xmath123 by @xmath124 for @xmath125 $ ] . the function @xmath126 \to \mathbb{r}$ ] defined by @xmath127 is , by construction , slowly varying . however , the sequence given in is unbounded . ( we are grateful to fredrik ekstrm for providing this example ) . if in the definition of the norm @xmath64 , one replaces the norm @xmath128 by the _ essential supremum norm _ @xmath129 , then by appropriately adapting the proofs given in the sequel , one can obtain a proof of theorem [ thm : main1 ] where the uniform convergence on compact subsets of @xmath79 $ ] is replaced by uniform convergence almost everywhere on compact subsets of @xmath4 . the first part of the proof of theorem [ thm : main1 ] [ i ] and [ ii ] are inspired by the first paragraph in the proof of ( * ? ? ? * theorem 10.4 ) . the structure of this paper is as follows . in section [ sec : pre ] we collect basic properties of @xmath0-farey maps and their corresponding transfer operators . in section [ j_diploma_thesis ] we provide a proof of theorem [ thm : main2 ] . this proof is inspired by arguments originally presented in @xcite . then in section [ sec : main1 ] we present the proof of theorem [ thm : main1 ] , breaking the proof into three constituent parts . in section [ section : counterexamples2.2 ] we obtain part [ i ] and give explicit examples of observables satisfying the given properties . in section [ mre_alpha_delta_in_051 ] we prove part [ ii ] , for explicit examples of observables which satisfy the pre - requests of theorem [ thm : main1 ] [ ii ] we refer the reader to remark [ rmk : rmk2 ] . finally we conclude with section [ section : counterexamples ] where part [ iii ] is proven using a constructive argument . before we conclude this section with a series of remarks , remarks [ rmk : rmk1 ] to [ rmk : rmk3 ] , in which we comment on how theorem [ thm : main1 ] , and hence theorem [ thm : main2 ] , complement the results obtained in @xcite , we introduce the _ perron - frobenius operator _ @xmath130 ) \to \mathcal{l}_{\lambda}^{1}([0 , 1])$ ] which is defined by @xmath131 where @xmath132 denotes the right derivative of @xmath16 and where @xmath133 . ( note , by construction , if @xmath16 is @xmath1-expansive , then the right derivative of @xmath16 at zero is equal to one . ) a useful relation between the operators @xmath134 and @xmath135 is that @xmath136 we refer the reader to @xcite for a proof of the equality in . [ rmk : rmk1 ] for certain interval maps @xmath137 \to [ 0,1]$ ] with two monotonically increasing , differentiable branches whose invariant measure has infinite mass and whose tail probabilities are regularly varying with exponent @xmath138 , thaler @xcite discerned the precise asymptotic behaviour of iterates of the associated perron - frobenius operator @xmath139 , namely , that for all riemann integrable functions @xmath140 with domain @xmath32 $ ] , one has that @xmath141 uniformly almost everywhere on compact subsets of @xmath142 $ ] . here , @xmath143 denotes the associated invariant density and @xmath144 denotes the wandering rate of @xmath145 . however , @xmath0-farey maps do not fall into this class of interval maps . using the relationship between the transfer and the perron - frobenius operator , theorem [ thm : main1 ] [ ii ] together with the assumption that the banach space of functions of bounded variation[page : bv - banach ] with the norm @xmath146 satisfies certain functional analytic conditions ( namely , conditions ( h1 ) and ( h2 ) given in section [ sec : pre ] ) , show that thaler s result can be extended to @xmath1-expansive @xmath0-farey maps . results of this form have also been obtained in @xcite for afn maps . ( note , an @xmath0-farey map is also not an afn map . ) [ rmk : rmk2 ] kessebhmer and slassi @xcite showed that for the classical farey map the convergence given in holds uniformly almost everywhere on @xmath147 $ ] for convex . likewise , for a @xmath1-expansive @xmath0-farey map , theorems [ thm : main1 ] [ ii ] implies that if @xmath140 is a convex @xmath148-observable , then the convergence in holds uniformly on compact subsets of @xmath142 $ ] . to see that a convex @xmath148-observable satisfies the requirements of theorem [ thm : main1 ] [ ii ] , one employs arguments similar to those used in example [ ex : ex1 ] together with and . [ rmk : rmk3 ] the consequences of theorem [ thm : main1 ] go even further , in that for a map , we are able to obtain that the convergence given in holds uniformly on compact subsets of @xmath79 $ ] , for certain non - riemann integrable observables which are not necessarily bounded . for instance , if @xmath149 is an observable such that @xmath150 , @xmath151 and @xmath152 , for some @xmath93 , then , as we will see in lemma [ lem : partiiclaim3 ] , since @xmath153 , this observable fulfils the conditions of theorem [ thm : main1 ] [ ii ] and it is neither riemann integrable nor is it bounded . we use the symbol @xmath154 between the elements of two sequences of real numbers @xmath155 and @xmath156 to mean that the sequences are asymptotically equivalent , namely that @xmath157 . we use the landau notation @xmath158 , if @xmath159 . the same notation is used between two real - valued function @xmath41 and @xmath160 , defined on the set of real numbers @xmath161 , positive real numbers @xmath162 , natural numbers @xmath163 or non - negative integers @xmath164 . specifically , if @xmath165 , then we will write @xmath166 , and if @xmath167 , then we will write @xmath168 . the map @xmath169 defined by @xmath170 is called the _ first return map _ and it is well known that @xmath171 is conservative , ergodic and measure preserving on @xmath172 , see for instance ( * ? ? ? * propositions 1.4.8 and 1.5.3 ) . from this point on , we write @xmath28 for both @xmath28 and @xmath173 and @xmath30 for both @xmath30 and @xmath174 . also , throughout , unless otherwise stated , we assume that @xmath16 is @xmath1-expansive . we denote the open unit disk in @xmath175 by @xmath176 , its closure by @xmath177 and its boundary by @xmath178 . given @xmath179 , define @xmath180 ) \to \mathcal{l}_{\mu_{\alpha}}^{1}([0,1])$ ] by @xmath181 it is an easy exercise to show that @xmath182 is a version of the transfer operator of the map @xmath171 . namely , for all @xmath183)$ ] and all measurable functions @xmath48 with @xmath184 finite , we have that @xmath185 we will see in proposition [ prop : conditions_h1_h2 ] that @xmath186 is a banach space , that the operators @xmath187 and @xmath182 map @xmath62 into itself and that the following properties are fulfilled . ( h1 ) : : there exists a constant @xmath188 such that the operator @xmath189 is a bounded linear operator with @xmath190 , for all @xmath21 . ( here , the operator norm @xmath191 is taken with respect to the banach space @xmath186 . ) ( h2 ) : : a result that will be crucial in the proof of theorem [ thm : main1 ] is ( * ? ? ? * theorem 2.1 ) . in order to see how this result reads in our situation , note , for a @xmath1-expansive @xmath0-farey map , that @xmath192 , which is essential in the proof of ( * ? ? ? * theorem 2.1 ) given in @xcite . further , since @xmath193 and @xmath194 , for all @xmath21 , karamata s tauberian theorem for power series ( * ? ? ? * corollary 1.7.3 ) implies that , for @xmath71 , @xmath195 here , @xmath196 . [ thm : mt2011:thm2.1 ] assuming the above setting , in particular that conditions ( h1 ) and ( h2 ) are satisfied , we have that @xmath197 in the sequel , we will also use of the following auxiliary results , where we set @xmath198 and for each @xmath21 and for each word @xmath199 we let @xmath200 \to [ 0 , 1]$ ] denote the function @xmath201 . if @xmath202 is equal to the empty word , then we set @xmath203 to be equal to the identity map . [ lamma : partiiclaim1 ] let @xmath2 \to [ 0 , 1]$ ] denote an arbitrary @xmath0-farey map . for each @xmath120 , we have that @xmath204 where the constants @xmath205 are given recursively by @xmath206 in particular , letting @xmath207 , we have that @xmath208 , for each @xmath120 . we proceed by induction on @xmath209 . the start of the induction is an immediate consequence of . suppose that the statement is true for some @xmath120 . we then have that @xmath210 this completes the proof of . the remaining assertion is proven by a straight forward inductive argument , using the defining relations given in . [ lem : partiiclaim3 ] for each @xmath21 , we have that @xmath211 and hence , by the definition of the norm , @xmath212 . for @xmath213 the result is immediate . for @xmath214 , we have , by lemma [ lamma : partiiclaim1 ] , that , on @xmath215 , @xmath216 to complete the proof , we need to evaluate the function @xmath217 at the point @xmath115 for @xmath218 . by lemma [ lamma : partiiclaim1 ] , we have that @xmath219 this completes the proof . we will now show that conditions ( h1 ) and ( h2 ) are satisfied for every @xmath1-expansive @xmath0-farey system and for the banach space @xmath186 . [ prop : conditions_h1_h2 ] the pair @xmath186 forms a banach space and for a @xmath1-expansive @xmath0-farey system , the operators @xmath187 and @xmath182 map @xmath62 into itself . moreover , ( h1 ) and ( h2 ) are satisfied . in the proof of the above proposition we will make use of the following lemma . [ lemma : c_10 ] for any @xmath0-farey map @xmath16 , we have that @xmath220 , where @xmath221 , for each @xmath21 . by construction of the @xmath0-farey map @xmath16 , we have that @xmath222 $ ] and that @xmath223 $ ] , for all integers @xmath224 . thus , @xmath225 we will now show by induction on @xmath7 that , for each @xmath120 , @xmath226 from , we have that @xmath227 , for each @xmath120 . suppose that the statement in is true for some @xmath21 . from , we have that @xmath228 for each @xmath120 , which gives @xmath229 this completes the proof of the statement in . setting @xmath230 in , we obtain that @xmath231 , for all @xmath21 . combining this with , completes the proof . it is shown in ( * ? ? ? * section 1 ) that the pair @xmath186 forms a banach space . we now prove that condition ( h1 ) holds and the invariance of @xmath62 . for this , let @xmath232 and fix @xmath120 . applying lemmas [ lamma : partiiclaim1 ] and [ lemma : c_10 ] we have that @xmath233 hence , by definition of the partition @xmath60 , we have that @xmath234 , and so , the operator @xmath235 maps @xmath62 into itself . further , by definition of @xmath182 , this gives that @xmath236 and so , the operator @xmath182 maps @xmath62 into itself . linearity of @xmath235 and @xmath182 follows from the linearity of @xmath135 . for the proof of property ( h2)(i ) , observe that @xmath171 is a piecewise linear expansive map with the following properties . 1 . on the set @xmath237 , the absolute value of the derivative of @xmath171 is equal to @xmath238 . moreover , since @xmath14 is a positive monotonically decreasing sequence which is bounded above by @xmath115 , it follows that there exists a constant @xmath239 with @xmath240 , for all @xmath21 . 2 . the partition @xmath60 is a countable - infinite partition of @xmath6 and @xmath241 and @xmath242 if @xmath218 , and hence , @xmath243 , for all @xmath21 . moreover , the @xmath31-algebra generated by @xmath244 is equal to the borel @xmath31-algebra on @xmath63 . 3 . for each @xmath21 and @xmath245 , we have that @xmath246 ) = \overline{\ { \phi = n \ } } \quad \text{and } \quad \frac{\mathrm{d } \mu_{\alpha } \circ f_{\alpha , \psi}}{\mathrm{d } \mu_{\alpha } } = t_{n } - t_{n+1 } = a_{n}. \ ] ] given these properties , ( h2)(i ) is a consequence of ( * ? ? ? * theorem 1.6 ) : the proof of which is based on the _ theorem on the difference of two norms _ by ionescu - tulcea and marinescu @xcite . for the proof of property ( h2)(ii ) , we distinguish between the cases @xmath247 and @xmath248 . _ case 1_. ( @xmath247 ) : : sarig showed in ( * ? ? ? * section 3 ) that @xmath249 where the operators @xmath250 ) \to \mathcal{l}_{\mu_{\alpha}}^{1}([0 , 1])$ ] are defined by @xmath251 by way of contradiction , suppose that @xmath115 is an eigenvalue of @xmath252 restricted to @xmath62 . then there exists a non - zero measurable function @xmath253 such that @xmath254 . substituting this into shows that @xmath48 is equal to zero @xmath28-almost everywhere , which gives a contradiction . _ case 2_. ( @xmath248 ) : : we will now show that @xmath115 is not an eigenvalue of @xmath252 . ( this part of the proof is based on the proof of ( * ? ? ? * lemma 6.7 ) . ) since @xmath248 , there exists a @xmath255 such that @xmath256 . suppose that @xmath257 , for some non - zero @xmath67 . let @xmath258 denote the class of complex - valued measurable functions @xmath41 with domain @xmath63 for which @xmath259 is @xmath28-integrable , and let it be equipped with the standard @xmath260-inner product , @xmath261 for each @xmath262 set @xmath263 . further , set @xmath264 and define @xmath265 by @xmath266 . noting that @xmath267 and using , we have that , for all @xmath268 and @xmath269 , @xmath270 further , @xmath271 and , as @xmath171 preserves the measure @xmath28 restricted to @xmath63 , we have that @xmath272 combining and , it follows that @xmath273 vanishes @xmath28-almost everywhere on @xmath63 . thus , by taking the modulus , the ergodicity of @xmath171 implies that @xmath42 is equal to a constant , everywhere on @xmath63 . as @xmath41 does not vanish @xmath28-almost everywhere , this constant is non - zero , and so , we obtain that @xmath274 almost everywhere on @xmath63 . now , for each @xmath21 , let @xmath275 be the interval of positive measure , such that @xmath276 and let @xmath277 since @xmath278 almost everywhere , and since the map @xmath171 is linear and expanding , we have that @xmath279 . in particular , the set @xmath280 is non - empty . we claim that there exists @xmath281 such that @xmath282 , for each @xmath283 . by way of contradiction , suppose that @xmath284 , for all @xmath285 . since @xmath41 is constant almost everywhere on @xmath63 , we have that @xmath41 is constant almost everywhere on @xmath286 , which gives an immediate contradiction to the assumption that @xmath284 , for all @xmath285 . therefore , we have that there exists @xmath287 , such that @xmath288 , @xmath282 and @xmath289 , for each @xmath283 . hence , we have that @xmath290 , for all @xmath291 , contradicting the initial choice of @xmath292 . finally , in preparation for the proof of theorem [ thm : main1 ] , let us make note of the following well known properties of slowly varying functions . [ lem : powerslowly ] let @xmath293 \to \mathbb{r}$ ] be a positive slowly varying function , for some @xmath294 . 1 . [ sv(i ) ] for a compact interval @xmath295 we have that @xmath296 holds uniformly with respect to @xmath297 , and hence , for a fixed @xmath298 , @xmath299 2 . [ sv(ii ) ] for a fixed @xmath298 we have that @xmath300 3 . [ sv(v ) ] if @xmath301 is continuous , strictly increasing and @xmath302 then , for a fixed @xmath303 , @xmath304 4 . [ sv(vi ) ] if @xmath305 is defined to be the linear interpolation of the function @xmath306 then @xmath307 is a slowly varying function and @xmath308 let us first recall that , for @xmath309 $ ] and @xmath291 , @xmath310 which gives @xmath311 we proceed by induction as follows . the start of the induction is given by the assumption in the theorem . for the inductive step , assume that the statement holds for @xmath312 , for some @xmath313 . then consider some arbitrary @xmath314 , and let @xmath315 denote the unique element in @xmath316 such that @xmath317 . using , the fact that @xmath318 and the inductive hypothesis in tandem with the assumption that @xmath319 , we obtain that @xmath320 where the last equality is a consequence of the eigenequation @xmath321 . using an analogous proof to that given above , one can obtain that the result of theorem [ thm : main2 ] holds for other interval maps , such as gibbs - markov maps , thaler maps and pomeau - manneville maps . throughout this section , we let @xmath37 , \mathscr{b } , \mu_{\alpha } , f_{\alpha})$ ] be a @xmath115-expansive @xmath0-farey system . in order to prove theorem [ thm : main1 ] [ i ] , we will use the following auxiliary results ( lemmas [ lem : convergnecelemma ] and [ lem : jsigman ] ) . before which we require the following notation . define the function @xmath322 by @xmath323,\\ t_{n+1 } ( x -n ) + w_{n } & \text{if } \ ; x \in [ n , n+1 ] , \ ; \text{for } \ ; n \in \mathbb{n}. \end{cases } \ ] ] note that @xmath324 is the linear interpolation of the function @xmath325 defined on @xmath164 , where @xmath326 . further , for @xmath327 , define @xmath328 for all @xmath329 . [ lem : convergnecelemma ] for a given @xmath327 , we have that @xmath330 . for @xmath327 , we have that @xmath331 where the last equality follows from the fact that @xmath324 is a positive , strictly monotonically increasing function and lemma [ lem : powerslowly ] [ sv(v ) ] . here and in the sequel we will use the following notation . for @xmath76 we let @xmath332 denote the largest integer not exceeding @xmath78 . [ lem : jsigman ] let @xmath333 denote a sequence of positive real numbers such that @xmath334 . if the wandering rate is moderately increasing , then @xmath335 without loss of generality , assume that @xmath336 and let @xmath327 be fixed . by definition of @xmath324 , we have , for @xmath337 , that @xmath338 by definition of @xmath339 , we have that @xmath340 . further , by lemma [ lem : powerslowly ] [ sv(vi ) ] and since @xmath341 is regularly varying sequence of order @xmath342 , we have that , @xmath343 if @xmath344 , then this completes the proof . otherwise , note that , by lemma [ lem : powerslowly ] [ sv(ii ) ] , we have that @xmath324 is a slowly varying function . also , since @xmath324 is an unbounded monotonically increasing function we have that @xmath345 and , by lemma [ lem : convergnecelemma ] , we have that @xmath346 . the above three statements in tandem with the assumptions that @xmath334 and that the wandering rate is moderately increasing , yield the following : @xmath347 this completes the proof in the case @xmath348 . by theorem [ thm : mt2011:thm2.1 ] and proposition [ prop : conditions_h1_h2 ] , we have for each @xmath21 that there exists @xmath349 such that @xmath350 and @xmath351 set @xmath352 , for @xmath21 and @xmath353 . by , we have on @xmath63 that @xmath354 since @xmath355)$ ] , it follows that in the final line of the third term converges to zero . to see that the first and the second term in the final line of converge to to zero , observe that 1 . since @xmath98 , we have that @xmath82)$ ] and , moreover , @xmath356 2 . since @xmath98 we have that @xmath357 is finite , and so the sequence @xmath358 is a bounded sequence ; 3 . using lemma [ lem : partiiclaim3 ] together with the fact that @xmath135 is positive and linear and the fact that if @xmath98 , then @xmath357 is finite , we have that @xmath359 ; 4 . given @xmath360 , there exists @xmath361 such that @xmath362 , for all @xmath363 . combining these observations with lemma [ lem : jsigman ] and , we have that the first and the second term in the final line of converge to to zero . since the arguments given above are independent of a given point in @xmath63 , an application of theorem [ thm : main2 ] now finishes the proof . in the proof of theorem [ thm : main1 ] [ i ] we have not used the specific structure of @xmath62 . we only used that @xmath62 is a banach space which satisfies conditions ( h1 ) and ( h2 ) . thus , we may replace @xmath62 by an arbitrary banach space which satisfies conditions ( h1 ) and ( h2 ) . for such alternative banach spaces see remark [ rmk : rmk1 ] . in doing such a substitution one may change the uniform convergence to almost everywhere uniform convergence . to conclude , we give examples of @xmath115-expansive @xmath0-farey systems and of observables which belong to the set @xmath69 and which satisfy the summability condition given in . [ ex : ex1 ] let @xmath37 , \mathscr{b } , \mu_{\alpha } , f_{\alpha})$ ] denote a @xmath115-expansive @xmath0-farey system with moderately increasing wandering rate . set @xmath364 , where @xmath365 , for @xmath366 ) \ ; \text{and } \ ; f \in c^{2}((0,1 ) ) \ ; \text{with } \ ; f ' > 0 \ ; \text{and } \ ; f''\leq 0\}.\ ] ] we claim that @xmath367 and moreover , that @xmath140 satisfies the summability condition given in . we first verify that @xmath367 . for this , we are required to show that @xmath368)$ ] , that @xmath369 and that @xmath370 , for all @xmath371 . by definition , any function belonging to @xmath372 is convex and continuous on @xmath4 , twice differentiable and @xmath28-integrable . thus , @xmath373)$ ] and @xmath369 . combining this with the fact that @xmath374 is @xmath28 integrable , non - negative and bounded , we have that @xmath368)$ ] and @xmath369 . let us now turn to the second assertion , namely that @xmath375 , for all @xmath21 . we immediately have that @xmath376 . for @xmath218 , note that , if @xmath160 is a differentiable lipschitz function on @xmath63 , then @xmath377 . thus , by lemma [ lem : partiiclaim3 ] and the chain rule , we have that , for each integer @xmath218 , @xmath378 since @xmath379 , we have that @xmath380 and that @xmath381 , for all @xmath382 . therefore , since @xmath383 , it follows that there exists @xmath188 , such that @xmath384 combining this with and using the facts that the sequence @xmath385 is summable and that @xmath386 and @xmath387 are finite , the summability condition in follows . hence it follows that @xmath367 . recall that @xmath37 , \mathscr{b } , \mu_{\alpha } , f_{\alpha})$ ] is a @xmath1-expansive @xmath0-farey system and that @xmath389 . from and , we have that , for all @xmath21 , @xmath390 combining this with the assumptions of the theorem , there exists a constant @xmath391 such that @xmath392 , for all @xmath21 . as in the proof of theorem [ thm : main1 ] [ i ] we have , by theorem [ thm : mt2011:thm2.1 ] and proposition [ prop : conditions_h1_h2 ] , that there exists @xmath393 \to \mathbb{r}$ ] such that @xmath394 and , for each @xmath395 , @xmath396 set @xmath397 , for each @xmath21 and @xmath398 . by a calculation similar as in , we have on @xmath63 that @xmath399 as @xmath400)$ ] , the third summand on the rhs of tends to zero . to see that the second term on the rhs of converges to zero , recall that the sequence @xmath401 is positive , monotonically decreasing and bounded above by one and that by assumption @xmath20 . further , by , given @xmath402 , there exist constants @xmath403 and @xmath404 such that , for all @xmath405 with @xmath406 , @xmath407 this implies that given @xmath408 , we have for all @xmath21 such that @xmath409 , @xmath410 ( recall that @xmath411 is the value given in condition ( b ) of theorem [ thm : main1 ] [ ii ] ) . a simple calculation shows that the rhs of the latter inequality converges to zero as @xmath7 tends to infinity . further , since all of the summand are positive , we have that the limit as @xmath7 tends to infinity exists and equals zero . this together with lemma [ lem : powerslowly ] [ sv(i ) ] implies that , for each @xmath412 , @xmath413 furthermore , @xmath414 since @xmath415 are arbitrary , it follows that @xmath416 and hence , the second term on the rhs of converges to zero as @xmath7 tends to infinity . we now show that the first term on the rhs of converges to zero . by and the fact that @xmath394 , given @xmath402 there exists @xmath417 such that , for all @xmath418 , @xmath419 moreover , there exists a constants @xmath420 such that , for all @xmath395 , @xmath421 furthermore , since @xmath16 is @xmath1-expansive , by condition ( b ) in theorem [ thm : main1 ] [ ii ] , we have that the sequence @xmath422 is summable . these properties together with lemma [ lem : partiiclaim3 ] and an argument similar to that presented in , imply the existence of a constant @xmath423 such that @xmath424 since @xmath402 was chosen arbitrarily , the result follows and , since the arguments given above are independent of a given point in @xmath425 , an application of theorem [ thm : main2 ] finishes the proof . in this section we provide a constructive proof of theorem [ thm : main1 ] [ iii ] . the proof is divided into several parts . first , we define a class of observables @xmath427 . second , in proposition [ prop : l1andlinfty ] we will show that if @xmath428 , then @xmath149 is bounded , of bounded variation , riemann integrable and belongs to @xmath39)$ ] . third , in proposition [ prop : balpha ] we will show that if @xmath428 , then it belongs to the space @xmath69 , and in proposition [ prop : summand ] we will show that the summability condition given in is satisfied for all @xmath428 . finally , in proposition [ prop : liminfandlimsup ] we will show that , if @xmath428 , then @xmath429 combing these results will then yield a proof of theorem [ thm : main1 ] [ iii ] let us now begin by defining the set @xmath427 . we let @xmath427 denote the class of observable @xmath430 \to \mathbb{r}$ ] which are of the following form : @xmath431 where @xmath432 and where @xmath433 , @xmath434 and @xmath435 denote three positive constants , depending on @xmath1 , such that [ ex : exc1c2c31 ] for @xmath80 , choose @xmath440 and @xmath441 . then it is clear that @xmath442 and @xmath443 satisfy the conditions ( c1 ) and ( c2 ) . with these choices one immediately verifies that ( c3 ) is equivalent to @xmath444 , for @xmath445 . hence , by choosing @xmath446 sufficiently small , it follows that the conditions ( c1 ) , ( c2 ) and ( c3 ) can be fulfilled simultaneously . the main reason why we require the sequence @xmath451 is to ensure that @xmath149 is of bounded variation . further , condition ( c3 ) is only required in the proof of the second statement of proposition [ prop : liminfandlimsup ] , specifically when lemma [ lemma : keylemma ] is used . by ( c1 ) and ( c2 ) , we have that @xmath460 and @xmath461 . this implies that , for all @xmath120 , @xmath462 using ( c1 ) and ( c2 ) once more immediately verifies that the latter term is strictly greater than one . clearly the observable @xmath149 is riemann integrable . moreover , @xmath149 is measurable , as each of the atoms of @xmath0 is measurable and @xmath149 is the sum of indicator functions of atoms of @xmath0 . further , the range of @xmath149 is equal to @xmath463 , and thus , @xmath464 . by lemma [ lem : nknk-1 ] , we have that @xmath459 , and so the variation of @xmath149 is equal to @xmath465 , which is finite , as @xmath466 and as @xmath435 is positive . this shows that @xmath149 is of bounded variation . it remains to show that @xmath149 is @xmath28-integrable . for this recall that @xmath467 , for each @xmath120 . choose a positive constant @xmath468 and recall that @xmath193 . by lemma [ lem : powerslowly ] [ sv(ii ) ] , there exists a constant @xmath188 such that @xmath469 , for each @xmath21 . therefore , by lemma [ lemmamvt ] and lemma [ lem : nknk-1 ] , we have that @xmath470 the latter series converges , since @xmath468 , @xmath471 and @xmath472 , for all @xmath120 . by proposition [ prop : l1andlinfty ] , we have that @xmath54)$ ] and that @xmath473 . moreover , by lemma [ lem : partiiclaim3 ] , we have on @xmath32 $ ] , that , for each @xmath371 , @xmath474 therefore , @xmath475 , for all @xmath371 , and hence , it follows that @xmath98 . 1 . using the facts that @xmath426 and @xmath360 , that the sequence @xmath479 is not bounded above and is strictly monotonically increasing , that @xmath480 and that @xmath481 is a fixed natural number , we have that @xmath482 2 . for each @xmath120 , we have that @xmath483 using condition ( c3 ) with the facts that @xmath426 , @xmath360 and that @xmath484 and @xmath435 are positive , it follows that @xmath485 3 . there exist constants @xmath486 such that , for all @xmath120 sufficiently large , @xmath487 here , the first inequality follows from the facts that @xmath488 is a slowly varying function and that @xmath489 together with lemma [ lem : powerslowly ] [ sv(i ) ] . the second inequality follows from lemma [ lem : powerslowly ] [ sv(ii ) ] , which guarantees the existence of the constant @xmath490 such that @xmath491 , for all @xmath21 . by theorem [ thm : mt2011:thm2.1 ] and proposition [ prop : conditions_h1_h2 ] , we have uniformly on @xmath63 that @xmath493 thus , given @xmath402 , there exists @xmath404 such that , for all @xmath406 on @xmath63 , @xmath494 we will first show the second statement in . for this , observe that by it is sufficient to show that , on @xmath6 , @xmath495 in order to see this , let @xmath496 be fixed and let @xmath497 denote the smallest integer for which @xmath498 . since @xmath135 is a positive linear operator , we have , for all @xmath499 , that @xmath500 now , lemma [ lem : powerslowly ] [ sv(i ) ] implies that @xmath501 . as the sequence @xmath502 is positive and since @xmath503 the value @xmath504 is finite and strictly greater than zero . hence , by , and and the fact that @xmath193 , we have on @xmath63 that , for each @xmath120 sufficiently large , @xmath505 by lemma [ lemma : keylemma ] , the latter term diverges . all that remains to show is that the first statement of holds . for this , observe that , by positivity and linearity of @xmath506 , theorem [ thm : mt2011:thm2.1 ] , proposition [ prop : conditions_h1_h2 ] and , we have on @xmath63 that , for each @xmath120 , @xmath507 since @xmath120 was arbitrary , the above inequalities imply that on @xmath63 , @xmath508 suppose that the latter inequality is strict , namely , suppose that there exists a constant @xmath188 such that on @xmath63 , @xmath509 this assumption together with implies that , given @xmath402 , there exists @xmath417 such that , for all @xmath510 and @xmath99 , @xmath511 thus , by karamata s tauberian theorem for power series ( * ? ? ? * corollary 1.7.3 ) , it follows that , for all @xmath510 and @xmath99 , @xmath512 hence , @xmath513 this is a contradiction , since by and by combining theorem [ thm : mt2011:thm2.1 ] with karamata s tauberian theorem for power series ( * ? ? ? * corollary 1.7.3 ) , we have that the set @xmath63 is a set and therefore , by ( * ? ? ? * proposition 3.7.5 ) , the @xmath0-farey system is pointwise dual ergodic , meaning that , for @xmath28-almost every @xmath514 $ ] , we have that @xmath515

we study the asymptotics of iterates of the transfer operator for non - uniformly hyperbolic @xmath0-farey maps . we provide a family of observables which are riemann integrable , locally constant and of bounded variation , and for which the iterates of the transfer operator , when applied to one of these observables , is not asymptotic to a constant times the wandering rate on the first element of the partition @xmath0 . subsequently , sufficient conditions on observables are given under which this expected asymptotic holds . in particular , we obtain an extension theorem which establishes that , if the asymptotic behaviour of iterates of the transfer operator is known on the first element of the partition @xmath0 , then the same asymptotic holds on any compact set bounded away from the indifferent fixed point .

without doubt , most particles can be regarded as composite particles , such as molecules composed of atoms , atoms composed of electrons and nuclei , nuclei composed of nucleons , so on , it is important to recognize that physics must be invariant for the composite particles and their constituent particles , this requirement is called particle invariance in this paper . but difficulties arise immediately because for fermion we use dirac equation , for meson we use klein - gordon equation and for classical particle we use newtonian mechanics , while the connections between these equations are quite indirect . thus if the particle invariance is held in physics , i.e. , only one physical formalism exists for any particle , we can expect to find out the differences between these equations by employing the particle invariance . using this approach is one of the goals of this paper , consequently , several new relationships between them are found , the most important result is that the obstacles that cluttered the path from classical mechanics to quantum mechanics are found , it becomes possible to derive the quantum wave equations from relativistic mechanics after the obstacles are removed . another goal is just to discuss interactions between particles under the particle invariance , several new formulae of interactions are derived and discussed . the new results provide an insight into improving quark model . fermions satisfy fermi - dirac statistics , bosons satisfy bose - einstein statistics , there is a connection between the spin of a particle and the statistics . it is clear that the spin is a key concept for particle physics . in this section we shall show that the spin of a particle is one of the consequences of the particle invariance . according to newtonian mechanics , in a hydrogen atom , the single electron revolves in an orbit about the nucleus , its motion can be described by its position in an inertial cartesian coordinate system @xmath0 . as the time elapses , the electron draws a spiral path ( or orbit ) , as shown in fig.[dfig1](a ) in imagination . if the reference frame @xmath1 rotates through an angle about the @xmath2-axis in fig.[dfig1](a ) , becomes a new reference frame @xmath3 ( there will be a lorentz transformation linking the frames @xmath1 and @xmath3 ) , then in the frame @xmath3 , the spiral path of the electron tilts with respect to the @xmath4-axis with the angle as shown in fig.[dfig1](b ) . at one instant of time , for example , @xmath5 instant , the spiral path pierces many points at the plane @xmath5 , for example , the points labeled @xmath6 , @xmath7 and @xmath8 in fig.[dfig1](b ) , these points indicate that the electron can appear at many points at the time @xmath9 , in agreement with the concept of probability in quantum mechanics . this situation gives us a hint to approach quantum wave nature from relativistic mechanics . because the electron pierces the plane @xmath5 with 4-vector velocity @xmath10 , at every pierced point we can label a local 4-vector velocity . the pierced points may be numerous if the path winds up itself into a cell about the nucleus ( due to a nonlinear effect in a sense ) , then the 4-vector velocities at the pierced points form a 4-vector velocity field . it is noted that the observation plane selected for the piercing can be taken at an arbitrary orientation , so the 4-vector velocity field may be expressed in general as @xmath11 , i.e. the velocity @xmath10 is a function of 4-vector position . at every point in the reference frame @xmath3 the electron satisfies relativistic newton s second law of motion : @xmath12 the notations consist with the convention@xcite . since the cartesian coordinate system is a frame of reference whose axes are orthogonal to one another , there is no distinction between covariant and contravariant components , only subscripts need be used . here and below , summation over twice repeated indices is implied in all case , greek indices will take on the values 1,2,3,4 , and regarding the rest mass @xmath13 as a constant . as mentioned above , the 4-vector velocity @xmath10 can be regarded as a multi - variable function , then @xmath14 @xmath15 substituting them back into eq.([1 ] ) , and re - arranging these terms , we obtain @xmath16 using the notation @xmath17 eq.([4 ] ) is given by @xmath18 because @xmath19 contains the variables @xmath20 , @xmath21 , @xmath22 and @xmath23 which are independent from @xmath24 , then a main solution satisfying eq.([6 ] ) is given by @xmath25 ( in this paper we do not discuss the special solutions that @xmath26 , if they exist ) . according to green s formula or stokes s theorem , the above equation allows us to introduce a potential function @xmath27 in mathematics , further set @xmath28 , we obtain a very important equation @xmath29 where @xmath30 representing wave nature may be a complex mathematical function , its physical meanings is determined from experiments after the introduction of the planck s constant @xmath31 . the magnitude formula of 4-vector velocity of particle is given in its square form by @xmath32 which is valid at every point in the 4-vector velocity field . multiplying the two sides of the above equation by @xmath33 and using eq.([8 ] ) , we obtain @xmath34 \nonumber \\ & = & ( -i\hbar \partial _ \mu -qa_\mu ) ( -i\hbar \partial _ \mu -qa_\mu ) \psi \nonumber \\ & & -[-i\hbar \psi \partial _ \mu ( mu_\mu ) ] \label{10}\end{aligned}\ ] ] according to the continuity condition for the electron motion @xmath35 we have @xmath36 it is known as the klein - gordon equation . on the condition of non - relativity , schrodinger equation can be derived from the klein - gordon equation @xcite(p.469 ) . however , we must admit that we are careless when we use the continuity condition eq.([11 ] ) , because , from eq.([8 ] ) we obtain @xmath37 where we have used lorentz gauge condition . thus from eq.([9 ] ) to eq.([10 ] ) we obtain @xmath38 this is of a complete wave equation for describing the motion of the electron accurately . the klein - gordon equation is a linear wave equation so that the principle of superposition is valid , however with the addition of the last term of eq.([14 ] ) , eq.([14 ] ) turns to display chaos . in the following we shall show dirac equation from eq.([8 ] ) and eq.([9 ] ) . from eq.([8 ] ) , the wave function can be given in integral form by @xmath39 where @xmath40 is an integral constant , @xmath41 and @xmath42 are the initial and final points of the integral with an arbitrary integral path . since maxwell s equations are gauge invariant , eq.([8 ] ) should preserve invariant form under a gauge transformation specified by @xmath43 where @xmath44 is an arbitrary function . then eq.([15 ] ) under the gauge transformation is given by @xmath45 the situation in which a wave function can be changed in a certain way without leading to any observable effects is precisely what is entailed by a symmetry or invariant principle in quantum mechanics . here we emphasize that the invariance of velocity field is held for the gauge transformation . suppose there is a family of wave functions @xmath46 which correspond to the same velocity field denoted by @xmath47 , they are distinguishable from their different phase angles @xmath40 as in eq.([15 ] ) . then eq.([9 ] ) can be given by @xmath48 suppose there are four matrices @xmath49 which satisfy @xmath50 then eq.([18 ] ) can be rewritten as @xmath51[a_{\mu jk}p_\mu \psi ^{(k)}-i\delta _ { jk}mc\psi ^{(k ) } ] \nonumber \\ & & \label{20}\end{aligned}\ ] ] where @xmath52 is kronecker delta function , @xmath53 . for the above equation there is a special solution given by @xmath54\psi ^{(k)}=0 \label{21}\ ] ] there are many solutions for @xmath49 which satisfy eq.([19 ] ) , we select a set of @xmath49 as @xmath55 where @xmath56 and @xmath57 are the matrices defined in dirac algebra@xcite(p.557 ) . substituting them into eq.([21 ] ) , we obtain @xmath58\psi = 0 \label{23}\ ] ] where @xmath30 is an one - column matrix about @xmath59 . then eq.([23 ] ) is just the dirac equation . the dirac equation is a linear wave equation , the principle of superposition is valid for it . let index @xmath60 denote velocity field , then @xmath61 , whose four component functions correspond to the same velocity field @xmath60 , may be regarded as the eigenfunction of the velocity field @xmath60 ( it may be different from the eigenfunction of energy ) . because the velocity field is an observable in a physical system , in quantum mechanics we know , @xmath62 constitute a complete basis in which arbitrary function @xmath63 can be expanded in terms of them @xmath64 obviously , @xmath63 satisfies eq.([23 ] ) . then eq.([23 ] ) is just the dirac equation suitable for composite wave function . alternatively , another method to show the dirac equation is more traditional : at first , we show the dirac equation of free particle by employing plane waves , we easily obtain eq.([23 ] ) on the condition of @xmath65 ; next , adding electromagnetic field , the plane waves are still valid in any finite small volume with the momentum of eq.([8 ] ) when we regard the field to be uniform in the volume , so the dirac equation eq.([23 ] ) is valid in the volume even if @xmath66 , the plane waves constitute a complete basis in the volume ; third , the finite small volume can be chosen to locate at anywhere , then anywhere have the same complete basis , therefore the dirac equation eq.([23 ] ) is valid at anywhere . of course , on the condition of non - relativity , schrodinger equation can be derived from the dirac equation @xcite(p.479 ) . by further calculation , the dirac equation can arrive at klein - gordon equation with an additional term which represents the effect of spin , this term is just the last term of eq.([14 ] ) approximately . but , do not forget that the dirac equation is a special solution of eq.([20 ] ) , therefore we believe there are some quantum effects beyond the dirac equation . with this consequence , it is easy to understand why some problems of quantum electrodynamics can not been completely explained by the dirac equation . eq.([20 ] ) originates from the magnitude formula of 4-vector velocity of particle , the formula is suitable for any particle , so it satisfies the particle invariance . the dirac equation is regarded as an approximation to eq.([20 ] ) , the approximation brings out many troubles with the spin concept . from the dirac equation we can predict that a composite particle and an its constituent both have their own spins , but this prediction is not true for mesons because pion has zero spin while its constituent quark has 1/2 spin , in other words , due to the approximation the dirac equation does not involve some states such as zero spin state . that is why we want to classify particles into fermions and mesons by spin and use different equations . if we can find a precise solution of eq.([20 ] ) instead of the dirac equation , then the classification is not necessary . it is noted that eq.([20 ] ) is nonlinear while the dirac equation is linear , this reminders us that we can never find any precise solutions in a linear equation which satisfy eq.([20 ] ) . therefore , for this problem , a good solution depends on how much precision we can reach for our requirement . in one hand , it is rather remarkable that klein - gordon equation and dirac equation can be derived from relativistic newton s second law of motion approximately , in another hand , all particles , such as fermions , bosons and classical particles , satisfy the relativistic newton s second law ( it will be further clear later ) , thus it is a natural choice that only the relativistic newton s second law is independent and necessary . only one formalism is necessary for any particle , this is just the particle invariance , we arrive at the aim . as mentioned above , the spin is one feature hidden in the relativistic newton s second law , but more features will turn out from the relativistic newton s second law in the following sections . in this section we discuss how to determine the planck s constant that emerges in the preceding section . in 1900 , m. planck assumed that the energy of a harmornic oscilator can take on only discrete values which are integral multiples of @xmath67 , where @xmath68 is the vibration frequency and @xmath69 is a fundamental constant , now either @xmath69 or @xmath70 is called as planck s constant . the planck s constant next made its appearance in 1905 , when einstein used it to explain the photoelectric effect , he assumed that the energy in an electromagnetic wave of frequency @xmath71 is in the form of discrete quanta ( photons ) each of which has an energy @xmath72 in accordance with planck s assumption . from then , it has been recognized that the planck s constant plays a key role in quantum mechanics . according to the previous section , no mater how to move or when to move in minkowski s space , the motion of a particle is governed by a potential function @xmath27 as @xmath73 for applying eq.([p1 ] ) to specific applications , without loss of generality , we set @xmath74 , then eq.([p1 ] ) is rewritten as @xmath75 the coefficient @xmath76 is subject to the interpretation of @xmath30 . there are three mathematical properties of @xmath30 worth recording here . first , if there is a path @xmath77 joining initial point @xmath41 to final point @xmath78 , then @xmath79 second , the integral of eq.([p3 ] ) is independent from the choice of path . third , the superposition principle is valid for @xmath80 , i.e. , if there are @xmath81 paths from @xmath41 to @xmath42 , then @xmath82 @xmath83 @xmath84 where @xmath85 is the average momentum . to gain further insight into physical meanings of this equations , we shall discuss two applications . as shown in fig.[pfig1 ] , suppose that the electron gun emits a burst of electrons at @xmath41 at time @xmath86 , the electrons arrive at the point @xmath42 on the screen at time @xmath87 . there are two paths for the electron to go to the destination , according to our above statement , @xmath30 is given by @xmath88 where we use @xmath89 and @xmath90 to denote the paths @xmath91 and @xmath92 respectively . multiplying eq.([p7 ] ) by its complex conjugate gives @xmath93 \nonumber \\ & = & 2 + 2\cos [ \frac p\kappa ( l_1-l_2 ) ] \label{p8}\end{aligned}\ ] ] where @xmath94 is the momentum of the electron . we find a typical interference pattern with constructive interference when @xmath95 is an integral multiple of @xmath96 , and destructive interference when it is a half integral multiple . this kind of experiments has been done since a long time age , no mater what kind of particle , the comparison of the experiments to eq.([p8 ] ) leads to two consequences : ( 1 ) the complex function @xmath30 is found to be probability amplitude , i.e. , @xmath97 expresses the probability of finding a particle at location @xmath42 in the minkowski s space . ( 2 ) @xmath76 is the planck s constant . the integral of time component in the above calculation has been autimatically canceled because the experimental pattern is stable . let us consider the modification of the two slit experiment , as shown in fig.[pfig2 ] . between the two slits there is located a tiny solenoid s , designed so that a magnetic field perpendicular to the plane of the figure can be produced in its interior . no magnetic field is allowed outside the solenoid , and the walls of the solenoid are such that no electron can penetrate to the interior . like eq.([p7 ] ) , the amplitude @xmath30 is given by @xmath98 and the probability is given by @xmath99 \nonumber \\ & = & 2 + 2\cos [ \frac p\kappa ( l_1-l_2)+\frac 1\kappa \oint_{(l_1+\overline{l_2}% ) } qa_\mu dx_\mu ] \nonumber \\ & = & 2 + 2\cos [ \frac p\kappa ( l_1-l_2)+\frac{q\phi } \kappa ] \label{p10}\end{aligned}\ ] ] where @xmath100 denotes the inverse path to the path @xmath90 , @xmath101 is the magnetic flux that passes through the surface between the paths @xmath89 and @xmath100 , and it is just the flux inside the solenoid . now , constructive ( or destructive ) interference occurs when @xmath102 where @xmath103 is an integer . when @xmath76 takes the value of the planck s constant , we know that this effect is just the aharonov - bohm effect which was shown experimentally in 1960 . in this section we shall correct a mistake about coulomb s force and gravitational force in physical education , which cluttered the path from classical mechanics to quantum mechanics . we also shall discuss maxwell s equations in detail . in the world , almost every young person was educated to know that the coulomb s force and gravitational force act along the line joining a couple of particles , but this knowledge is incorrect in the theory of relativity . in relativity theory , the 4-vector velocity @xmath10 of a particle has components @xmath104 , the magnitude of the 4-vector velocity @xmath10 is given by @xmath105 the above equation is valid so that any force can never change @xmath10 in the magnitude but can change @xmath10 in the direction . we therefore conclude that the coulomb s force and gravitational force on a particle always act in the direction orthogonal to the 4-vector velocity of the particle in the 4-dimensional space - time , rather than along the line joining a couple of particles . alternatively , any 4-vector force @xmath106 satisfy the following perpendicular or orthogonal relation @xmath107 this simple inference clearly tells us that the forces are not centripetal or centrifugal forces about their sources , even if in 3-dimensional space [ see eq.([e7 ] ) ] , this character provides a internal reason for accounting for the quantum behavior of particle or chaos . thus the derivations in terms of 4-vector velocity field in the preceding section become reasonable . in the present paper , eq.([e02 ] ) has been elevated to an essential requirement for definition of force , which brings out many new aspects for coulomb s force and gravitational force . we assume that coulomb s law remains valid only for two particles both at rest in usual 3-dimensional space . suppose there are two charged particle @xmath108 and @xmath109 locating at positions @xmath42 and @xmath110 in a cartesian coordinate system @xmath1 and moving at 4-vector velocities @xmath10 and @xmath111 respectively , as shown in fig.[afig1 ] , where we use @xmath112 to denote @xmath113 . the coulomb s force @xmath106 acting on particle @xmath108 is perpendicular ( orthogonal ) to the velocity direction of @xmath108 , as illustrated in fig.[afig1 ] , like a centripetal force , the force @xmath106 should make an attempt to rotate itself about its path center , the center may locate at the front or back of the particle @xmath109 , so the force @xmath106 should lie in the plane of @xmath111 and @xmath112 , then @xmath114 where @xmath115 and @xmath116 are unknown coefficients , the possibility of this expansion will be further clear in the next subsection in where the expansion is not an assumption [ see eq.([g3 ] ) ] . using the relation @xmath117 , we get @xmath118 we rewrite eq.([e1 ] ) as @xmath119 \label{e3}\ ] ] it follows from the direction of eq.([e3 ] ) that the unit vector of the coulomb s force direction is given by @xmath120 \label{e4}\ ] ] because @xmath121 \nonumber \\ & = & \frac 1{c^2r}[(u\cdot r)u^{\prime } -(u\cdot u^{\prime } ) r ] \nonumber \\ & = & -[(\widehat{u}\cdot \widehat{r})\widehat{u}^{\prime } -(\widehat{u}\cdot \widehat{u}^{\prime } ) \widehat{r } ] \nonumber \\ & = & -\widehat{u}^{\prime } \cosh \alpha + \widehat{r}\sinh \alpha \label{e5a}\end{aligned}\ ] ] @xmath122 where @xmath123 refers to the angle between @xmath10 and @xmath124 , @xmath125 , @xmath126 , @xmath127 , @xmath128 , @xmath129 . suppose that the magnitude of the force @xmath106 has classical form @xmath130 combination of eq.([e6 ] ) with ( [ e4 ] ) , we obtain a modified coulomb s force @xmath131 \nonumber \\ & = & \frac{kqq^{\prime } } { c^2r^3}[(u\cdot r)u^{\prime } -(u\cdot u^{\prime } ) r ] \label{e7}\end{aligned}\ ] ] this force is in the form of lorentz force for the two particles , relating with the ampere s law and biot - savart - laplace law . is perpendicular to the 4-vector velocity @xmath10 of @xmath108 , and lies in the plane of @xmath111 and @xmath112 with the retardation with respect to @xmath109 . ] it follows from eq.([e7 ] ) that the force can be rewritten in terms of 4-vector components as @xmath132 where we have used the relations @xmath133 from eq.([e8c ] ) , because of @xmath134 , we have @xmath135 it is known as the lorentz gauge condition . to note that @xmath124 has three degrees of freedom on the condition @xmath136 , so we have @xmath137 @xmath138 from eq.([e8b ] ) , we have @xmath139 where we define @xmath140 . from eq.([e8b ] ) , by exchanging the indices and taking the summation of them , we have @xmath141 the eq.([e14 ] ) and ( [ e15 ] ) are known as the maxwell s equations . for continuous media , they are valid as well . from the maxwell s equations , we know there is a retardation time for action to propagate between the two particles , the retardation effect is measured by @xmath142 as illustrated in fig.[afig1 ] . then @xmath143 obviously , eq.([e18 ] ) is known as the lienard - wiechert potential for a moving particle . the above formalism clearly shows that maxwell s equations can be derived from the classical coulomb s force and the perpendicular ( orthogonal ) relation of force and velocity . in other words , the perpendicular relation is hidden in maxwell s equation . specially , eq.([e3 ] ) directly accounts for the geometrical meanings of curl of vector potential , the curl contains the perpendicular relation . since the perpendicular relation of force and velocity is one of the consequences from relativistic newton s second law , it is also one of the features from the particle invariance . the above formalism has a significance on guiding how to develope the theory of gravity . in analogy with the modified coulomb s force of eq.([e7 ] ) , we directly suggest a modified universal gravitational force as @xmath144 \nonumber \\ & = & -\frac{gmm^{\prime } } { c^2r^3}[(u\cdot r)u^{\prime } -(u\cdot u^{\prime } ) r ] \label{g1}\end{aligned}\ ] ] for a couple of particles with masses @xmath13 and @xmath145 respectively . comparing with some incorrect statements about coulomb s force and gravitational force in most textbooks , and for emphasizing its feature , the perpendicular ( orthogonal ) relation of force and velocity was called the direction adaptation nature of force in the author s previous paper@xcite . we emphasize that the perpendicular relation of force and velocity must be valid if gravitational force can be defined as a force . it follows from eq.([g1 ] ) that we can predict that there are gravitational radiation and magnet - like components for the gravitational force . particularly , the magnet - like components will act as a key role in the geophysics and atmosphere physics . if we have not any knowledge but know there exists the classical universal gravitation @xmath146 between two particles @xmath13 and @xmath145 , what form will take the 4-vector gravitational force @xmath106 ? suppose that @xmath147 is at rest at the origin , using @xmath148 , @xmath149 and @xmath150 , we have @xmath151 @xmath152 \nonumber \\ & = & \frac 1{icu_4}[(u^{\prime } \cdot u)\mathbf{f}-(\mathbf{u}\cdot \mathbf{f}% ) u^{\prime } ] \nonumber \\ & = & \frac{|\mathbf{f}|}{icu_4|\mathbf{r}|}[(u^{\prime } \cdot u)\mathbf{r}-(% \mathbf{u}\cdot \mathbf{r})u^{\prime } ] \nonumber \\ & = & \frac{|\mathbf{f}|}{icu_4|\mathbf{r}|}[(u^{\prime } \cdot u)r-(u\cdot r)u^{\prime } ] \label{g3}\end{aligned}\ ] ] where @xmath125 , @xmath153 . if we rotate our frame of reference to make @xmath145 not to be at rest , eq.([g3 ] ) will still be valid because of covariance . then we find the 4-vector gravitational force goes back to the form of eq.([g1 ] ) , like lorentz force , having the magnet - like components . it is noted that the perpendicular relation of force and velocity is valid for any force : strong , electromagnetic , weak and gravitational interactions , therefore there are many new aspects remaining for physics to explore . under the invariance of particle , the most simple model of particle is that all particles are composed of identical constituents , the constituent is regarded as the most elementary and most small particle in the world . since quarks have never been observed , our speculation leads us to propose a better model to organize known data . for this challenging purpose , in the present paper , we introduce a fictitious elementary particle , given a name dollon for our convenience , to assemble other particles such as fermions , mesons or classical particles , the dollon is regarded as the most elementary and most small particle in the world . our work focuses on conceptual development . consider a dollon moving in minkowski s space @xmath154 with 4-vector velocity @xmath148 , the motion of the dollon satisfies the magnitude formula of 4-vector velocity of particle @xmath155 differentiating the above equation with respect to the proper time interval @xmath156 of the dollon gives @xmath157 where the result has been written in the two parts by defining a 3-dimensional vector @xmath146 . defining a 4-vector @xmath158 then from eq.([r1 ] ) we have readily @xmath159 it means that @xmath10 and @xmath106 are orthogonal with each other . consider two particles bob and alice located at @xmath42 and @xmath110 in the 4-dimensional space respectively , they are composed of many dollons , the number of dollons in alics is @xmath160 , and in bob is @xmath13 , when bob and alice move with 4-vector velocities @xmath10 and @xmath111 respectively , following eq.([r2 ] ) , they can be assigned two sets of motion equations as @xmath161 @xmath162 now we have a question : what is the interaction between bob and alice ? obviously , the form of eq.([r5 ] ) seems to be relativistic newton second law for bob , @xmath146 seems to be a 3-vector force , @xmath163 seems to be the rate at which the force does work on bob . for seeking for further answers , we need to recall the newton s first law of motion , the law is valid in theory of relativity and reads _ first law : an object at rest will remain at rest and an object in motion will continue to move in a straight line at constant speed forever unless some net external force acts to change this motion . _ if the object is a composite system composed of many dollons , then we can understand the first law with three consequences . consequence 1 : let @xmath1 denote the number of dollons in a composite system , the average velocity of the system is defined as @xmath164 where @xmath165 is the 4-vector velocity of the ith dollon . the average velocity represents the motion of the center of the system . the fist law only means that the center of the system remains at rest or in motion , i.e. , rotation about its center is permited . consequence 2 : the total number of dollons in the system must be unchanged , i.e. , the conservation of dollon number must be held , otherwise any creation or annihilation of dollon will lead to a sudden shift of the center of the system . consequence 3 : when two bodies are seperated from an infinite distance , the interaction between them must vanish . otherwise , no body can be at rest , bacause a rest body will always be affected by the motion of a far distance body , whereas the far distance bodies are innumerable as a background . now we go back to consider the whole system composed of bob and alics , without loss of generality , suppose that the center is at rest at the origin of the frame of reference , then the center has a 4-vector velocity @xmath166 , the at rest refers to being at rest in usual 3-dimensional space . from eq.([r4 ] ) , the quantity @xmath106 must be orthogonal with the 4-vector velocity @xmath10 of bob , likewise for alice , we have @xmath167 they set up a rule for the interaction between bob and alice in the composite system . we specially choose to study the interaction which happens at such instant that the position vector @xmath112 of bob with respect with alice ( i.e. , @xmath168 ) is orthogonal to @xmath10 and @xmath111 simultaneously . @xmath169 the existence of such instant of time will become clear in the subsection [ x ] . from eq.([r8 ] ) and eq.([r9 ] ) , we get parallel relations @xmath170 for bob , using notation @xmath171 , @xmath172 , vector - multiplying eq.([r5 ] ) by @xmath173 , because @xmath146 parallels @xmath173 , we have @xmath174 it means @xmath175 where @xmath176 is an integral constant . likewise for alice . from eq.([r10 ] ) we can expand @xmath177 in a taylor series in @xmath178 , this gives @xmath179 from eq.([r5 ] ) we obtain @xmath180 where @xmath181 is an integral constant . now consider eq.([r13 ] ) , it means that bob moves around alice ( no matter by attractive or repulsive interaction ) , when @xmath182 , bob may access alice as close as possible at perihelion point , at the perihelion point we find @xmath183|_{perihelion } \nonumber\\ \label{r16}\end{aligned}\ ] ] since @xmath184 are the coefficients that are independent from distance @xmath185 , integral constant @xmath69 and integral constant @xmath181 , they take the same values for various cases which have various man - controlled parameters @xmath186 and @xmath181 . now we consider two extreme cases . _ first case : bob is at rest forever . _ according to the newton s first law of motion , the interaction between them must completely vanishes . since bob speed @xmath187 should not depend on the distance , according to eq.([r15 ] ) , a reasonable solution may be @xmath188 , @xmath189 , @xmath190 and @xmath191 . to note that the values of @xmath192 and @xmath193 do not depend on this extreme case . _ second case : with _ @xmath182 , _ bob passes the perihelion point about alice with a distance _ @xmath194__. _ _ according to eq.([r16 ] ) , a reasonable solution may be that the all coefficients @xmath184 are zero but except @xmath195 . therefore we obtain @xmath196 where the subscript of @xmath195 has been dropped . likewise for alice , we have @xmath197 where @xmath6 and @xmath198 are coefficients . differentiating eq.([r7 ] ) with respect to time interval @xmath199 gives the center acceleration as @xmath200 where we have used @xmath201 , @xmath202 . substituting eq.([r17 ] ) and eq.([r19 ] ) into eq.([r21 ] ) , we get @xmath203 this equation leads to @xmath204 where @xmath205 is a constant . then eq.([r5 ] ) and eq.([r6 ] ) may be rewritten as @xmath206 @xmath207 if @xmath205 takes a negative constant , then , the above equations show that bob is attracted by alice with newton s universal gravitation force . _ but we do not want to make this conclusion at once , because there are still a few problems among them . _ in this subsection , we study coulomb s force by based on our the most simple model : all particles are composed of identical constituents dollons . from the above subsection , now we can manifestly interpret the quantity @xmath106 as the 4-vector force exerting on a dollon of bob . it is a natural idea to think of that dollon has two kinds of charges : positive and negative . if bob and alice are separated by a far distance , and @xmath106 is the force acting on a positive dollon in bob , then @xmath208 is the force acting on a negative dollon in bob . regardless of the internal forces in bob , it follows from eq.([r2 ] ) that the motion of the ith dollon is governed by @xmath209 where @xmath210 , @xmath165 and @xmath211 denote the proper time interval , 4-vector velocity and 4-vector force acting on the ith dollon , respectively . taking sum over all dollons in bob , we get @xmath212=\frac{d(mu_c)}{dt } \label{c2 } \\ \sum\limits_{i=1}^m\frac{icf^{(i)}}{u_4^{(i ) } } & \simeq & q\frac{icf_c}{u_{c4 } } \label{c3}\end{aligned}\ ] ] where @xmath213 is the 4-vector velocity of the center of bob , @xmath214 denotes its 4th component , @xmath108 denotes the net charges of bob , @xmath215 denotes the 4-vector force acting on the dollon located at the center of bob ( this dollon may be virtual one because it features the average action ) . combining eq.([c2 ] ) and eq.([c3 ] ) with eq.([c1 ] ) , we obtain @xmath216 where we neglect the approximation in eq.([c3 ] ) . like that in the above subsection , the first law must be valid for the composite system of bob and alice , in other words , when they are separated from a infinite distance they are isolated , whereas they go to nearest points they should not touch each other , these requirements lead to @xmath217 where @xmath218 @xmath7 and @xmath6 are coefficients . without loss of generality , we have @xmath219 substituting eq.([c5 ] ) and eq.([c6 ] ) into eq.([c7 ] ) , we get @xmath220 where @xmath109 denotes the net charges of alice . this equation leads to @xmath221 where @xmath222 is a constant . then the motions of bob and alice are governed by @xmath223 @xmath224 the 4th component equations corresponding to the above equations express the energy change rates of bob and alice , they are not independent components . the eq.([c10 ] ) and eq.([c11 ] ) are known as the coulomb s forces . if bob and alice are two atoms with neutral net charges , the coulomb s force between them will vanish off . but , precisely , this is not true , the inspection of eq.([c3 ] ) tells us that the net interaction between them still remains when the atoms are considered as composite systems . in this paper , planet , stone , molecule , atom and nucleus are all regarded as composite systems composed of dollons , the constituents of the composite systems move about their centers . if bob and alice are two planets with neutral net charges , then it is reasonable _ to assume that the net force acting on bob is proportional to the number of dollons in bob _ , eq.([c3 ] ) reads @xmath225 where @xmath226 is a very very small proportional coefficient . then the motion of bob is given by @xmath227 in analogy with the above subsections , we may obtain the motion equations of bob and alice , they are governed by @xmath228 @xmath229 where @xmath230 is a constant proportional to @xmath226 . the @xmath13 and @xmath160 has been identified or defined as the masses by employing dollon mass as a unit when we count the dollon numbers in bob or alice . the eq.([c14 ] ) and eq.([c15 ] ) are known as the newton s universal gravitational forces . why is the net force of bob attractive ? this may be explained as that electrons with light masses move always around massive nuclei , the attraction is a little bigger than the repulsion between two atoms separated by a far distance . in this formulation , the gravitational force possesses statistic meanings . we use the most simple model all particles are composed of identical dollons to study nuclear force , to fulfill the conceptual development boosted by the newton s first law of motion . now consider that bob and alice are two nucleons composed of dollons . if bob and alice go closely in a distance comparable with the sizes of them , then it is clear that eq.([c3 ] ) turns to be inadequate , their polarization can provide a strong interaction , while the effect of net charges between their centers becomes to be trivial . the strong interaction is regarded as the nuclear force in this paper . therefore , the strong nuclear force is charge - independent , it only comes into play when the nucleons are very close together , and it drops rapidly to coulomb s force for far distance , we know from experiments that the sensitive distance is about @xmath231 . as mentioned above , the ith dollon in bob is governed by @xmath232 then the motion of bob is given by @xmath233 ^ 2 } \label{c18}\end{aligned}\ ] ] where @xmath234 where @xmath213 can be understood as the velocity of momentum center ( see eq.([c19 ] ) , but @xmath213 is not the _ relativistic _ velocity of the geometrical center of bob , the _ relativistic _ velocity of the geometrical center of bob is defined by using its geometrical center proper time , i.e. , @xmath235 , thus we have to establish their relation by introducing a correcting factor @xmath236 so that @xmath237 , i.e. , @xmath238 in the following we drop the subscript center when without confusion , then above equations can be rewritten as @xmath239 ^ 2 } \label{c22}\end{aligned}\ ] ] to note that the right side of eq.([c22 ] ) is the rate at which the forces do works on bob , then the quantity @xmath240 in the left side should be energy , thus we can define the energy as @xmath241 where @xmath242 , @xmath187 is the classical speed of the geometrical center of bob , @xmath243 is the relativistic mass , while @xmath13 is the rest mass . eq.([c23 ] ) is known as the energy mass relationship . but eq.([c23 ] ) has a little difference from einstein s mass - energy relationship . our energy formula contains a factor @xmath236 that represents the internal motion of dollons in bob , obviously , @xmath244 , this can be seen clearly from eq.([c20 ] ) , in other words , even if the center is at rest , the internal constituents can still have relativistic energies . in dealing with nuclear reaction , in many textbooks , mass defect is understood as the decrease in total relativistic mass , even if all nuclei seem to be at rest before or after the nuclear reaction the total relativistic masses should not have apparent change . we have been puzzled by these statements for a long time . now the reasons are clear , no relativistic masses change but @xmath236 changes in these cases , in other words , the internal energy of particle has changed . @xmath236 is a physical quantity sensitive to the internal structure of a particle , is a criteria for particle being elementary or not . consider that a hadron possesses net charge @xmath108 , we can naturally image that the charge distributes in several parts inside the hadron , assuming three parts , the three parts have net charges denoted by @xmath245 , @xmath246 , and @xmath247 respectively , then @xmath248 comparing with the gell - mann - nishijima relation @xmath249 we can understand the conservations of isospin @xmath250 , baryon number @xmath116 and strangeness number @xmath1 with four remarks : ( 1 ) the three parts inside the hadron are insulated from one another , no charge transports from one to another . ( 2 ) during collision of hadrons , only identical parts impact or touch each other , with exchanging net charges . ( 3 ) the mass of the hadron seems to depend primarily on the masses of the parts inside the hadron , weakly on the net charges of the parts . ( 4 ) if we assign the quantum states of quarks @xmath10 , @xmath60 and @xmath251 to the three parts , the quark model seems to be improved in a manner that we can avoid the fractional charges of the quarks . in the preceding subsections , we have mentioned that the interaction between bob and alice we studied happens at such instant that their relative position in the minkowski s space is denoted by a 4-vector @xmath252 , @xmath112 satisfies the orthogonal relation simultaneously @xmath253 the purpose of choosing this instant is to meet the convenience that @xmath112 parallels to @xmath106 and @xmath254 simultaneously , because @xmath255 see eq.([r8])-([r11 ] ) . eq.([c30 ] ) can be rewritten in the form of inner product of two vectors as @xmath256 this leads to two solutions given by @xmath257 eq.([c33 ] ) again leads to two solutions given by @xmath258 the first solution expresses that the force acting on bob is retarded by time @xmath259 , the second one expresses that the action is preceded . our choice is the first one which gives an effect that follows the cause . we know , this retarded time is just the time needed for the propagation of interaction from alice to bob , the propagation speed is @xmath8 , no mater what kind of interaction . eq.([c35 ] ) represents the orthogonal relationship . therefore , the interaction happens at such instant that either in retarded state or in orthogonal state , or mixture . in preceding sections , we have realized that relativistic newton s second law and forces can be derived from newton s first law and the magnitude formula of 4-vector velocity of particle . the formula is given by @xmath260 in a minkowski s space . it is noted that all particles satisfy the above equation , it then is regarded as the origin of the particle invariance . we wonder at what is the essence of the minkowski s space . in this section we shall discuss the minkowski s space , for this purpose we need to establish a standard method for describing the motion of particle in space - time . our construction follows four steps . suppose alice is a pretty girl being famous for her fast running records , we state some her records here in a story ( in imagination ) . \(1 ) jan . , 1 , 2001 , 10:00 am , sportsground in buaa , beijing . in a time interval @xmath261 alice ran a straight line distance @xmath262 at a constant speed @xmath263 . this data can be given in physical terms by @xmath264 it can be rewritten either as @xmath265 or as @xmath266 where @xmath42 and @xmath267 denote the coordinate system fixed at the sportsground . by defining a imaginary quantity @xmath268 , the data is given by @xmath269 we appreciate the simplicity and beauty of its form . it is also our favorite manner to mark the running process in a graph with three mutually perpendicular axes @xmath270 and @xmath271 . the distance from the starting point to the final point in this coordinate system equals to zero because of eq.([m4 ] ) . this graph we called as 20010101 graph . \(2 ) jan . , 2 , 2001 , 10:00 am , sportsground in buaa , beijing . in a time interval @xmath261 alice ran a straight line distance @xmath272 at a constant speed @xmath273 . this data is given in physical terms by @xmath274 we directly mark this day running process in the yesterday s 20010101 graph , we are lazy to draw a new graph . \(3 ) jan . , 3 , 2001 , 10:00 am , sportsground at buaa , beijing . in a time interval @xmath261 alice ran a straight line distance @xmath275 at a constant speed @xmath276 . we also directly mark the running process in the 20010101 graph . bob was also a good runner , in a time interval @xmath261 bob ran a straight line distance @xmath277 at a constant speed @xmath278 . we also directly mark the running process in the 20010101 graph . in fact , their running records all are marked in the 20010101 graph . because of laziness , we only use the 20010101 graph to record the running data , it has actually become a temporary standard frame , all motions can be marked or calculated in the graph , it is much connivent for describing any movement . to note that @xmath271 , the @xmath279 axis in the graph has involved the speed @xmath280 created by alice on jan 1 , 2001 . thus we find that the geometrical distance @xmath281 from the starting point to the final point in the graph on the second day for alice is not equal to zero . @xmath282 it is clear after comparing with eq.([m5 ] ) . so do for bob , the distance @xmath283 for bob in the 20010101 graph is given by @xmath284 @xmath285 dividing the two sides of the above equation by @xmath286 , we get @xmath287 defining modified velocity @xmath288 where @xmath289 , we have dropped the subscript @xmath7 that indicates bob , then eq.([m9 ] ) is given by @xmath290 the modified velocity of bob in the 20010101graph is based on the alice s best speed record @xmath291 . in fact , any one , any body or any particle , their modified velocity in the 20010101 graph satisfies eq.([m13 ] ) . because of convenience , it has become a habit for us to use the 20010101 graph to mark the all motions of any body . it is sure that not all scientists in the world like alice , then we gradually recognize that we need a permanent runner for establishing a standard graph . now we had to face a new task : to look for a new hero . it was said that the light , an element of the nature , is the fastest runner , whenever and whereever its speed is @xmath292 . we do not hesitate to use the light speed to replace alice s speed , and setup a new frame called standard graph , the standard graph contains four mutually perpendicular axes @xmath42 , @xmath267 , @xmath293 and @xmath294 ( we can draw several partial frames to assemble the whole frame ) . from then , any motions can be described in the standard graph with the space - time @xmath295 or @xmath296 . in analogy with eq.([m10])-([m13 ] ) , defining modified velocity @xmath297 where @xmath298 , we obtain @xmath299 the standard graph is just the minkowski s space , the 4-vector velocity @xmath300 is known as the relativistic velocity . we immediately recognize that physics holds its validity only in the standard graph ( involving with the light speed ) , rather than in the 20010101 graph ( involving with the alice s speed ) , this situation can be explained by the fact that all physical quantities and their measurements are defined on facilities whose principles are based on the light directly or indirectly , for example , the meter and second are defined on the light speed directly . if we do not hope that one graph has advantage over than another , then the transformation between the standard graph and 20010101 graph will be given by @xmath301 where the subscript 1 denotes in the 20010101 graph . it means we need to redefine all physical quantities such as rod and clock in the 20010101 graph , do not use the light . a complete inner product space is called a hilbert space . our experience in the preceding sections tells us that it is an easy thing to put dynamics into the hilbert space if we have an invariant quantity . the formalism of the interaction can be derived from some basic laws , it is strongly based on concrete instances . in the section [ basicf ] , the newton s first law of motion means that the 4-vector average velocity of an isolated system remains at rest or in motion . this explanation is based on the definition of average velocity given by @xmath302 where @xmath1 denotes the number of dollons in the system as in eq.([r7 ] ) . the newton s first law of motion becomes a sort of strong constraint , inevitably leads to action reaction law or momentum conversation law being valid inside the system , for example , for a rest system composed of two dollons alice and bob we have @xmath303 the above equation means that the action and reaction are equal in magnitude and reverse in directions on the line joining the two particle . but we immediately wonder at that alice and bob have to adjust their proper times @xmath304 and @xmath305 from time to time to meet the requirement of eq.([d2 ] ) . that is why we say the first law is a strong constraint for the system . obviously , the geometrical center of a system is defined by @xmath306 the relativistic 4-vector velocity of the geometrical center of the system is given by @xmath307 as mentioned in the section [ nuclf ] , @xmath308 . immediately , we find the newton s first law of motion can be newly explained by based on the relativistic 4-vector velocity of the geometrical center of the system , i.e. , the newton s first law of motion means that the 4-vector velocity of the geometrical center of an isolated system remains at rest or in motion . this new explanation implies that the action reaction law for the relativistic 4-vector forces inside the system are not held [ comparing to eq.([d2 ] ) ] , but the following expansions for alice and bob become possible . @xmath309 where @xmath115 , @xmath116 , @xmath310 and @xmath311 are unknown coefficients . all conclusions we obtained in the preceding sections can be retained or modified by retracing the route of the paper , in accordance with the section [ direction ] . the new explanation seems to be much reasonable , but it is worth further studying the action reaction law and momentum conversation law which are confronting with serious troubles , they need special treatment like that for ampere s force in electromagnetism . another topic we would like to discuss briefly is su(n ) group . each infinitesimal transformation of the su(n ) group can be written in the form @xmath312 as usual , repeated indices must be summed over . where the real parameter @xmath313 are treated as small quantities , @xmath314 and @xmath315 are matrices which satisfy the definition of the group @xmath316 we recall from eq.([20 ] ) that dirac equation was derived from the following equation @xmath317[a_{\mu jk}p_\mu \psi ^{(k)}-i\delta _ { jk}mc\psi ^{(k)}]=0 \label{d9}\ ] ] it is much impressive that eq.([d8 ] ) and eq.([d9 ] ) have a similar form , especially when we let a matrix @xmath318 to absorb the right side of eq.([d8 ] ) , i.e. @xmath319 from this comparison we may understand why the su(n ) group could embed in quantum mechanics in a obscure way . this situation arouses our interest to measure a new group whose matrices satisfy @xmath320 we believe this new group has even more direct relations with quantum mechanics . it is important to recognize that physics must be invariant for composite particles and their constituent particles , only one physical formalism exists for any particle , this requirement is called particle invariance . under the particle invariance , it is rather remarkable to find that klein - gordon equation and dirac equation can be derived from the relativistic newton s second law of motion on different conditions respectively , thus only one formalism is necessary for particle , the relativistic newton s second law is regarded as one which suitable for any kinds of particles . we point out that the coulomb s force and gravitational force on a particle always act in the direction orthogonal to the 4-vector velocity of the particle in 4-dimensional space - time , rather than along the line joining a couple of particles . this inference is obviously supported from the fact that the magnitude of the 4-vector velocity is kept constant . maxwell s equations can be derived from classical coulomb s force and the magnitude formula of 4-vector velocity of particle . our speculation on the quarks model leads to introduce a new elementary particle called dollon to assemble particles such as baryons , mesons and other composite particles . instead of quark model , the dollon model is better in organizing known data , specially in modelling interactions . it is found that relativistic newton s second law and various interactions can be derived from the newton s first law of motion and the magnitude formula of 4-vector velocity of particle . the structure of minkowski s space is discussed in detail , it indicates that the magnitude formula of 4-vector velocity of particle is only a geometrical distance formula ( pythagoras s theorem ) , so that it is completely free from any particle property . any dynamics or dynamical characteristics originated from the magnitude formula of 4-vector velocity of particle will completely preserve the particle invariance , i.e. , the dynamics do not distinguish particle species . thus the magnitude formula of 4-vector velocity of particle is regarded as the origin of the particle invariance .

since the particles such as molecules , atoms and nuclei are composite particles , it is important to recognize that physics must be invariant for the composite particles and their constituent particles , this requirement is called particle invariance in this paper . but difficulties arise immediately because for fermion we use dirac equation , for meson we use klein - gordon equation and for classical particle we use newtonian mechanics , while the connections between these equations are quite indirect . thus if the particle invariance is held in physics , i.e. , only one physical formalism exists for any particle , we can expect to find out the differences between these equations by employing the particle invariance . as the results , several new relationships between them are found , the most important result is that the obstacles that cluttered the path from classical mechanics to quantum mechanics are found , it becomes possible to derive the quantum wave equations from relativistic mechanics after the obstacles are removed . an improved model is proposed to gain a better understanding on elementary particle interactions . this approach offers enormous advantages , not only for giving the first physically reasonable interpretation of quantum mechanics , but also for improving quark model . + + pacs numbers : 11.30.ly , 12.90.+b , 03.65.ta . + +

a large number of physical systems , in particular in plasma physics , are described in terms of a variable evolving in a deterministic velocity field under the effect of a random perturbation . this is described by a stochastic differential equation of the form @xmath0 where @xmath1 is the velocity field and the perturbation is the white noise @xmath2 this general form has been invoked in several applications . more recently , in a series of works devoted to the explanation of the intermittent behavior of the statistical characteristics of the turbulence in magnetically confined plasma it has been developed a formalism based on barrier crossing . previous works that have discussed subcritical excitation of plasma instabilities are refs . @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . we offer a comparative presentation of two functional integral approaches to the determination of the statistical properties of the system s variable @xmath3 for the case where the space dependence of @xmath4 is characterized by the presence of three equilibrium points , @xmath5 . we will take @xmath6 in studying the stochastic processes the functional methods can be very useful and obtain systematic results otherwise less accessible to alternative methods . the method has been developed initially in quantum theory and it is now a basic instrument in condensed matter , field theory , statistical physics , etc . in general it is based on the formulation of the problem in terms of an action functional . there are two distinct advantages from this formulation : ( 1 ) the system s behavior appears to be determined by all classes of trajectories that extremize the action and their contributions are summed after appropriate weights are applied ; ( 2 ) the method naturally includes the contributions from states close to the extrema , so that fluctuations can be accounted for . there are technical limitations to the applicability of this method . in the statistical problems ( including barrier - type problems ) it is simpler to treat cases with white noise , while colored noise can be treated perturbatively . in the latter case , the procedure is however useful since the diagrammatic series can be formulated systematically . the colored noise can be treated by extending the space of variables : the stochastic variable with finite correlation is generated by integration of a new , white noise variable . the one dimensional version can be developed up to final explicit result . since however the barrier type problem is frequently formulated in two - dimensions , one has to look for extrema of the action and ennumerate all possible trajectories . it is however known that , in these cases , the behavior of the system is dominated by a particular path , `` the optimum escape path '' , and a reasonnable approximation is to reduce the problem to a one - dimensional one along this system s trajectory . we will briefly mention the steps of constructiong the msr action functional , in the jensen path integral reformulation . we begin by choosing a particular realisation of the noise @xmath7 . all the functions and derivatives can be discretised on a lattice of points in the time interval @xmath8 $ ] ( actually one can take the limits to be @xmath9 ) . the solution of the equation ( [ eqcanoi ] ) is a `` configuration '' of the field @xmath10 which can be seen as a point in a space of functions . we extend the space of configurations @xmath3 to this space of functions , including all possible forms of @xmath10 , not necessarly solutions . in this space the solution itself will be individualised by a functional dirac @xmath11 function . @xmath12 any functional of the system s real configuration ( _ i.e. _ solutions of the equations ) can be formally expressed by taking as argument an arbitrary functional variable , multiplying by this @xmath11 functional and integrating over the space of all functions . we will skip the discretization and the fourier representation of the @xmath13 functions , followed by reverting to the continuous functions . the result is the following functional @xmath14 \emph{d}\left [ k\left ( t\right ) \right ] \exp \left [ i\int_{-t}^{t}dt\left ( -k\overset{\cdot } { x}+kv\left ( x\right ) + \sqrt{2d}k\xi \right ) \right]\ ] ] the label @xmath15 means that the functional is still defined by a choice of a particular realization of the noise . the generating functional is obtained by averaging over @xmath15 . @xmath16 \emph{d}\left [ k\left ( t\right ) \right ] \exp \left [ i\int dt\left ( -k\overset{\cdot } { x}+kv\left ( x\right ) + idk^{2}\right ) % \right ] \label{zfaraj}\ ] ] we add a formal interaction with two currents @xmath17 \emph{d}\left [ k\left ( t\right ) \right ] \exp \left [ i\int dt\left ( -k\overset{\cdot } { x}% + kv\left ( x\right ) + idk^{2}+j_{1}x+j_{2}k\right ) \right ] \label{zj}\ ] ] in view of future use to the determination of correlations . this functional integral must be determined explicitely . the standard way to proceed to the calculation of @xmath18 is to find the saddle point in the function space and then expand the action around this point to include the fluctuating trajectories . this requires first to solve the euler - lagrange equations @xmath19 the simplest case should be examined first . we assume there is no deterministic velocity ( @xmath20 ) in order to see how the purely diffusive behavior is obtained in this framework @xmath21 the equations can be trivially integrated @xcite , @xcite @xmath22 where @xmath23 \\ \delta _ { 12}\left ( t , t^{\prime } \right ) & = & \theta \left ( t - t^{\prime } \right ) \\ \delta _ { 21}\left ( t , t^{\prime } \right ) & = & \theta \left ( t^{\prime } -t\right ) \\ \delta _ { 22}\left ( t , t^{\prime } \right ) & = & 0\end{aligned}\ ] ] with the symmetry @xmath24 the lowest approximation to the functional integral @xmath18 is obtained form this saddle point solution , by calculating the action along this system s trajectory . we insert this solutions in the expression of the generating functional , for @xmath25 @xmath26 \emph{d}\left [ k\left ( t\right ) \right ] \exp \left [ i\int_{-t}^{t}dt\left ( -k\overset{\cdot } { x}+idk^{2}+j_{1}x+j_{2}k\right ) % \right ] \right| _ { x^{\left ( 0\right ) } , k^{\left ( 0\right ) } } \\ & = & \exp \left [ \frac{1}{2}i\int_{-t}^{t}dt\int_{-t}^{t}dt^{\prime } j_{i}\left ( t\right ) \delta _ { ij}\left ( t , t^{\prime } \right ) j_{j}\left ( t\right ) \right]\end{aligned}\ ] ] the dispersion of the stochastic variable @xmath10 can be obtained by a double functional derivative followed by taking @xmath27 . we obtain @xmath28 \right| _ { j_{1,2}=0 } \\ & = & d\min \left ( t , t^{\prime } \right)\end{aligned}\ ] ] which is the diffusion . the same mechanism will be used in the following , with the difference that the equations can not be solved in explicit form due to the nonlinearity . in general the nonlinearity can be treated by perturbation expansion , if the amplitude can be considered small . this is an analoguous procedure as that used in the field theory and leads to a series of terms represented by feynman diagrams . we can separate in the lagrangian the part that can be explicitely integrated and make a perturbative treatment for the non - quadratic term . this is possible when we assume a particular ( polynomial ) form of the deterministic velocity , @xmath1 . obviously , this term is @xmath29 in eq.([zj ] ) . the functional integral can be written , taking account of this separation @xmath30 \emph{z}_{j}^{\left ( q\right ) } \label{zjnl}\ ] ] where the remaining part in the lagrangian is _ quadratic _ @xmath31 \emph{d}\left [ k\left ( t\right ) \right ] \exp \left [ i\int dt\left ( -k% \overset{\cdot } { x}+akx+idk^{2}+j_{1}x+j_{2}k\right ) \right ] \label{zq}\ ] ] the euler - lagrange equations are @xmath32 the solutions can be expressed as follows @xmath33 with @xmath34 \theta \left ( t^{\prime } -t\right ) + \left [ \exp \left ( -2at^{\prime } \right ) -1\right ] \theta \left ( t - t^{\prime } \right ) \right\ } \notag \\ & & \times \exp \left ( at^{\prime } \right ) \notag \\ \delta _ { 21}\left ( t , t^{\prime } \right ) & = & \exp \left ( -at\right ) \theta \left ( t^{\prime } -t\right ) \exp \left ( at^{\prime } \right ) \notag \\ \delta _ { 12}\left ( t , t^{\prime } \right ) & = & \exp \left ( at\right ) \theta \left ( t - t^{\prime } \right ) \exp \left ( -at^{\prime } \right ) \notag \\ \delta _ { 22}\left ( t , t^{\prime } \right ) & = & 0 \notag\end{aligned}\ ] ] the form of the generating functional derived from the quadratic part is @xmath35 the occurence of the first term in the exponent is the price to pay for not making the expansion around @xmath36 . however , such an expansion would have produced two non - quadratic terms in the lagrangian density : @xmath37 and @xmath38 . this would render the perturbative expansion extremly complicated since we would have to introduce two vertices : one , of order four , is that shown in eq.([zjnl ] ) and another , of order three , related to the first of the nonlinearities mentioned above . even in the present case , the calculation appears very tedious . we have to expand the vertex part of @xmath18 as an exponential , in series of powers of the vertex operator . in the same time we have to expand the exponential in eq.([zjq ] ) as a formal series . then we have to apply term by term the first series on the second series . the individual terms can be represented by diagrams . in this particular case we have a finite contribution even at the zero - loop order ( the `` tree '' graph ) . it is however much more difficult to extract the statistics since we will need at least the diagrams leaving two free ends with currents @xmath39 . in the case we examine here , the perturbative treatment is not particularly useful since the form of the potential ( from which the velocity field is obtained ) supports topologically distinct classes of saddle point solutions and this can not be represented by a series expansion . to make comparison with other approaches , we take @xmath40 and integrate over the functional variable @xmath41 . @xmath42 \emph{d}\left [ k\left ( t\right ) \right ] \exp \left\ { \int_{0}^{t}dt\left [ -dk^{2}+i\left ( -k% \overset{\cdot } { x}+kv\right ) \right ] \right\ } \\ & = & \int \emph{d}\left [ x\left ( t\right ) \right ] \emph{d}\left [ k\left ( t\right ) \right ] \exp \left\ { -\int_{0}^{t}dt\left [ \sqrt{d}k+\frac{i% \overset{\cdot } { x}-iv}{2\sqrt{d}}\right ] ^{2}+\int_{0}^{t}dt\frac{\left ( i% \overset{\cdot } { x}-iv\right ) ^{2}}{4d}\right\}\end{aligned}\ ] ] @xmath43 \exp \left\ { -% \frac{1}{4d}\int_{0}^{t}dt\left ( \overset{\cdot } { x}-v\right ) ^{2}\right\}\ ] ] in other notations @xmath43 \exp \left [ -% \frac{s}{d}\right]\ ] ] where @xmath44 this is eq.(25 ) of the reference lehmann , riemann and hnggi , pre62(2000)6282 . in this reference it is called the onsager - machlup action functional and the analysis is based on this formula . however , we can go further and we will find inconsistencies . we now take account of the fact that the velocity is derived form a potential @xmath45 & = & -\frac{du\left [ x\left ( t\right ) \right ] } { dx } \\ & \equiv & -u^{\prime } \left [ x\left ( t\right ) \right]\end{aligned}\ ] ] @xmath46 \\ & = & \frac{1}{2}\left [ u\left ( t\right ) -u\left ( 0\right ) \right ] + \int_{0}^{t}dt\left [ \frac{1}{4}\left ( \overset{\cdot } { x}^{2}+u^{\prime 2}\right ) \right]\end{aligned}\ ] ] this leads to the form of the generating functional @xmath47 } { \exp \left [ -\frac{u\left ( 0\right ) } { 2d}\right ] } k\left ( x , t;x_{i},t_{i}\right)\ ] ] with @xmath48 \exp \left ( -\frac{s}{d}\right)\ ] ] @xmath49 \label{sfara}\ ] ] these are almost identical to the formulas ( 2 - 5 ) of the reference @xcite ( except that @xmath50 ) . also , it is quite close of the eqs.(7a-7c ) of the ref.@xcite . * however there is an important difference*. there is a term missing in eq.([sfara ] ) which however is present in the two above references . the full form of the action @xmath51 , instead of eq.([sfara ] ) is @xmath52 @xmath53 this term comes from the _ jacobian _ that is hidden in the functional @xmath13 integration . in our approach the most natural way of proceeding with a stochastic differential equation is to use the msr type reasonning in the jensen reformulation . the equation is discretized in space and time and selectd with @xmath11 functions in an ensemble of functions ( actually in sets of arbitrary numbers at every point of discretization ) . the result is a functional integral . there is however a particular aspect that needs careful analysis , as mentioned in the previous subsection . it is the problem of the jacobian associated with the @xmath11 functions . this problem is discussed in ref.@xcite . the equation they analyse is presented in most general form as @xmath54 + \theta _ { j}\ ] ] where the number of stochastic equations is @xmath55 , @xmath56 is functional of the fields , @xmath57 is the streaming term which obeys a current - conserving type relation @xmath58 \exp \left\ { -h% \left [ \phi \right ] \right\ } = 0\ ] ] the noise is @xmath59 . the following generating functional can be written @xmath60 \exp \int dt% \left [ l_{j}\phi _ { j}\left ( t\right ) \right ] \prod_{j , t}\delta \left ( \frac{% \partial \phi _ { j}\left ( t\right ) } { \partial t}+k_{j}\left [ \phi \left ( t\right ) \right ] -\theta _ { j}\right ) j\left [ \phi \right]\ ] ] the functions @xmath61 are currents , @xmath62 \equiv -\left ( \gamma _ { 0}\right ) _ { jk}\frac{\delta h}{\delta \phi _ { k}\left ( t\right ) } + v_{j}\left [ \phi \left ( t\right ) \right]\ ] ] and @xmath63 $ ] is the jacobian associated to the dirac @xmath11 functions in each point of discretization . the jacobian can be written @xmath64 } { \delta \phi _ { k}}\right ) \delta \left ( t - t^{\prime } \right ) \right]\ ] ] up to a multiplicative constant @xmath65 \right)\ ] ] or @xmath66 \right)\ ] ] since the operator @xmath67 is retarded , only the lowest order term survives after taking the trace @xmath68 } { \delta \phi _ { j}\left ( t\right ) } \right]\ ] ] the factor @xmath69 comes from value of the @xmath70 function at zero . in the treatment which preserves the dual function @xmath71 associated to @xmath72 in the functional , there is a part of the action @xmath73\ ] ] then a @xmath71 and a @xmath72 of the same coupling term from @xmath74 $ ] close onto a loop.since @xmath75 is retarded , all these contributions vanish except the one with a single propagator line . this cancels exactly , in all orders , the part coming from the jacobian . then it is used to ignore all such loops and together with the jacobian . we can now see that in our notation this is precisely the term needed in the expression of the action . @xmath76 & \rightarrow & u^{\prime } \left [ x\left ( t\right ) \right ] \\ \frac{\delta k_{j}\left [ \phi \left ( t\right ) \right ] } { \delta \phi _ { j}\left ( t\right ) } & \rightarrow & -u^{\prime \prime } \left [ x\left ( t\right ) \right]\end{aligned}\ ] ] and the action ( [ sfara ] ) is completed with the new term @xmath77 now the generating functional is @xmath47 } { \exp \left [ -\frac{u\left ( 0\right ) } { 2d}\right ] } k\left ( x , t;x_{i},t_{i}\right)\ ] ] with @xmath48 \exp \left ( -\frac{s}{d}\right)\ ] ] and @xmath78 @xmath79 now the two expressions are identical with those in the references cited . this will be the starting point of our analysis . in conclusion we have compared the two starting points in a functional approach : the one that uses _ dual functions _ @xmath10 and @xmath80 , closer in spirit to msr ; and the approach based on onsager - machlup functional , traditionally employed for the determination of the probabilities @xcite , @xcite . either we keep @xmath81 and ignore the jacobian ( the first approach ) or integrate over @xmath80 and include the jacobian . the approaches are equivalent and , as we will show below , lead to the same results . a final observation concerning the choice of one or another method : in the msr method , the trajectories include the diffusion from the direct solution of the euler - lagrange equations . in the onsager - machlup method the paths extremizing the action are deterministic and the diffusion is introduced by integrating on a neighborhood in the space of function , around the deterministic motion . the equations for the saddle point trajectory are in complex so we extend also the variable in complex space @xmath82 the obtain a system of four nonlinear ordinary differential equations which can be integrated numerically . a typical form of the solution @xmath10 is similar to the _ kink instanton _ ( _ i.e. _ the @xmath83 function ) . the function @xmath84 spends very much time in the region close to the equilibrium point ; then it performs a fast transition to the neighbour equilibrium point , where it remains for the rest of the time interval . ( unstable ) equilibrium point in the effective potential @xmath85,width=377 ] ( unstable ) equilibrium point with a position close to the right ( stable ) equilibrium point to in the effective potential @xmath85,width=377 ] the action functional eqs.([news ] ) and ( [ w ] ) leads to the following differential equation ( which replaces eqs.([eqel ] ) ) @xmath86 multiplying by @xmath87 and integrating we have @xmath88 ^{1/2 } \label{eqxpoint}\ ] ] we are interested in the functions @xmath10 that has the following physical property : they stay for very long time stuck to the equilibrium points and perform a fast jump between them at a certain moment of time . then we can take @xmath89 . the solution can be obtained form the integration @xmath90 the upper limit @xmath91 will be specified later . for the next calculation it will be taken as the smallest of the roots of the polynomial under the square root . the details of the calculations in terms of elliptic functions can be found in _ byrd and friedman _ @xcite . the roots of the forth degree polynomial will be noted @xmath92 where @xmath93 such as to have @xmath94 ; then we will use @xmath95 . the notations are @xmath96 the following substitutions are required @xmath97 a new variable is introduced identifying the lower limit of the integral , @xmath98 @xmath99 the integral can be written @xmath100 this integral can be expressed in terms of _ elliptic _ functions . we take @xmath101@xmath102\end{aligned}\ ] ] here the notations are @xmath103 @xmath104\end{aligned}\ ] ] @xmath105\end{aligned}\ ] ] and @xmath106 the symbol @xmath107 represents the legendre s incomplete elliptic integral of the third kind and @xmath108 is the amplitude of @xmath109 . the symbols @xmath110 , @xmath111 , @xmath112 represent the jacobi elliptic functions . there are several well known examples of instantons . they appear in physical systems whose lowest energy state is degenerate and the minima of the action functional ( or the energy , for stationary solutions ) are separated by energy barriers . instantons connect these minima by performing transitions which are only possible in imaginary time ( the theory is expressed in euclidean space , with uniform positive metric ) . it is only by including these instantons that the action functional is correctly calculated and real physical quantities can be determined . from this calculation we can obtain the explicit trajectories that extremize the action functional and in the same time reproduce the jump of the system between the two distant equilibrium positions . these trajectories will be necessary in the calculation of the functional integral . however , since we have eliminated the external currents and integrated over the dual functional variable @xmath113 we can not derive the statistical properties of @xmath10 from a generating functional . in the approach based on the onsager - machlup action the instanton is not used in its explicit form ( elliptic functions ) in the calculation of the action . the reason is that the result can be proved to depend essentially on local properties of the potential @xmath1 . this will be shown later . in the approach with dual functions , one can reduce the instanton to its simplest form , an instantaneous transition between two states , a jump appearing at an arbitrary moment of time . using this form as a first approximation we will calculate the solutions of the euler - lagrange equations and then the action . using the onsager - machlup action we have @xmath114 k\left ( x , t;x_{i},t_{i}\right)\ ] ] the new function @xmath115 has the expression @xmath116 \exp \left ( -\frac{1}{d}\int_{t_{i}}^{t}d\tau \left [ \frac{% \overset{\cdot } { x}^{2}}{4}+w\left ( x\right ) \right ] \right ) \label{kcar}\ ] ] the integrand at the exponent can be considered as the lagrangean density for a particle of mass @xmath69 moving in a potential given by @xmath117 in a semiclassical treatment ( similar to the quantum problem , where @xmath118 is the _ small _ diffusion coefficient @xmath119 of the present problem ) , the most important contribution comes from the neighborhood of the classical trajectories , @xmath120 that extremalizes the action @xmath51 . the `` classical '' equation of motion is @xmath121 to take into account the trajectories in a functional neighborhood around @xmath122 we expand the action to second order introducing the new variables @xmath123 this gives @xmath124 \\ & & \times \int_{y\left ( t_{i}\right ) = 0}^{y\left ( t\right ) = 0}\emph{d}\left [ y\left ( \tau \right ) \right ] \exp \left\ { -\frac{1}{d}\int_{t_{i}}^{t}d\tau % \left [ \frac{1}{4}\overset{\cdot } { y}^{2}+\frac{1}{2}y^{2}w^{\prime \prime } \left ( x_{c}\left ( \tau \right ) \right ) \right ] \right\}\end{aligned}\ ] ] the deviation of the action from that obtained at the extremum @xmath125 , can be rewritten @xmath126 y\left ( \tau \right)\ ] ] the functional integration can be done since it is gaussian and the result is @xmath127 ^{1/2}}\ ] ] in order to calculate the determinant , one needs to solve the eigenvalue problem for this operator @xmath128 y_{n}\left ( \tau \right ) = \lambda _ { n}y_{n}\left ( \tau \right)\ ] ] withe the eigenfunctions verifying the conditions @xmath129 @xmath130 the formal result for @xmath115 is ( also van vleck ) @xmath131\ ] ] where @xmath55 is a constant that will be calculated by normalizing @xmath132 . another way to calculate @xmath55 is to fit this result to the known _ harmonic oscillator _ problem . it has been shown ( coleman ) that the factor arising from the determinant can be written in the form @xmath133 ^{1/2}}\ ] ] where the function @xmath134 is the solution of @xmath135 with the boundary conditions @xmath136 in the case where there are degenerate minima in @xmath137 the particle can travel from one minimum to another . these solutions are called instantons . consider for example the potential with two degenerate maxima of @xmath138 at @xmath139 and with a minimum at @xmath140 . we want to calculate the probability @xmath141 . the classical solution connecting the point @xmath142 to the point @xmath143 is a _ the energy of this solution is exponentially small @xmath144\end{aligned}\ ] ] this solution spends quasi - infinite time in both harmonic regions around @xmath145 where it has very small velocity ; and travels very fast , in a short time @xmath146 between these points ( this is the time - width of the instanton ) . the special effect of the translational symmetry in time is seen in the presence of the parameter representing the center of the instanton . it can be any moment of time between @xmath147 and @xmath148 . this case must be treated separately and we note that this corresponds to the lowest eigenvalue in the spectrum , since the range of variation of the coefficient in the expansion of any solution in terms of eigenfunction is the inverse of the eigenvalue @xmath149 the widest interval , for the variation of the center of the instanton , must be associated with the smallest eigenvalue and this and its eigenfunction must be known explicitely . instead of a precise knowledge of the lowest eigenvalue and its corresponding eigenfunction we will use an approximation , exploiting the fact the function @xmath150 is very close of what we need . we start by noting that @xmath150 is a solution of the eigenvalue problem for the operator of second order functional expansion around the instanton . this eigenfunction corresponds to the eigenvalue @xmath151@xmath152 and has boundary conditions @xmath153 very close to @xmath151 , which is be the exact boundary condition we require from the eigenfunctions of the operator . so the difference between @xmath154 and the true eigenfunction are very small . since @xmath150 corresponds to eigenvalue @xmath151 we conclude that , by continuity , the true eigenfunction will have an eigenvalue @xmath155 very small , exponentially small . then the range of important values of the coefficient @xmath156 is very large and the gaussian expansion is invalid since the departure of such a solution from the classical one ( the instanton ) can not be considered small . the degeneracy in the moment of time where the center of the instanton is placed ( _ i.e. _ the moment of transition ) can be solved treating this parameter as a colective coordinate . the result is @xmath157 ^{1/2}\int_{-t/2}^{t/2}d\theta \left\ { \frac{s\left [ x_{i}\left ( \tau -\theta \right ) \right ] } { 4\pi d}\right\ } ^{1/2 } \\ & & \times \exp \left\ { -\frac{1}{ds\left [ x_{i}\left ( \tau -\theta \right ) % \right ] } \right\}\end{aligned}\ ] ] the parameter @xmath158 in the expression of the instanton solution shows is the current time variable along the solution that is used to calculate the action . the integartion is performed on the intermediate transition moment @xmath159 . we also note that @xmath160 = \underset{t\rightarrow \infty } { \lim } s\left [ x_{i}\left ( \tau \right ) \right ] = s_{0}\ ] ] since , except for the very small intervals ( approx . the width of the instanton ) at the begining and the end of the interval @xmath161 , the value of the action @xmath51 is not sensitive to the position of the transition moment . @xmath162 ^{1/2}\left ( \frac{s_{0}}{4\pi d}% \right ) ^{1/2}t\exp \left ( -\frac{s_{0}}{d}\right)\ ] ] the instanton degeneracy introduces a _ linear _ time dependence of the probability . in general , for a function @xmath163 that has two minima separated by a barrier ( a maximum ) the potential @xmath137 calculated form the action will have three minima and these are not degenerate . the inverse of this potential , @xmath164 , which is appears in the equation of motion , will have three maxima in general nondegenerate and the differences in the values of @xmath164 at these maxima is connected with the presence of the term containing @xmath119 . since we assume that @xmath119 is small , the non - degeneracy is also small . the previous discussion in which the notion of instanton was introduced and @xmath115 was calculated , take into consideration the degenerate maxima and the instanton transition at equal initial and final @xmath165 . let us consider the general shape for @xmath164 with three maxima , at @xmath166 , @xmath151 and @xmath143 . the heigths of these maximas are @xmath167 this is because the extrema of @xmath164 @xmath168 } { dx}=0\ ] ] coincides according to the equation of motion to the pointes where @xmath169 and the constant can not be taken other value but zero @xmath170 then , since we have approximately that @xmath171 ( for small @xmath119 ) then we have that at these extrema of @xmath172 we have @xmath173 and only the second term in the expression of @xmath137 remains . this justifies eq.([dupp ] ) . it will also be assumed that @xmath174 . we want to calculate , on a kramers time scale , @xmath175 the probability @xmath176 we have to find the classical solution of the equation of motion connecting @xmath177 with @xmath178 . this is the trivial solution , particle sitting at @xmath179@xmath180 we have to calculate explicitely the form of the propagator in this case @xmath131\ ] ] we use the formulas given before @xmath133 ^{1/2 } } \label{colem}\ ] ] where the function @xmath134 is the solution of @xmath181 with the boundary conditions @xmath182 we use simply @xmath183 and @xmath184 for the respective functions calculated at the fixed point @xmath179 . we have @xmath185 \\ & \approx & \frac{1}{2}\left ( u^{\prime \prime } \right ) ^{2}\;+\;\text{terms of order } d\end{aligned}\ ] ] the equation for the eigenvalues becomes @xmath186 @xmath187 and it results form the boundary conditions @xmath188 \\ a_{2 } & = & -\frac{1}{2\left| u^{\prime \prime } \right| } \exp \left [ \left| u^{\prime \prime } \right| t_{i}\right]\end{aligned}\ ] ] then @xmath189 -\exp \left [ -\left| u^{\prime \prime } \right| \left ( t - t_{i}\right ) \right ] \right\}\ ] ] * note*. this is the @xmath190 which is obtained in the calculation of the ground level splitting by quantum tunneling for a particle in two - well potential . now we can calculate @xmath191 ^{1/2}}=\frac{1}{\left ( 4\pi d\right ) ^{1/2}}\left [ 2\left| u^{\prime \prime } \right| \right ] ^{1/2}% \frac{\exp \left [ -\frac{1}{2}\left| u^{\prime \prime } \right| \left ( t - t_{i}\right ) \right ] } { \left ( 1-\exp \left [ -2\left| u^{\prime \prime } \right| \left ( t - t_{i}\right ) \right ] \right ) ^{1/2}}\ ] ] it remains to calculate the action for this trivial trajectory @xmath192@xmath193 \\ & = & \int_{t_{i}}^{t}dtw\left ( x_{c}\right ) \\ & = & \int_{t_{i}}^{t}dt\left [ \frac{1}{4}u^{\prime 2}\left ( x_{c}\right ) -% \frac{d}{2}u^{\prime \prime } \left ( x_{c}\right ) \right]\end{aligned}\ ] ] since the position of the extremum of @xmath165 is very close ( to order @xmath119 ) of the position where @xmath1 is zero , and since @xmath194 , we can take with good approximation the first term in the integrand zero . then @xmath195 we have to put together the two factors of the propagator and take into account that at @xmath179 we have @xmath196@xmath197 ^{1/2}}\exp \left [ -\frac{s_{c}}{d}\right ] \\ & = & \left ( \frac{u^{\prime \prime } } { 2\pi d}\right ) ^{1/2}\frac{\exp \left [ -% \frac{1}{2}u^{\prime \prime } \left ( t - t_{i}\right ) \right ] } { \left ( 1-\exp % \left [ -2u^{\prime \prime } \left ( t - t_{i}\right ) \right ] \right ) ^{1/2}}\exp \left ( \frac{1}{2}u^{\prime \prime } \left ( t - t_{i}\right ) \right)\end{aligned}\ ] ] @xmath198 \right ) ^{1/2}}\ ] ] the contribution to the action is @xmath199 ^{1/2 } } \\ & \simeq & \left ( \frac{u_{b}^{\prime \prime } } { 2\pi d}\right ) ^{1/2}\;\;\text{% ( at large } t\text{)}\end{aligned}\ ] ] it should be noticed that no other classical solution exists since there are no turning points permitting the solution to come back to @xmath179 . we apply the procedure described for the purely diffusive case to the case @xmath200 and for this we need the solution of the euler - lagrange equations . an approximation is possible if the diffusion coefficient is small . in this case the diffusion will take place around the equilibrium positions @xmath201 and @xmath202 . taking the equilibrium @xmath203 in the equation for @xmath80 we have @xmath204 where @xmath205 \theta \left ( t^{\prime } -t\right ) \exp \left [ -\left ( -a+3bx_{0}^{2}\right ) t^{\prime } \right ] \label{delta21}\ ] ] the symbol @xmath70 stands for the heaviside function . in the equation for @xmath3 we expand around the equilibrium position @xmath206 and solve the equation @xmath207 taking account of eq.([k1 ] ) @xmath208 where @xmath209 \label{delta11 } \\ & & \hspace*{-1cm}\times \left\ { \frac{1}{2\left ( -a+3bx_{0}^{2}\right ) } % \left ( \exp \left [ 2\left ( -a+3bx_{0}^{2}\right ) t\right ] -1\right ) \theta \left ( t^{\prime } -t\right ) \right . \notag \\ & & \hspace*{-1cm}\left . + \frac{1}{2\left ( -a+3bx_{0}^{2}\right ) } \left ( \exp % \left [ 2\left ( -a+3bx_{0}^{2}\right ) t^{\prime } \right ] -1\right ) \theta \left ( t - t^{\prime } \right ) \right\ } \notag \\ & & \times \exp \left [ \left ( a-3bx_{0}^{2}\right ) t^{\prime } \right ] \notag\end{aligned}\ ] ] and @xmath210 \theta \left ( t - t^{\prime } \right ) \exp \left [ \left ( -a+3bx_{0}^{2}\right ) t^{\prime } \right ] \label{delta12}\ ] ] we note that at the limit where no potential would be present , @xmath211 , the propagators @xmath212 become @xmath213\ ] ] @xmath214 _ i.e. _ the propagators of a purely diffusive process ( see @xcite ) . using the solutions eqs.([k1 ] ) and ( [ x1 ] ) we can calculate the action along this path . @xmath215\ ] ] we will insert the expansion eq.([expx ] ) , perform an integration by parts over the first term and take into account the equations , _ i.e. _ the first line of eq.([eqel ] ) and eq.([eqeps ] ) @xmath216 \varepsilon + j_{2}k\right\}\end{aligned}\ ] ] @xmath217 now we express the two solutions , for @xmath113 and @xmath218 in terms of the propagators @xmath212@xmath219 where summation over @xmath220 is assumed , and @xmath221 . we now dispose of the generating functional of the system _ when this is in a region around _ @xmath222 , the fixed equilibrium points . to see what is the effect of the diffusion in this case we calculate for the variable @xmath10 the average and the dispersion . @xmath223 which was to be expected . and @xmath224 @xmath225 \\ & & \hspace*{-1cm}\times \left [ x_{0}+\frac{1}{2}\int_{0}^{t}dt^{\prime \prime } \delta _ { 11}\left ( t^{\prime } , t^{\prime \prime } \right ) j_{1}\left ( t^{\prime \prime } \right ) + \frac{1}{2}\int_{0}^{t}dt^{\prime \prime } \delta _ { 12}\left ( t^{\prime } , t^{\prime \prime } \right ) j_{2}\left ( t^{\prime \prime } \right ) + \frac{1}{2}\int_{0}^{t}dtj_{2}\left ( t\right ) \delta _ { 21}\left ( t , t^{\prime } \right ) \right ] \\ & & \times \exp \left ( i\emph{s}_{j}\right)\end{aligned}\ ] ] this gives the result @xmath226 d \label{xtxtp } \\ & & \times \left\ { \frac{1}{2\left ( -a+3bx_{0}^{2}\right ) } \left ( \exp \left [ 2\left ( -a+3bx_{0}^{2}\right ) t\right ] -1\right ) \theta \left ( t^{\prime } -t\right ) \right . \notag \\ & & \left . + \frac{1}{2\left ( -a+3bx_{0}^{2}\right ) } \left ( \exp \left [ 2\left ( -a+3bx_{0}^{2}\right ) t^{\prime } \right ] -1\right ) \theta \left ( t - t^{\prime } \right ) \right\ } \notag \\ & & \times \exp \left [ \left ( a-3bx_{0}^{2}\right ) t^{\prime } \right ] \notag\end{aligned}\ ] ] in the absence of the potential @xmath227 and @xmath228 , this is simply @xmath229 _ i.e. _ the diffusion around the position @xmath36 . in the present case , we note that @xmath230 with @xmath231 . fixing the parameters , we take @xmath232 and obtain @xmath233 \exp \left ( -u_{0}^{\prime \prime } t^{\prime } \right)\ ] ] for @xmath234 ( the dispersion ) we obtain ( with @xmath235 ) @xmath236 \label{x2canoi}\ ] ] ) ( continuous line ) and numerical integration of the stochastic equation.,width=377 ] it is straightforward to calculate the higher order statistics for this process , since the functional derivatives can easily be done . we have to remember that this derivation was based on the approximation consisting in taking the equilibrium position in the euler - lagrange equation for @xmath81 . however , since @xmath237 and @xmath238 are small quantities and we can suppose that the instantons , even if they are not exact solutions of the equations of motion , can give a contribution to the action . the instantons connects the points of maximum not of @xmath164 ( because they are not equal ) but of a different potential , a _ corrected _ @xmath164 to order one in @xmath119 that has degenerate maxima . this potential will be called @xmath239 and it will be considered in the calculation of the contribution of the instantons and antiinstantons . @xmath240 where @xmath241 is the point corresponding to the minimum situated between the two maximas . it is introduced the family of trajectories @xmath242 which leaves @xmath166 at time @xmath243 and reach @xmath140 at time @xmath244 . they have all the same energy @xmath245 and the classical action @xmath246 . we now consider the travel from @xmath166 to @xmath140 made by an instanton @xmath247 with the center located at time @xmath248 ; next the return made by an antiinstanton which is actually an instanton @xmath249 starting from @xmath140 and going to @xmath166 with the center located at time @xmath248 . with these two instantons we create a single classical solution @xmath250 the contribution to the action of this _ assambled _ solution is @xmath251 ^{2}}{2}\end{aligned}\ ] ] the contributions in this formula comes from the potential energy and the kinetic energy along the trajectory . we should remember that the action is the integral on time of the density of lagrangian , where there is the kinetic energy term and minus the potential energy . @xmath252 \right\}\ ] ] if the particle would have remained in @xmath166 imobile for all time @xmath253 then the contribution from the potential would have been @xmath254 the instanton spends @xmath255 time in the point @xmath151 before returning to @xmath179 . then it accumulates the action equal with the difference in potential between @xmath179 and @xmath151 multiplied with this time interval . @xmath256 \left ( t_{1}-t_{0}\right ) = \left ( w_{0}-w_{b}\right ) \left ( t_{1}-t_{0}\right)\ ] ] ( since @xmath257 ) . define the kinetic energy and the energy @xmath258 \\ \overset{\cdot } { x}\left ( t\right ) & = & \sqrt{2\left ( e+w\left ( x\right ) \right ) } \end{aligned}\ ] ] then @xmath259 in this formula we have to replace the expression of the trajectory @xmath260 and integrate . in our case the energy has the value of the initial position . here the velocity is zero and the potential is @xmath261@xmath262 and the potential is actually the current value of the zeroth -order potential @xmath263 -\left [ -w^{\left ( 0\right ) } \left ( x\right ) \right ] \\ & = & w^{\left ( 0\right ) } \left ( x\right ) -w\left ( b\right)\end{aligned}\ ] ] if we want to calculate the integral of the _ kinetic energy _ along the trajectory , we have to consider separately the intervals where the kinetic energy is strongly determined by the velocity , _ i.e. _ the region where the instanton transition occurs , from the rest of the trajectory where , the velocity being practically zero , the potential is a better description and can be easily approximated . the approximation will relay on the fact that the particle is practically imobile in @xmath179 or in @xmath151 , after the transition has been made . so we will use both expressions for the kinetic energy @xmath264 ^{1/2}\ ] ] the terms must be considered twice and the interval of integration can be extended for the region of transition , since in any case it is very small @xmath265 ^{1/2}-2\int_{\frac{t_{1}+t_{0}}{2}}^{\infty } d\tau \frac{1}{2}\overset{% \cdot } { x}^{2}\left ( \tau -t_{0}\right)\ ] ] the term @xmath246 is @xmath266 ^{1/2}\ ] ] using an approximation for the form of the instanton , it results @xmath267\end{aligned}\ ] ] where @xmath268 and @xmath269 ^{1/2}}-\frac{1}{\left [ w_{h}^{\left ( 0\right ) } \left ( x\right ) -w_{b}\right ] ^{1/2}}\right\}\ ] ] the local harmonic approximation to @xmath270 is @xmath271@xmath272 * note * since the trajectory @xmath273 is not an _ exact _ solution of the equation of motion in the potential @xmath274 , the expansion of the action @xmath51 will not be limitted to the zeroth and the second order terms . it will also contain a firts order term , @xmath275 . it can be shown that this contribution is negligible in the order @xmath276 . the calculation of the contribution to the functional integral from the second order expansion around @xmath277 is done as usual by finding the eigenvalues of the determinant of the corresponding operator . as before , the product of the eigenvalues should not include the first eigenvalue since this is connected with the translational symmetry of the instanton solution . this time there will be two eigenvalues , one for @xmath248 ( the transition performed by the @xmath228 instanton ) and the second for @xmath278 ( the transition prtformed by the antiinstanton , or the transition @xmath279 ) . another way of expressing this invariance to the two time translations is to say that the pair of instantons has not a determined central moment and , in addition , there is an _ internal _ degree of freedom of the _ breathing _ solution , which actually is this pair instanton - antiinstanton . to take into account these modes , whose eigenvalues are zero , we need to integrate in the functional integral , over the two times , @xmath248 and @xmath278 . the measure of integration for the two translational symmetries @xmath248 and @xmath280 is @xmath281 this quantity is the jacobian of the change of variables in the functional integration over the fluctuations around the instanton solution . the fluctuation that corresponds to the lowest ( almost zero ) eigenvalue is replaced in the measure of integration with the differential of the time variable representing the moment of transition . then it results this jacobian . it can be shown that in the approximation given by exponentially small terms , the contribution to the path integral of the small fluctuations around the classical instanton - antiinstanton @xmath277 solution is the _ product _ of the fluctuation terms around the instanton and antiinstanton separately . @xmath282 ^{1/2}\exp \left [ -\frac{s_{ia}\left ( t;t_{0},t_{1}\right ) } { d}% \right]\end{aligned}\ ] ] where @xmath283 represent the lowest eigenvalue of the operator arising from the second order expansion of the action , defined on the time intervals @xmath284 the notation @xmath285 and respectively @xmath286 represents @xmath287 @xmath288 this corresponds to the formula of coleman which replaces the infinite product of eigenvalues with @xmath289 where @xmath134 is calculated at the end of the interval of time , where @xmath134 verifies the boundary conditions ( [ psibo ] ) . the result is @xmath290 ^{1/2}\frac{\left ( x_{m}-b\right ) \left| x_{m}\right| w_{0}^{\prime \prime } w_{b}^{\prime \prime } } { 2\pi ^{2}d^{2}}\exp \left [ -\frac{\varepsilon _ { 0}^{\left ( b\right ) } t}{d}\right ] \notag \\ & & \times \exp \left [ -\frac{2s_{b0}}{d}+\delta _ { bm}\left ( 2w_{b}^{\prime \prime } \right ) ^{1/2}+\delta _ { bm}\left ( 2w_{0}^{\prime \prime } \right ) ^{1/2}\right ] \notag \\ & & \hspace*{-1cm}\times \int_{-t/2}^{t/2}dt_{0}\int_{t_{0}}^{t/2}dt_{1}\exp \left\ { -\frac{\varepsilon _ { 0}^{\left ( 0\right ) } -\varepsilon _ { 0}^{\left ( b\right ) } } { d}\left ( t_{1}-t_{0}\right ) + \frac{c}{d}\exp \left [ -\left ( 2w_{0}^{\prime \prime } \right ) ^{1/2}\left ( t_{1}-t_{0}\right ) \right ] \right\ } \notag\end{aligned}\ ] ] where @xmath291 is the lowest eigenvalue of the schrodinger equation associated with the fokker - planck diffusion equation in the local harmonic approximation of the potential @xmath165 in the well @xmath292 . @xmath293 \label{constc}\ ] ] and @xmath294 ^{1/2}}-\frac{1}{\left [ w_{h}^{\left ( 0\right ) } \left ( x\right ) -w_{b}\right ] ^{1/2}}\right\}\ ] ] there is a very important problem with this formula : the coefficient @xmath295 is positive and the contribution of this part in the @xmath278 integration comes from time intervals @xmath296 since this is a very small time interval , it results that the contributions are due to states where the instanton and the antiinstanton are very close one of the other , which is unphysical . it will be necessary to calculate in a particular way this part , introducing acontour in the complex @xmath297 plane , with an excursion on the imaginary axis . * the calculation of the propagator * @xmath298 . here @xmath299 means that only one pair of instanton and anti - instanton is considered . we particularize the formula above using the expression for @xmath300 given in terms of @xmath163 . now we have @xmath301 the expression of @xmath246 is @xmath302 @xmath303 where @xmath304\ ] ] we will use the functional approach in the setting that has been developed by us to the calculation of the probability of transition from one minimum to the same minimum with an intermediate stay at the unstable maximum point ( symmetric potential ) . the initial equation is @xmath305 the action functional with external current added is @xmath306 @xmath307 the equations of motion are @xmath308 we have to solve these equations , and replace the solutions @xmath84 and @xmath113 in the action functional . then the functional derivatives to the external current @xmath309 will give us the correlations for the stochastic variable @xmath10 . we can also calculate the probability that the particle , starting from one point at a certain time will be found at another point at other time . this will be done below . the first step is to obtain an analytical solution to the euler - lagrange equation . the method to solve these equation is essentially a successive approximation , as we have done above , for the diffusive motion around a stable position in the potential ( harmonic region ) . we know that the classical trajectory must be of the type of a transition between the initial point , taken here as the left minimum of the potential and the final point , the unstable maximum of the potential . from there , the particle will return to the left minimum by an inverse transition . we have to calculate simultaneously @xmath10 and @xmath113 , but we have sufficient information to find an approximation for @xmath10 , by neglecting the effect of diffusion ( the last term in the differential equation for @xmath10 ) . @xmath310 this gives @xmath311 \right\ } ^{1/2}}\ ] ] this solution is a transition between either of the minima @xmath312 and @xmath151 . it shows the same characteristics as found numerically or by integrating the elliptic form of the equation , in the case of onasger - machlup action . the particle spends long time in the initial and final points and makes a fast transition between them at an arbitrary time @xmath313 . the width of transition is small compared to the rest of quasi - imobile stays in the two points , especially if @xmath143 is large . then we will make an approximation , taking the solution as @xmath314 where @xmath201 and @xmath151 are the initial and final positions , the indice @xmath315 means that this is the calssical solution ( extremum of the action ) and @xmath299 means the first part of the full trajectory , which will also include the inverse transition , from @xmath151 to @xmath201 . the following structure of the total trajectory is examined : the total time interval is between @xmath316 and @xmath317 ( later the parameter @xmath317 will be identified with @xmath147 for comparison with the results from the literature ) . the current time variable is @xmath158 and in the present notations , @xmath318 is any moment of time in the interval @xmath319 . at time @xmath248 the particle makes a jump to the position @xmath140 and remains there until @xmath320 . at @xmath278 it performs a jump to the position @xmath201 , where it remains for the rest of time , until @xmath317 . with this approximative solution of the euler - lagrange equations ( since we have neglected the term with @xmath119 in the equation of @xmath321 ) we return to the equation for @xmath322 . we first calculate @xmath323 we need the integration of this quantity in the inverse direction starting from the end of the motion toward the initial time @xmath324 \theta \left ( t - t_{0}\right ) + \left [ -at+3at_{0}-2at\right ] \theta \left ( t_{0}-t\right)\ ] ] we now introduce the second part of the motion : at time @xmath325 the particle makes the inverse transition @xmath326 with the similar quantities . after explaining the steps of the calculation , we change to work with the full process , assembling the two transitions and the static parts into a single trajectory @xmath327 @xmath328 and @xmath329 we have introduce the notation @xmath330 then the solution of the equation for the dual variable is @xmath331 \left\ { k_{t}+\int_{t}^{\tau } dt^{\prime } \left [ -j_{1}\left ( t^{\prime } \right ) \right ] \exp \left [ w\left ( t^{\prime } \right ) \right ] \right\}\ ] ] according to the procedure explained before we will need to express the solutions as bilinear combinations of currents , so we identify @xmath332 \exp \left [ w\left ( t^{\prime } \right ) % \right ] \label{del21new}\ ] ] and the solution can be rewritten @xmath333 + \int_{-t}^{t}dt^{\prime } \delta _ { 21}\left ( t , t^{\prime } \right ) j_{1}\left ( t^{\prime } \right ) \label{kdel21}\ ] ] using these first approximations for the extremizing path @xmath334 we return to the euler lagrange equations and expend the variable @xmath335 as @xmath336 whose equation is @xmath337 and the solution @xmath338 \left\ { b_{0}+\int_{0}^{t}dt^{\prime } 2idk_{\eta } \left ( t^{\prime } \right ) \exp % \left [ -\int_{0}^{t^{\prime } } dt^{\prime \prime } \left ( \frac{dv}{dx}\right ) _ { x_{\eta } \left ( t^{\prime \prime } \right ) } \right ] \right\}\ ] ] here @xmath339 is a constant to be determined by the condition that @xmath340 vanishes at the final point . we note that here all integrations are performed forward in time . we also notice that this expression will contain the current @xmath39 and we will introduce the propagator @xmath341 . performing the detailed calculations we obtain @xmath342 + \exp % \left [ w\left ( t\right ) \right ] \int_{-t}^{t}dt^{\prime } \left ( 2idk_{t}\right ) \theta \left ( t - t^{\prime } \right ) \exp \left [ -w\left ( t^{\prime } \right ) \right ] \label{delxp } \\ & & + \int_{-t}^{t}dt^{\prime } \delta _ { 11}\left ( t , t^{\prime } \right ) j_{1}\left ( t^{\prime } \right ) \notag\end{aligned}\ ] ] the first two terms depend on constants of integrations and the propagator is @xmath343 \int_{-t}^{t}dt^{\prime \prime } 2id\theta \left ( t - t^{\prime \prime } \right ) \delta _ { 21}\left ( t^{\prime \prime } , t^{\prime } \right ) \exp \left [ -w\left ( t^{\prime \prime } \right ) \right ] \label{del11}\ ] ] using the solutions ( [ xexpa ] ) and ( [ kdel21 ] ) we can calculate the action along this trajectory . @xmath344 \\ & = & i\int_{-t}^{t}dt\left ( \frac{1}{2}x\overset{\cdot } { k}+\frac{1}{2}% kv+j_{1}x\right ) \\ & = & i\int_{-t}^{t}dt\left [ \frac{1}{2}x\left ( -j_{1}+k\frac{dv}{dx}\right ) + % \frac{1}{2}kv+j_{1}x\right ] \\ & = & i\int_{-t}^{t}dt\left\ { \frac{1}{2}j_{1}x+\frac{1}{2}k\left [ v-\left ( \frac{dv}{dx}\right ) x\right ] \right\}\end{aligned}\ ] ] [ it becomes obvious that when the potential is linear ( which means that the lagrangean is quadratic the two functional variables @xmath10 and @xmath113 ) the potential does not contribute to the action along the extremal path ] . using the solutions we have @xmath345 \\ & & + i\int_{-t}^{t}dt\frac{1}{2}\left [ v\left ( x_{\eta } \right ) -\left ( \frac{% dv}{dx}\right ) _ { x_{\eta } } x_{\eta } \right ] i\int_{-t}^{t}dt^{\prime } \delta _ { 21}\left ( t , t^{\prime } \right ) j_{1}\left ( t^{\prime } \right)\end{aligned}\ ] ] it will become clear later that the parts containing constants are not significative for the final answer , the probability . we will focus on the terms containing explicitely the current @xmath39 since the statistical properties are determined by functional derivatives to this parameter . the action is @xmath346 \int_{-t}^{t}dt^{\prime } \delta _ { 21}\left ( t , t^{\prime } \right ) j_{1}\left ( t^{\prime } \right ) \right\}\end{aligned}\ ] ] the generating functional of the correlations ( at any order ) is @xmath347 = \exp \left\ { \emph{s}\left [ j_{1}\right ] \right\ } \label{zands}\ ] ] and we can calculate any quantity by simply parforming functional derivatives and finally taking @xmath348 . instead of that and in order to validate our procedure , we will calculate the probability for the process : a particle in the initial position @xmath349 at time @xmath316 can be found at the final position @xmath350 at the time @xmath351 . ( later we will particularize to @xmath352 and @xmath353 ) . the calculation of this probability @xmath132 can be done in the functional approach in the following way . we have defined the statistical ensemble of possible particle trajectories starting at @xmath354 at @xmath355 and reaching an arbitrary point @xmath335 at time @xmath318 . in the martin - siggia - rose - jensen approach it is derived the generating functional as a functional integration over this statistical ensemble of a weight measure expressed as the exponential of the classical action . now we restrict the statistical ensemble by imposing that the particle is found at time @xmath351 in the point @xmath350 . by integration this will give us precisely the probability required . the condition can be introduced in the functional integration by a dirac @xmath13 function which modifies the functional measure @xmath356 \rightarrow \emph{d}\left [ x\left ( \tau \right ) \right ] \delta \left [ x\left ( t_{c}\right ) -x_{c}\right]\ ] ] and we use the fourier transform of the @xmath11 function @xmath357 \delta \left [ x\left ( t_{c}\right ) -x_{c}\right ] \exp \left ( \emph{s}\right ) \\ & = & \frac{1}{2\pi } \int d\lambda \exp \left ( -i\lambda x_{c}\right ) \int \emph{d}\left [ x\left ( \tau \right ) \right ] \exp \left [ s+i\lambda x\left ( t_{c}\right ) \right]\end{aligned}\ ] ] it is convenient to write @xmath358 + i\int_{-t}^{t}d\tau \lambda x\left ( \tau \right ) \delta \left ( \tau -t_{c}\right ) \\ & = & i\int_{-t}^{t}d\tau \left [ \emph{l}+\lambda x\left ( \tau \right ) \delta \left ( \tau -t_{c}\right ) \right]\end{aligned}\ ] ] we can see that the new term plays the same rle as the external current @xmath359 and this sugests to return to the eq.([zands ] ) and to perform the modification @xmath360 obtaining @xmath361 \right| _ { j_{1}=0}\ ] ] using the previous results we have @xmath362 \right| _ { j_{1}=0 } & = & \exp \left\ { i\frac{1}{2}\lambda x_{\eta } \left ( t_{c}\right ) \right . \label{zlamb1 } \\ & & + i\frac{1}{2}\lambda ^{2}\delta _ { 11}\left ( t_{c},t_{c}\right ) \notag \\ & & \left . + i\frac{1}{2}\left [ v-\left ( \frac{dv}{dx}\right ) x\right ] _ { t = t_{c}}\lambda \delta _ { 21}\left ( t_{c},t_{c}\right ) \right\ } \right| _ { j_{1}=0 } \notag\end{aligned}\ ] ] this expression will have to be calculated at @xmath363 and @xmath364 . @xmath365 so the last term is zero . _ { j_{1}=0}=\left . \exp \left [ -i\frac{1}{2}\lambda \sqrt{\frac{a}{b}}+i\frac{% 1}{2}\lambda ^{2}\delta _ { 11}\left ( t , t\right ) \right ] \right| _ { j_{1}=0 } \label{zdel}\ ] ] we need the propagator @xmath367 in the expression for @xmath341 , @xmath368 \exp \left [ w\left ( t\right ) % \right]\ ] ] @xmath369 then the intration giving the propagator @xmath341 is @xmath370 \exp \left [ -w\left ( t^{\prime } \right ) \right ] \\ & = & 2id\int_{-t}^{t}dt^{\prime } \exp \left [ -2w\left ( t^{\prime } \right ) % \right ] \\ & = & 2id\left\ { \exp \left [ -4at+6at_{1}-6at_{0}\right ] \int_{-t}^{t_{0}}dt% \exp \left ( 4at\right ) \right . \\ & & + \exp \left [ -4at+6at_{1}\right ] \int_{t_{0}}^{t_{1}}dt\exp \left ( -2at\right ) + \\ & & \left . + \exp \left [ -4at\right ] \int_{t_{1}}^{t}dt\exp \left ( 4at\right ) \right\}\end{aligned}\ ] ] we introduce the notation @xmath371 \left\ { \exp \left [ 2a\left ( t_{1}-t_{0}\right ) \right ] -1\right\ } \\ & & -\exp \left [ -8at\right ] \exp \left [ 6a\left ( t_{1}-t_{0}\right ) \right]\end{aligned}\ ] ] and we obtain the expression @xmath372 returning to ( [ zdel ] ) @xmath373 = \exp \left [ -i\frac{% \lambda } { 2}\sqrt{\frac{a}{b}}-\frac{\lambda ^{2}d}{4a}y\right]\ ] ] and the probability is @xmath374 \emph{z}^{\left ( \lambda \right ) } \left [ j_{1}=0\right ] \\ & = & \frac{1}{2\pi } \int_{-\infty } ^{\infty } d\lambda \exp \left [ \frac{1}{2}% i\lambda \sqrt{\frac{a}{b}}-\frac{\lambda ^{2}d}{4a}y\left ( t_{0},t_{1}\right ) \right]\end{aligned}\ ] ] we have made manifest the dependence of the probability on the arbitrary times of jump @xmath375 . integrating on @xmath376@xmath377 this expression contains an arbitrary parameter , which is the duration of stay in the intermediate position , _ i.e. _ at the unstable extremum of the potantial , @xmath140 . integrating over this duration , @xmath378 , adimensionalized with the unit @xmath379 , for all possible values between @xmath151 and all the time interval , @xmath380 , will give the probability . in doing this time integration we will make a simplification to consider that @xmath317 is a large quantity , such that the exponential terms in @xmath381 will vanish . then @xmath381 reduces to @xmath382 and the integration is trivial @xmath383 p\left ( t_{0},t_{1}\right ) \\ & = & 2ta\sqrt{\frac{a}{\pi d}}\exp \left ( -\frac{a^{2}}{4bd}\right)\end{aligned}\ ] ] this is our answer . to compare with the result of caroli et al . , we first adopt their notation @xmath384 and we write the potential as @xmath385 with @xmath386 we note that @xmath387 which allows to write @xmath388\ ] ] which corresponds to the result of caroli et al . for a single instanton - anti - instanton pair , where @xmath389 the only problem is the need to consider the reverse of sign due to the virtual reflection at the position @xmath140 . in the tratment of caroli , it is included by an integration in complex time plane , after the second jump , @xmath280 . the problem arises at the examination of the eq.([k(1)raw ] ) where it is noticed that the constant @xmath295 multiplying the exponential term _ in the exponential _ in the integrations over @xmath248 and @xmath278 is positive ( see eq.([constc ] ) ) . then the dependence of the integrand on @xmath297 makes that the most important contribution to the integration over the variable @xmath278 to come from @xmath390 here the integrand is of the order @xmath391 this would imply , in physical terms , that the most important configurations contributing to the path integral would be pairs of instanton anti - instanton very closely separated , _ i.e. _ transitions with very short time of stay in the point @xmath151 . this is not correct since we expect that in large time regime the trajectories assembled from instantons and anti - instantons with arbitrary large separations ( duration of residence in @xmath151 before returning to @xmath179 ) should contribute to @xmath115 . this problem can be solved by extending the integration over the variable representing the time separation between the transitions @xmath392 in complex . * note*. the quantity @xmath393 is the frequency of the harmonic oscillations in the parabolic approximation of the potential . the contour _ c _ is defined as @xmath394 the form of the trajectory @xmath277 connecting the point @xmath179 to itself , @xmath179 , assembled from one instanton @xmath395 and one anti - instanton @xmath396 can now be described as follows . the trajectory starts in the point @xmath179 and follows the usual form of the instanton solution , which means that for times @xmath397 it practically conicides with the fixed position @xmath179 . at @xmath248 it makes the transition to the point @xmath151 , the duration of the transition ( width of the instanton ) being @xmath146 . then it remains almost fixed at the point @xmath398 . the problem is that there is _ no _ turning point at @xmath140 since there the potential @xmath164 is smaller than at @xmath166 . then we do not have the classical instanton which takes an infinite time to reach exactly @xmath140 , etc . also there is no exact instanton solution which starts from @xmath140 to go back to @xmath166 . in the assambled solution @xmath399 , there is a discontinuity at @xmath140 in the derivative , the velocity simply is forced ( assumed ) to change from almost zero positive ( directed toward @xmath140 ) to almost zero negative ( directed toward @xmath166 ) . what we do is to introduce a turning point , which limits effectively the transition solution , but for this we have to go to complex time , to compensate for the negative value of the potential . making time imaginary changes the sign of the velocity squared ( kinetic energy ) and correspondingly can be seen as a change of sign of the potential which now becomes positive and presents a wall from where the solution can reflect . this is actually a local euclidean - ization of the theory , around the point @xmath140 . at time @xmath400 the trajectory starts moving along the imaginary @xmath401 axis . it reaches an imaginary turning point at @xmath402 and returns to the point @xmath403 , @xmath404 . this trajectory can be considered the limiting form of the trajectory of a fictitious particle that traverses a part of potential ( forbidden classically ) when this part goes to zero . then the expression is @xmath405 where the minus sign is due to the reflection at the turning point . * calculation of the transitions implying the points * @xmath179 and @xmath143 this is the term @xmath406 and corresponds to the transitions @xmath407 again we have to introduce a new potential replacing @xmath165 . the real potential @xmath137 as it results from the langevin equation , depende on @xmath119 which is supposed to be small . then the `` expansion '' in this small parameter @xmath119 has as zerth order the new potential @xmath408 that will be used in the computations below . this is @xmath409 @xmath410 where the new point @xmath411 is in the region of the right - hand minimum of @xmath412 . this point will correspond to the center of the instanton connecting @xmath151 to @xmath143 . we introduce the new family of instanton solution @xmath413 in the potential @xmath414 that leave @xmath140 at @xmath243 and reach @xmath415 at time @xmath244 . with this instanton and the previously defines ones , we assamble a function that provide approximately the transitions conncetin @xmath179 to @xmath179 via @xmath143 . this , obviously , is not a classical solution of the equation of motion , although it is very close to a solution . @xmath416 these trajectories are rather artificially , being assambled form pieces of true solutions . they have singularities : in the middel , at @xmath417 the function is continuous but the derivative is discontinuous ; in the points where one instanton arrives and another must continue ( _ i.e. _ @xmath400 and @xmath418 , both the function and the derivatives are discontinuous . this is corrected by extending the trajectories in complex time plane . one defines the variables @xmath419 and the three integration contours @xmath420 together these contours must assure the continuity of the function @xmath277 in the region @xmath151 , from which the condition arises @xmath421 = -% \widetilde{x}_{i}\left [ \func{re}\left ( \frac{t_{0}-t_{1}}{2}\right ) \right]\ ] ] which gives for large separations @xmath297@xmath422 the exprsssion of @xmath115 at this moment is @xmath423 \\ & & \times \int_{-t/2}^{t/2}dt_{0}\int_{\emph{c}_{1}}dz_{1}\int_{\emph{c}% _ { 2}}dz_{2}\int_{\emph{c}_{3}}dz_{3}\exp \left [ f\left ( z_{1}\right ) + g\left ( z_{2}\right ) + f\left ( z_{3}\right ) \right]\end{aligned}\ ] ] where @xmath424 @xmath425 and @xmath426 @xmath427 it is necessary to carry out the complex time integrals . * the * @xmath428 * * integral**. here we have @xmath429 along the real axis of @xmath401 , we can substitute @xmath430 and obtain @xmath431 this simply results in the constant @xmath432 * the * @xmath433 * * integral * * ; the @xmath433 integral involves only the exponential of @xmath434 and the latter function does not contain a term proportional with @xmath435 ( comapre with ( [ fz ] ) ) , but only the exponential term . the function @xmath434 is very close to @xmath382 as soon as @xmath436 where @xmath437 is the suzuki time for the region @xmath143@xmath438 then , with very good approximation @xmath439 = \int_{0}^{% \frac{t}{2}-t_{1}}dz_{2}\exp \left [ g\left ( z_{2}\right ) \right ] \simeq \frac{t}{2}\ ] ] * the * @xmath440 * integral*. this is similar to the integral over @xmath428 and will imply again @xmath441 as in ( [ rezc3 ] ) . finally we note that only the constant @xmath442 relative to the intermediate point @xmath151 appears , and the one for the farthest point of the trajectory , @xmath143 , is absent . there will be then a time integration over the moment of the first transition , @xmath248 , of an integrand that contains the time @xmath318 arising from the integration on the contour @xmath443 , in the complex plane of the variable @xmath444 . this variable represents the duration between the last transition to @xmath143 and the first transition from @xmath143 , _ i.e. _ is the duration of stay in @xmath143 . this time integration will produce a term with @xmath445 . replacing in the expressions resulting from the integrations on @xmath440 and @xmath446 the constant @xmath442 from ( [ co ] ) the factors @xmath241 and @xmath411 disappear . the result is @xmath447 the objective is to calculate @xmath448 on the basis of the above results . we have to sum over any number of independent pseudomolecules of the four species @xmath449 where the factor which is associated to each pseudomolecule is @xmath450 a particular path containing @xmath451 @xmath452 and @xmath453 @xmath454 has the form @xmath455 it was taken into account that a path @xmath456 contains the same number @xmath457 of @xmath458 and of @xmath459 . the results from the previous calculations are listed below . * the probability that a particle initially at * @xmath460 * will be found at * @xmath178 @xmath461 \right\}\ ] ] * the probability that a particle initially at * @xmath460 * will be found at * @xmath462@xmath463 \right\}\ ] ] * the probability that a particle initially at * @xmath464 * will be found at * @xmath178@xmath465 \right\}\ ] ] * the probability that a particle initially at * @xmath464 * will be found at * @xmath462@xmath466 \right\}\ ] ] * the long time limit of the probability density * @xmath467 the formula has been derived for the case where the two positions @xmath335 and @xmath468 belong to the harmonic regions @xmath469 and @xmath470 . this is because asymptotically these regions will be populated . consider the case where @xmath471 the first trajectory contributing to the path integral is the direct connection between @xmath179 and @xmath143 . the contribution to @xmath115 is @xmath472 since the points of start and/or arrival @xmath473 are different of @xmath474 , there is finite ( _ i.e. _ non exponentially small ) slope of the solution , and there is nomore degeneracy with respect to the translation of the center of the instantons . then there will be not a proportionality with time in the one - instanton term . the terms connecting @xmath36 with @xmath335 with onely one intermediate step in either @xmath179 or @xmath143 have comparable contributions to the action . now the term with two intermediate steps @xmath475 in the very long time regime , @xmath476 can be factorized @xmath477 this equation is independent of @xmath478 and @xmath479 as long as @xmath480 then @xmath481\ ] ] there is also the approximate equality @xmath482 it results that all the contributions are included if in the expression of @xmath483 we replace the middle factor , which has become @xmath484 by the full @xmath485 . it is obtained @xmath486 p\left ( a,\frac{t}{2};b,-\frac{t}{2}% \right ) \\ & = & \exp \left [ -\frac{u\left ( x\right ) -u_{a}}{2d}\right ] \left ( \frac{% u_{a}^{\prime \prime } } { 2\pi d}\right ) ^{1/2}\frac{\alpha _ { b}}{\alpha _ { a}+\alpha _ { b}}\left\ { 1-\exp \left [ -t\left ( \alpha _ { a}+\alpha _ { b}\right ) \right ] \right\}\end{aligned}\ ] ] * the case where both * @xmath335 * and * @xmath36 * belong to the harmonic region * @xmath487 the same calculation shows that @xmath488 \left ( \frac{u_{b}^{\prime \prime } } { 2\pi d}% \right ) ^{1/2}\frac{1}{\alpha _ { a}+\alpha _ { b}}\left\ { \alpha _ { a}+\alpha _ { b}\exp \left [ -t\left ( \alpha _ { a}+\alpha _ { b}\right ) \right ] \right\}\ ] ] in the following we reproduce the graphs of the time - dependent probability distribution of a system governed by the basic langevin equation . various initialisations are considered , showing rapid redistribution of the density of presence of the system . the speed of redistribuition is , naturally , connected to the asymmetry of the potential . each figure consists of a set of graphs : * the potential functions @xmath489 , @xmath490 and @xmath491 * four probabilities of passage from and between the two equilibrium points @xmath143 and @xmath179 ; * the average value of the position of the system , as function of time , for the two most characteristic initializations the two parameters @xmath143 and @xmath119 take different values , for illustration . ( we apologize for the quality of the figures . better but larger ps version can be downloaded from http://florin.spineanu.free.fr/sciarchive/topicalreview.ps ) and @xmath492,width=340 ] and @xmath493,width=415 ] and @xmath494,width=415 ] and @xmath493,width=415 ] and @xmath494,width=415 ] and @xmath493,width=415 ] and @xmath494,width=415 ] and @xmath495,width=415 ] and @xmath494,width=415 ] the stochastic motion is described in terms of the _ conditional probability _ @xmath496 that the particle initially @xmath497 at @xmath498 to be at the point @xmath335 at time @xmath318 . we will use the notation that suppress the @xmath151 as the initial time , @xmath499 . the conditional probability obeys the following fokker - planck equation @xmath500 + d\frac{\partial ^{2}p}{\partial x^{2}}\ ] ] where the velocity function is here derived from the potential @xmath501@xmath502 \equiv -u^{\prime } \left ( x\right)\ ] ] since there is no drive . the initial condition for the probability function is @xmath503 the solution of the fokker - planck equation is given in terms of the following _ path - integral _ @xmath506 } { \exp \left [ -u\left ( x_{i}\right ) /2d\right ] } k\left ( x , t;x_{i}\right)\ ] ] where @xmath507 \exp \left [ -\frac{s}{d}\right ] \label{kfunc}\ ] ] where the functional integration is done over all trajectories that start at @xmath508 and end at @xmath509 . the _ action functional _ is given by @xmath510\ ] ] with the notation @xmath511 in order to calculate explicitely the functional integral we look first for the paths @xmath10 that extremize the action . they are provided by the euler - lagrange equations , which reads @xmath512 with the boundary condition @xmath513 the first thing to do after finding the extremizing paths is to calculate the _ action _ functional along them , @xmath514 . after obtaining these extremum paths we have to consider the contribution to the functional integral of the paths situated in the neighborhood and this is done by expanding the action to second order . the argument of the expansion is the difference between a path from this neigborhood and the extremum path . the functional integration over these differences can be done since it is gaussian and the result is expressed in terms of the _ determinant _ of the operator resulting from the second order expansion of @xmath51 . then the expression ( [ kfunc ] ) becomes @xmath515 where @xmath516 the constant @xmath55 is for normalization . in order to find the determinant , one has to solve the eigenvalue problem @xmath517 \phi _ { n}\left ( \tau \right ) = 0 \label{eig}\ ] ] with boundary conditions for the _ differences _ between the trajectories in the neighborhood and the extremum trajectory @xmath518 there is a fundamental problem concerning the use of the equation ( [ kexp ] ) . it depends on the possibility that all the eigenvalues from the eq.([eig ] ) are determined . however , a path connecting a point in the neigborhood of the maximum of @xmath163 , @xmath521 with a point in the neighborhood of one of the minima @xmath522 is a kinklike solution . this solution ( which will be replaced in the expression of the operator in eq.([eig ] ) contains a parameter , the `` center '' of the kink , @xmath248 . this is an arbitrary parameter since the moment of traversation is arbitrary . the equation for the eigenvalues has therefore a symmetry at time translations of @xmath248 and has as a consequence the appearence of a _ zero _ eigenvalue . then the expression of the determinant would be infinite and the rate of transfer would vanish . actually , the time translation invariance is treated by considering the arbitrary moment @xmath248 as a new variable and performing a change of variable , from the set of functions @xmath523 to the set @xmath524 . the trajectory that extremizes the action is a kinklike solution @xmath525 ( is not exactly a kink since the shape of the potential is not that which produces the @xmath83 solution ) connecting the points : @xmath526 this treatment then consists of considering @xmath248 as a collective coordinate ( * see rajaraman and coleman and gervais & sakita * ) . the new form of eq.([kexp ] ) is @xmath527 where @xmath528 is the action of the path @xmath529 . the fact that the zero eigenvalue has been eliminated is indicated by the @xmath530 . two limits of time are important . the first is @xmath532 for the times @xmath533 for a particle initially in the region of the _ unstable _ state , @xmath534 , the region around the _ stable _ state , @xmath535 , or : @xmath536 is insignificant for the calculation of the probability @xmath537 . then the expression ( [ kexp ] ) can be used . much more important is the subsequent time regime @xmath538 where the particles leave the _ unstable _ point and the equilibrium state with density around the _ stable _ positions @xmath535 is approached . then the general formula ( [ kexpbar ] ) should be used . the energy @xmath520 of the path @xmath120 connecting the points @xmath539 and @xmath509 can be approximated from the expression ( * see caroli * ) @xmath540 ^{1/2}}-\theta _ { 0}\left ( x,0\right)\ ] ] where @xmath541 the notations are @xmath542 @xmath543 the last expressions are the harmonic approximation of @xmath544 and the potential around the _ unstable _ point @xmath140 . the general expression of the function @xmath115 in the case where we include the time regimes beyond the limits given above . it has the form of a convolution @xmath555 here the approximations for the ratio @xmath556 ^{1/2}$ ] are obtained by the same method as before . the relation between the energy and the time is used for the two main regions : around the _ stable _ and the _ unstable _ ( initial ) positions @xmath557 ^{1/2}}-\theta _ { 0}\left ( x_{m},x_{i}\right)\ ] ] @xmath558 ^{1/2}}-\theta _ { 1}\left ( x , x_{m}\right)\ ] ]

we review the analytical methods of solving the stochastic equations for barrier - type dynamical behavior in plasma systems . the path - integral approach is examined as a particularly efficient method of determination of the statistical properties .

quantum coherence is one of the fundamental features which distinguish quantum world from classical realm . it is the origin for extensive quantum phenomena such as interference , laser , superconductivity @xcite and superfluidity @xcite . it is an important subject related to quantum mechanics , from quantum optics @xcite , solid state physics @xcite , thermodynamics @xcite , to quantum biology @xcite . coherence , together with quantum correlations like quantum entanglement @xcite , quantum discord @xcite , are crucial ingredients in quantum computation and information tasks @xcite . coherence shines its quantum merits in quantum metrology @xcite , quantum key distribution @xcite , entanglement creation @xcite , etc . unlike quantum entanglement and other quantum correlations , coherence , regarded as a physical resource @xcite , has been just investigated very recently in establishing the framework of quantifying coherence in the language of quantum information theory @xcite . due to its fundamental role in quantum physics and quantum information theory , it is still necessary to understand how coherence works in information processing and investigate the relations between coherence and quantum correlations . in this paper , we consider the creation of coherence of a quantum system a , which initially has zero coherence , with the help of quantum channels on another quantum system b. such creation of coherence depends on the correlations between a and b , as well as the quantum operations on system b. we establish an explicit relation among the creation of the coherence on system a , the quantum entanglement ( concurrence ) between a and b , and the quantum operations . remote creation of coherence ( rcc ) can be illustrated by a simple example . consider a two - qubit system ab , which is initially in the maximally entangled state @xmath0 . if b undergoes a projective measurement under basis @xmath1 , and tells measurement outcome , for example @xmath2 , to a , the system a s final state would be a superposition of @xmath3 and @xmath4 with some probability , if the basis @xmath2 is neither @xmath3 nor @xmath4 . the same analysis also holds for measurement outcome @xmath5 . namely , system a can gain an averaged coherence over all the outcomes . for general states , we first investigate what kind of states can be used to create remote coherence and present a necessary and sufficient condition . then for pure states , we give a necessary and sufficient condition that the operations must satisfy for nonzero rcc . and finally , we give upper bounds of rcc and investigate the relations between coherence and entanglement in our scenario . since a quantum state s coherence depends on the reference basis , throughout our paper , we fix the system a s reference basis to be the computational basis . a well defined and mostly used coherence measure is the @xmath6 norm coherence @xmath7 @xcite . the @xmath6 norm coherence of a quantum state is defined as the sum of all off - diagonal elements of the state s density matrix under the reference basis , i.e. , @xmath8 where @xmath9 is the absolute value of @xmath10 . in the following , we use the @xmath6 norm coherence when we discuss the upper bound of rcc and its relation to entanglement . let @xmath11 be a bipartite quantum state and @xmath12 a quantum operation acting on the subsystem b. let @xmath13/\operatorname{tr}[(\mathbb{i}\otimes \$ ) { \rho^{ab}}]$ ] be the reduced state of the system a after the operation . concerning the quantum operations used for remote creation of coherence with a general quantum state , we have the following theorem . [ th1 ] given a bipartite quantum state @xmath11 , for any quantum operation @xmath12 acting on the subsystem b , the coherence of the final subsystem a @xmath14 if and only if @xmath11 is an incoherent - quantum state @xmath15 . let @xmath16 be the quantum state of system ab under the computational basis , then the marginal state @xmath17 . after the quantum operation @xmath18 acting on b , @xmath19 becomes , @xmath20 where @xmath21 $ ] is the probability of getting the state @xmath22 . tracing over the system b , we get the final state of system a , @xmath23 where @xmath24 and @xmath25 . `` @xmath26 '' . from equations ( [ cl1 ] ) and ( [ mixrhoap ] ) , for any operation @xmath18 , @xmath27 means that for any @xmath28 and @xmath29 , @xmath30 , i.e. , @xmath31 for the arbitrary hermitian operator @xmath32 ( the operation @xmath18 is aribitray ) , @xmath33 and the diagonal entries of @xmath28 are arbitrary real numbers which are independent of the off diagonal entries . thus from ( [ mix1 ] ) , we get @xmath34 , @xmath35 . on the other hand , for all @xmath29 , @xmath36 set @xmath37 , where @xmath38 is the unit imaginary number . substituting it into the above equation , we get @xmath39=0 . \end{aligned}\ ] ] since @xmath40 @xmath41 is arbitrary , @xmath42 and @xmath43 are all independent . then equation ( [ mix2 ] ) implies that for all @xmath44 , @xmath45 , i.e. , @xmath46 , @xmath47 . therefore @xmath48 , @xmath49 . hence the initial state @xmath11 becomes @xmath50 which actually is an incoherent - quantum state @xcite . `` @xmath51 '' . if @xmath11 is an incoherent - quantum state , @xmath15 . it is easy to check that , for any operation @xmath18 acting on system b , the final state of the system a , @xmath13/\operatorname{tr}[(\mathbb{i}\otimes \$ ) { \rho^{ab}}]$ ] , has zero coherence . from the above proof one can see that @xmath52 also means that the initial coherence of system a is also zero , @xmath53 . the above theorem implies that any non - incoherent quantum state can be used for rcc under certain operations . note that the incoherent - quantum state actually is the classical - quantum correlated state with the fixed reference basis @xcite . thus the states , that can be used to create coherence , are not limited within entangled states . according to theorem [ th1 ] a non - incoherent - quantum separable state , with the system a having zero coherence , can also be used for rcc . interestingly , the condition in theorem [ th1 ] for creating nonzero rcc is the same as the distillable coherence of collaboration in @xcite . however , it should be noticed that our scenario is different from the asymptotic scenario in @xcite . in @xcite , they study the maximal distilled coherence of collaboration under local quantum - incoherent operations and bilateral classical communications for the asymptotic case . while our work investigates the system a s average coherence after system b going through a _ certain _ quantum channel ( see the rest of our paper ) , and the rcc in our scenario only requires one way classical communication and does not involve the maximal process @xcite . the next natural question is , for a non - incoherent quantum state , what is the exact form of the operation acting on system b for the creation of coherence ? obviously for operation @xmath54 acting on system b such that @xmath55 , where @xmath56 is an arbitrary real number bigger than 1 , the coherence can not be created , which can be seen by substituting @xmath57 into equation ( [ mixrhoap ] ) . one can check that the important quantum operations including depolarizing operations , phase flip operations , bit flip operations and bit - phase flip operations all belong to this form . moreover , all the trace preserving quantum channels will not create the coherence remotely , and measurements on the system b are necessary in order to create nonzero coherence . we give the necessary and sufficient condition for rcc in the following . let @xmath58 be a pure bipartite entangled quantum state with zero coherence of the system a. we have [ th2 ] after a quantum operation @xmath12 acts on the system b , the system a , with initial coherence being zero , gains coherence if and only if there is a computational basis @xmath59 with @xmath60\neq 0 $ ] , where @xmath61 is the initial state of ab , @xmath62 and @xmath63=ab - ba$ ] is the lie bracket . with the local computational basis , the state of system ab can be expressed as @xmath64 , with @xmath65 the normalization condition . one has @xmath66 . respect to @xmath53 , the rows of the coefficient matrix @xmath67 are mutually orthogonal . then the singular value decomposition of @xmath68 has a simple form , @xmath69 , where @xmath70 is a diagonal matrix with nonzero singular values @xmath71 ( zero singular values are trivial for our proof ) , @xmath72 is a unitary matrix . thus we get the schmidt decomposition , @xmath73 , @xmath74 . we also have @xmath75 . after the local operation acting on the system b , the final state of the system a has the form , @xmath76/p ' \\ = & \operatorname{tr}_b [ ( \mathbb{i}\otimes \$)\sum _ { ij } \sqrt{\omega_i\omega_j } |i\ra \la j|\otimes |\beta_i\ra \la \beta_j|]/p ' \\ = & \sum_{ij } \sqrt{\omega_i\omega_j } |i\ra \la j| \operatorname{tr}[\sum_n f_n(|\beta_i\ra \la \beta_j|)f_n^\dagger]/p'\\ = & \sum_{ij } \sqrt{\omega_i\omega_j } n_{ji } |i\ra \la j|/p ' , \end{aligned}\ ] ] where @xmath77 , @xmath78 and @xmath79 $ ] , the probability of getting the state @xmath80 . since @xmath81 @xmath82 , all the off diagonal entries of @xmath80 vanish if and only if @xmath83 , which means that @xmath84s are just the eigenvectors of @xmath28 . equivalently , @xmath85=0 $ ] , @xmath82 . thus @xmath86 if and only if there is a computational basis @xmath59 such that @xmath85\neq 0 $ ] , where @xmath87 . for a pure bipartite entangled state @xmath88 with a s initial coherence being zero , we have presented an operational way to determine whether @xmath12 on system b can create system a s coherence . in fact , from the proof we have also given the explicit form of the quantum operation that can not create coherence of system a. such operation @xmath28 has the form : @xmath89 , where @xmath90s are arbitrary real numbers in @xmath91 $ ] , @xmath92 . theorem [ th2 ] shows the necessary and sufficient condition for quantum operations that create nonzero coherence . now we study how much rcc can be created and the relation between rcc and the entanglement between a and b. [ le1 ] under a quantum operation @xmath93 , the remote created coherence for a pure bipartite state @xmath88 is bounded by @xmath94 where @xmath95 is the @xmath6 norm coherence , @xmath96 is the entanglement measure , concurrence , @xmath97 is the probability of getting the state @xmath80 , @xmath98 and @xmath99 is the matrix elements under @xmath88 s schmidt decomposition s basis of system b. from equations ( [ cl1 ] ) and ( [ purerhoap ] ) we have , @xmath100 by cauchy inequality @xmath101 , we have @xmath102 on the other hand , the concurrence of the state ( [ psiab ] ) is given by @xmath103)}=\sqrt{2 \sum_{i\neq j } \omega_i \omega_j}$ ] , which together with ( [ crhoap ] ) and ( [ neq1 ] ) complete the proof . under the operation @xmath18 acting on the system b , one interesting thing is that the upper bound of rcc ( [ bound1 ] ) is proportional to the entanglement while inversely proportional to the probability of getting the final state . here , one should note that @xmath99s depend on the local schmidt decomposition basis , hence the bound ( [ bound1 ] ) not only depends on the entanglement between a and b , but also the local basis of the system b , as the coherence is a basis dependent quantity . next , we investigate the relations among the average rcc under an operation on the system b for maximally entangled states , that for an arbitrarily given pure quantum state and the entanglement of this given state . consider a pure bipartite entangled state @xmath88 and a trace preserving quantum channel @xmath104 , @xmath105 . under the channel @xmath18 on the system b , the state @xmath88 becomes @xmath106 with probability @xmath107 $ ] . bob communicates each outcome to alice such that , under the channel @xmath18 , alice can gain an average coherence over all the outcomes , @xmath108/p_n')$ ] . [ th3 ] for a pure bipartite entangled state @xmath88 with zero coherence of system a , under a trace preserving quantum channel @xmath104 on system b , the average created coherence of system a satisfies the following relation : @xmath109 where @xmath110 is the maximal entangled state in the schmidt decomposition basis of @xmath88 , @xmath111 and @xmath112 are the average coherence of systems a under the channel @xmath18 for states @xmath88 and @xmath110 respectively , @xmath95 is the @xmath6 norm coherence , @xmath113 is the dimension of system a and @xmath96 is the concurrence . by setting @xmath114 , @xmath115 , in ( [ psiab ] ) we get the maximally entangled state with respect to @xmath88 s schmidt basis , latexmath:[\[\label{beta } @xmath113 is the dimension of system a. under the channel @xmath18 on system b , the final states of system a corresponding to @xmath88 and @xmath110 are given by @xmath117/p_n'\ ] ] and @xmath118/p_n'',\ ] ] where @xmath107 $ ] and @xmath119 $ ] are the probabilities of getting the states @xmath120 and @xmath121 respectively . employing equation ( [ purerhoap ] ) , we get the coherence @xmath122 and @xmath123 where @xmath62 and @xmath124 . utilizing the following relation @xmath125 and lemma [ le1 ] , we get the relation @xmath126 since @xmath18 is trace preserving , the average coherence satisfies @xmath127 which gives rise to the relation ( [ bound2 ] ) . theorem 3 shows that for a bipartite state @xmath88 with initial zero coherence of system a , under a tracing preserving channel of system b , the average increasing of the coherence of system a is bounded by the entanglement between systems a and b , and the average coherence of system a for the maximally entangled state @xmath110 under the same channel for the system b. in fact , ( [ bound2 ] ) is also valid for the system going through some non trace - preserving channels with certain probabilities . let @xmath128 be a set of quantum operations such that @xmath129 , i.e. , each @xmath130 is not a trace - preserving channel , but all @xmath130 together is . if the system b of a given state @xmath61 and the maximally entangled state @xmath110 go through the operation @xmath130 with probability @xmath131 $ ] and @xmath132 $ ] , respectively , then one can prove that the average coherence @xmath133/p_k')$ ] and @xmath134/p_k'')$ ] satisfy the relation ( [ bound2 ] ) . here , if each @xmath130 is given by one kraus operator , @xmath135 , then one recovers the result for a trace preserving channel . it should be noticed here that for the remote creation of the averaged coherence , the one - way classical communication is required . otherwise , one would end up with super - luminal signaling . _ remark _ averaging the @xmath130s for equation ( [ bound1 ] ) we can also obtain a tighter bound for the average rcc . @xmath136 where @xmath137 and @xmath138 , with @xmath139 as given in ( [ beta ] ) . this inequality provides a tighter bound than ( [ bound2 ] ) and could be used to estimate the average rcc for a pure entangled state and a set of operations . [ fig1 ] pure states with the form of equation ( [ psiab ] ) in @xmath140 system , of which the average rccs are under the phase damping channel @xmath141 acting on the system b. the average rccs , @xmath111 ( blue dots ) , get larger as the phase damping rate @xmath142 increases ( the different darkness of blue dots stands for different values of @xmath142 ) , while the ratio between the average rcc of a state and the average rcc of its corresponding maximal entangled state , @xmath143 ( red dots ) , is always equal to its entanglement for all @xmath142.,title="fig:",width=264 ] nevertheless , the theorem [ th3 ] gives a more explicit relation among the entanglement , rccs of the given state and the corresponding maximally entangled state . in particular , for two - qubit case , we have the following theorem : [ th4 ] for a pure @xmath144 entangled state @xmath88 with zero coherence of system a , under a trace preserving quantum channel @xmath104 on system b , the average rcc of @xmath88 equals to the product of the entanglement of @xmath88 and the average rcc of @xmath88 s corresponding maximally entangled state @xmath110 , i.e. , @xmath145 the proof for the above can be derived by noting that , in @xmath144 case , the summation in ( [ neq1 ] ) and ( [ neq2 ] ) only contains one entry which leads to that the equality holds in theorem [ th3 ] . the above factorization law manifests that , in a @xmath140 system , if the operations satisfy the nonzero rcc condition in theorem [ th2 ] , then the average rcc of the state is proportional to its entanglement . since the entanglement is smaller than 1 , its average rcc is always smaller than its corresponding maximal entangled state s average rcc . fig . 1 shows the relations between the average rcc and entanglement for @xmath144 pure states under phase damping channel . for higher dimensional systems , while the upper bound of the average rcc for a state is proportional to its entanglement , depending on the operations one chooses , its average rcc can exceeds its corresponding maximally entangled state s average rcc . here it should be noted that the coherence depends on the reference basis , while the entanglement is local unitary invariant . hence we fix the same basis for the maximally entangled state @xmath110 to the one of @xmath88 s schmidt basis . besides quantum entanglement , similar relations like ( [ bound2 ] ) or ( [ bound3 ] ) may also exist for other quantum correlations like quantum discord , as theorem [ th1 ] implies that all the non - incoherent - quantum states are useful for rcc , other quantum correlations could be responsible for rcc either . in conclusion , we have studied the coherence creation for a system a with zero initial coherence , with the help of quantum operations on another system b that is correlated to a and one - way classical communication . we have found that all the non - incoherent quantum states can be used for rcc and all the incoherent - quantum states can not . for pure states , the necessary and sufficient condition of rcc for the quantum operations on system b has been presented . the upper bound of average remote created coherence has been derived , which shows the relation among the entanglement and rcc of the given quantum state , and the rcc of the corresponding maximally entangled state . moreover , for two - qubit systems , a simple factorization law for the average remote created coherence has been given . * acknowledgments * we thank b. chen , y. k. wang and s. h. wang for useful discussions . this work is supported by the nsfc under numbers 11401032 , 11275131 , 11175094 , 11675113 and 91221205 , and the national basic research program of china ( 2015cb921002 ) . f. london and h. london , proc . a * 149 * , 71 ( 1935 ) . a. b. migdal , nucl . phys . * 13 * , 655 ( 1959 ) . l. mandel and e. wolf , _ optical coherence and quantum optics _ ( cambridge university press , cambridge , england , 1995 ) . j. aberg , phys . . lett . * 113 * , 150402 ( 2014 ) . g. s. engel , t. r. calhoun , e. l. read , t. k. ahn , t. manal , y. c. cheng , r. e. blakenship , and g. r. fleming , natrue ( london ) * 446 * , 782 ( 2007 ) . r. horodecki , p. horodecki , m. horodecki , and k. horodecki , rev . phys . * 81 * , 865 ( 2009 ) . k. modi , a. brodutch , h. cable , t. paterek , and v. vedral , rev . phys . * 84 * , 1655 ( 2012 ) . m. a. nielsen and l. chuang , _ quantum computation and quantum information _ ( cambridge university press , cambridge , england , 2000 ) . v. giovannetti , s. lloyd , and l. maccone , science * 306 * , 1330 ( 2004 ) . r. demkowicz - dobrzanski and l. maccone , phys . lett . * 113 * , 250801 ( 2014 ) . t. sashki , y. yamamoto , and m. koashi , natrue * 509 * , 475 ( 2014 ) . j. k. asboth , j. calsamiglia , and h. ritsch , phys . rev . lett . * 94 * , 173602 ( 2005 ) . a. streltsov , u. singh , h. s. dhar , m. n. bera , and g. adesso , phys . lett . * 115 * , 020403 ( 2015 ) . t. baumgratz , m. cramer , and m. b. plenio , phys . lett . * 113 * , 140401 ( 2014 ) . t. r. bromley , m. cianciaruso , and g. adesso , phys . lett . * 114 * , 210401 ( 2015 ) . e. chitambar , a. streltsov , s. rana , m. n. bera , g. adesso , and m. lewenstein , phys . lett . * 116 * , 070402 ( 2016 ) . y. yao , x. xiao , l. ge , and c. p. sun , phys . a * 92 * , 022112 ( 2015 ) . m. n. bera , t. qureshi , m. a. siddiqui , and a. k. pati , phys . a * 92 * , 012118 ( 2015 ) . d. girolami , phys . * 113 * , 170401 ( 2014 ) . a. mani and v. karimipour , phys . a * 92 * , 032331 ( 2015 ) . x. yuan , h. zhou , z. cao , and x. f. ma , phys . a * 92 * , 022124 ( 2015 ) .

we study remote creation of coherence ( rcc ) for a quantum system a with the help of quantum operations on another system b and one way classical communication . we show that all the non - incoherent quantum states are useful for rcc and all the incoherent - quantum states are not . the necessary and sufficient conditions of rcc for the quantum operations on system b are presented for pure states . the upper bound of average rcc is derived , giving a relation among the entanglement ( concurrence ) , the rcc of the given quantum state and the rcc of the corresponding maximally entangled state . moreover , for two - qubit systems we find a simple factorization law for the average remote created coherence .

this work is part of the research program of the stichting voor fundamenteel onderzoek der materie ( fom ) , which is financially supported by the nederlandse organisatie voor wetenschappelijk onderzoek ( nwo ) . de a. acknowledges support from the spanish mec ( mat2010 - 14885 and consolider nanolight.es ) . f. lpez - tejeira , s. g. rodrigo , l. martn - moreno , f. j. garca - vidal , e. devaux , t. w. ebbesen , j. r. krenn , i. p. radko , s. i. bozhevolnyi , m. u. gonzlez , j. c. webber , and a. dereux , nature phys . * 3 * , 324 ( 2007 ) .

we map the complex electric fields associated with the scattering of surface plasmon polaritons by single sub - wavelength holes of different sizes in thick gold films . we identify and quantify the different modes associated with this event , including a radial surface wave with an angularly isotropic amplitude . this wave is shown to arise from the out - of - plane electric dipole induced in the hole , and we quantify the corresponding polarizability , which is in excellent agreement with electromagnetic theory . time - resolved measurements reveal a time - delay of @xmath0fs between the surface plasmon polariton and the radial wave , which we attribute to the interaction with a broad hole resonance . the interaction of light , and in particular of surface waves such as surface plasmon polaritons ( spps ) , with sub - wavelength holes in metallic films leads to phenomena such as extraordinary optical transmission ( eot ) @xcite and negative index metamaterials @xcite . in this context , sub - wavelength holes are important in diverse fields of research , such as nanophotonics , enhanced nonlinear optics , and biosensing , and consequently , with a few notable exceptions @xcite , experiments focus on the macroscopic consequences of the light - hole interactions . conversely , much theoretical work has been devoted to their understanding from a microscopic viewpoint @xcite . that is , while the models intrinsically consider scattering of light and ssps from sub - wavelength defects , often in terms of their polarizabilities , most experimental studies address collective response features such as eot spectra and their dependence on changes in sample geometry , film thickness @xcite , and hole shape @xcite . in essence , a detailed investigation of the dynamics of plasmonic scattering remains incomplete . in this letter , we present a systematic experimental study of the scattering of spps from individual circular sub - wavelength holes of different diameters in optically thick gold films . we use phase - sensitive time - resolved near - field optical observations to measure the plasmonic scattering on a nanometer length- and femtosecond time - scale , determining the angularly resolved scattering amplitude for sub - wavelength holes of different sizes and quantifying the temporal dynamics of the scattering events . we show that these results can be explained in terms of the polarizability of the hole , in excellent agreement with electromagnetic theory . we investigate the scattering from individual holes , with diameters ranging from 50 to 980 nm , milled with a focused ion beam into a 200 nm thick gold film deposited on a glass substrate . we launch spps on the air - gold interface by illuminating a slit in the film with either 150 femtosecond fwhm pulses or a continuous wave laser source centered at a wavelength of 1550 nm . at this wavelength , a waveguide mode is supported by holes with diameters above 760 nm . these spps are efficiently directed towards the hole by a bragg grating that is milled into the film on the opposite side of the slit @xcite , as shown in fig . [ fig : sample ] . + we measure the complex electric field of this system , at a height of 20 nm , with near field microscopy @xcite . in fig . [ fig : example ] we present typical experimental results for a hole with a diameter of 875 nm . + we observe fringes in the amplitude that originate from the interference of the forward propagating spp with the scattered waves . here , the dark vertical stripes near 5 and 8 @xmath1 m are attributed to geometrical imperfections in the area of the launching slit . the salient features of the data are reproduced by first - principle calculations [ fig . [ fig : example](b ) ] performed using a modified version of a model presented elsewhere @xcite . this method , which involves a rigorous expansion of maxwell s equations , both in terms of plane waves on either side of the film and in cylindrical waves inside the hole , is used to calculate the electric and magnetic polarizabilities of the hole , which are in turn used to obtain the scattered fields @xcite . the different waves present in the scattering event are readily identifiable in _ k_-space @xcite after a fourier transform [ fig . [ fig : example](c ) ] . in _ k_-space , signatures of the forward propagating spp , evidenced by the peak near @xmath2 , and the scattered wave , which emerges as a circular pattern , are readily separable . these signatures are shown in fig . [ fig : example](d ) , in which both the data ( thick shaded curves ) and the fits for the different surface waves ( thin dark curves ) are presented . the distributions of both waves peak around a wave vector @xmath3 @xmath1m@xmath4 , which is in good agreement with the theoretical value of @xmath5m@xmath4 , obtained from the dielectric constant of gold at 1550 nm , @xmath6 @xcite . in the inset of fig . [ fig : example](d ) , we show the angular dependence of the in - plane electric field distribution . here , the outer curve represents the total integrated field amplitude while the inner curve represents the integrated amplitude that we can attribute to the surface waves . we find these curves by integrating up to @xmath7 @xmath1m@xmath4 under the data and fits , respectively , in each direction ( main part of the figure ) . first , we observe no angular dependence of the radial mode and attribute the slight left - right asymmetry found in our measurements to either a slightly asymmetric hole or near - field tip . second , for this relatively large hole the amplitude content of the radial wave is about 7@xmath8 of the amplitude of the forward propagating spp . as we show below ( fig . [ fig : size ] ) , for an 875 nm hole @xmath9 of the forward propagating spp amplitude is contained in the incident beam , and hence the amplitude of the radial wave is @xmath10 that of the _ incident _ spp . note that our incident gaussian beam is @xmath11 m width , which is larger than the hole ( see @xcite for a discussion on the scattering cross section of the hole . finally , from this data we can also estimate that approximately half of the radially scattered field amplitude is in the surface modes . the remaining half of the radial fields have in - plane wave vectors that are smaller than @xmath12 @xmath1m@xmath4 , corresponding to radiative modes of light , which propagate upward , away from the metal surface . qualitatively , these are seen as the regions of non - zero amplitude within the surface wave circle in fig . [ fig : example](c ) , or in the difference between the inner and outer curves ( other than near @xmath13 ) in fig . [ fig : example](d ) . in order to determine the nature of the observed scattered field , we consider its asymptotic form @xcite @xmath14 where the azimuthal scattering amplitude @xmath15 is given for excitation by a spp that travels along the @xmath16 direction . here , @xmath17 is the in - plane radial distance to the hole center , @xmath18 is the height above the film surface , @xmath19 is the out - of - plane component of the spp wave vector , @xmath20 and @xmath21 are out - of - plane and in - plane unit vectors , @xmath22 and @xmath23 are the in- and out - of plane electric dipoles , and @xmath24 is the in - plane magnetic dipole of the hole . consequently , the isotropic nature of the detected scattered field [ inset to fig . [ fig : example](d ) ] suggests that we are mostly sensitive to the contribution from @xmath23 . this is not entirely unexpected since the contributions of other dipole components peak near @xmath25 and @xmath26 and are therefore hard to separate from the incident spp field . however , we note that to accurately reproduce the field in all directions [ fig . [ fig : example](b ) ] the fields that arise due to @xmath24 , as well as those due to @xmath23 , are required @xcite . we image the @xmath23 dependent electric field by filtering out the forward propagating spp mode in _ k_-space and then fourier transforming back to real - space . in fig . [ fig : radial ] we show this field , both near @xmath27 where only a contribution from @xmath23 is expected , and in a 20 @xmath1m@xmath28 area around the hole ( inset ) . fit . the inset shows the two - dimensional image of the real part of the electric field , and the arrow indicates the line along which the data shown in the main part of the figure is taken.,title="fig:",width=313 ] + a fit of the form @xmath29 reproduces the experimental data with @xmath30 @xmath1m@xmath4 and @xmath31 . hence , the measured amplitude damping is in excellent agreement with the @xmath32 behavior predicted for circular waves by eq . ( [ eq : asympte ] ) , as required for energy conservation for radial propagation on a plane . the discrepancy between the data and fit close to the hole may be indicative of the presence of creeping waves , which can have substantial amplitudes at short distances @xcite . we can also use eq . ( [ eq : asympte ] ) to express the ratio of the scattered field to the incident field for @xmath33 in terms of the electric polarizability @xmath34 hence , we measure @xmath35 m@xmath36 , which is larger than , but comparable to , the value @xmath37 m@xmath36 calculated for a hole in a perfectly conducting film @xcite . in fig . [ fig : temporal ] we present the temporal dynamics of plasmonic scattering , showing the amplitudes of both the forward and scattered radial modes as a function of time . fs.,title="fig:",width=313 ] + the forward amplitude is averaged over a range of @xmath38 while the scattered amplitude is averaged over a similar range in the transverse directions ( @xmath39 ) to ensure no overlap between the modes . both amplitudes are normalized to the peak of the forward propagating spp , and the second peak of the double peak results from spps that are initially launched backward and subsequently reflect from the bragg grating towards the hole @xcite . most strikingly , we observe that the radial wave peaks one time - step _ after _ the forward propagating spp : there is a time - delay between the forward propagating spp pulse and the radial pulse resulting from the plasmonic scattering . by averaging over the time - delay that we measure between incident spp and the scattered wave , in all available directions , we quantify this shift as @xmath40fs . while the frequency dependence of the polarizability of the hole alone results in a shift only a few fs , it is able to explain the observed shift when combined with a modestly chirped incident pulse . that is , since the different plane waves that form our pulse scatter from the hole with different amplitudes and phases , their reconstitution into a beam results in the observed time shift . this shift , then , represents a first measurement of the complex spectral response of an individual hole . we study the effect of hole size on the plasmonic scattering for hole sizes ranging from 50 to 980 nm . the results are summarized in fig . [ fig : size ] . ( and hence the calculated scattered amplitude ) . inset : fraction of field contained in the forward propagating spp peak , normalized to unity for a featureless film , as a function of hole size . the vertical dashed lines show the cutoff of the lowest - order guided mode in a deep hole for 1550 nm light.,title="fig:",width=313 ] + the inset depicts the fraction of the field amplitude that is contained in the forward propagating spp mode , which we normalize to the field on films without a hole . as expected , less energy is found in this plasmonic wave as the hole size increases and hence scatters more energy . the main part of fig . [ fig : size ] depicts the dependence of the scattered field amplitude of the radial wave normalized to that of the forward plasmon field ( symbols ) on the hole diameter , @xmath41 . this dependence is accurately reproduced by the calculated electric polarizability of the hole ( curve ) @xcite . an offset of 0.017 is present in the data ( and added in the calculations ) due to measurement noise , and represents a signal - to - noise ratio of about 50 to 1 , while the error bar is mainly indicative of the asymmetry of our measurements . for the smaller holes , well below the cutoff , the field amplitude increases as @xmath42 , where the divergence from a perfect dipolar behavior is due to the finite size of the hole and the film thickness . as the hole size increases , and in particular once the cutoff is reached and waveguide modes in the hole become accessible , the scattered amplitude begins to saturate . we attribute this behavior to an increase in energy flow through the hole @xcite . finally , we note that the scattered amplitude peaks at about 10@xmath8 of the forward propagating spp amplitude . to summarize , we have captured the near - field interplay between the waves associated with plasmon scattering from sub - wavelength holes and mapped both their dependence on hole size and their temporal dynamics . we show that both the hole size dependent amplitude and the time - delay of the scattered mode can be understood in terms of the calculated polarizability of the hole . our results provide a comprehensive understanding of hole - plasmon scattering and should , in the future , allow for the optimization of the interaction between holes . consequently , the geometric properties of hole arrangements can be tuned to create electromagnetic hot spots . these can be used to , for example , enhance emission from dye molecules , address individual quantum emitters , or create near field distributions for sensing or imaging purposes . hence , these results have implications for a broad range of nanoplasmonic applications .

the cabibbo - kobayashi - maskawa ( ckm ) matrix element @xmath8 , the coupling of the @xmath9 quark to the @xmath10 quark , is a fundamental parameter of the standard model . it is one of the smallest and least known elements of the ckm matrix . with the increasingly precise measurements of decay - time - dependent @xmath11 asymmetries in @xmath12-meson decays , in particular the angle @xmath13 @xcite , improved measurements of the magnitude of @xmath8 will allow for stringent experimental tests of the standard model mechanism for @xmath11 violation @xcite . this is best illustrated in terms of the unitarity triangle , the graphical representation of the unitarity condition for the ckm matrix , for which the length of the side that is opposite to the angle @xmath13 is proportional to @xmath14 . the extraction of @xmath14 is a challenge , both theoretically and experimentally . experimentally , the principal challenge is to separate the signal @xmath15 decays from the 50 times larger @xmath16 background . this can be achieved by selecting regions of phase space in which this background is highly suppressed . in the rest frame of the @xmath12 meson , the kinematic endpoint of the electron spectrum is @xmath17 for the dominant @xmath18 decays and @xmath19 for @xmath20 decays . thus the spectrum above 2.3 @xmath21is dominated by electrons from @xmath20 transitions . this allows for a relatively precise measurement , largely free from background , in a 300 @xmath22interval that covers approximately 10% of the total electron spectrum for charmless semileptonic @xmath12 decays . in the @xmath23 rest frame , the finite momenta of the @xmath12 mesons cause additional spread of the electron momenta of @xmath24@xmath22 , extending the endpoints to higher momenta . the weak decay rate for @xmath25 can be calculated at the parton level . it is proportional to @xmath26 and @xmath27 , where @xmath28 refers to the @xmath9-quark mass . to relate the semileptonic decay rate of the @xmath12 meson to @xmath14 , the parton - level calculations have to be corrected for perturbative and non - perturbative qcd effects . these corrections can be calculated using various techniques : heavy quark expansions ( hqe ) @xcite and qcd factorization @xcite . both approaches separate perturbative from non - perturbative expressions and sort terms in powers of @xmath29 . hqe is appropriate for the calculations of total inclusive @xmath12 decay rates and for partial @xmath12 decay rates integrated over sufficiently large regions of phase space where the mass and momentum of the final state hadron are large compared to @xmath30 . qcd factorization is better suited for calculations of partial rates and spectra near the kinematic boundaries where the hadronic mass is small . in this region the spectra are affected by the distribution of the @xmath9-quark momentum inside the @xmath12 meson @xcite , which can be described by a structure or shape function ( sf ) , in addition to weak annihilation and other non - perturbative effects . extrapolation from the limited momentum range near the endpoint to the full spectrum is a difficult task , because the sf can not be calculated . to leading order , the sf should be universal for all @xmath31 transitions ( here @xmath32 represents a light quark ) @xcite . several functional forms for the sf , which generally depend on two parameters related to the mass and kinetic energy of the @xmath9-quark , @xmath33 or @xmath28 , and @xmath34 or @xmath35 , have been proposed . the values and precise definitions of these parameters depend on the specific ansatz for the sf , the mass renormalization scheme , and the renormalization scale chosen . in this paper , we present a measurement of the inclusive electron momentum spectrum in charmless semileptonic @xmath12 decays , averaged over charged and neutral @xmath12 mesons , near the kinematic endpoint . we report measurements of the partial branching fractions in five overlapping momentum intervals . the upper limit is fixed at 2.6 @xmath21 , while the lower limit varies from 2.0 @xmath21to 2.4 @xmath21 . by extending the interval for the signal extraction down to 2.0 @xmath21 , we capture about 25% of the total signal electron spectrum , but also much larger @xmath36 backgrounds . inclusive measurements of @xmath14 have been performed by several experiments operating at the @xmath37 resonance , namely argus @xcite , cleo @xcite , @xcite , and belle @xcite , and experiments operating at the @xmath38 resonance , namely l3 @xcite , aleph @xcite , delphi @xcite , and opal @xcite . this analysis is based on a method similar to the one used in previous measurements of the lepton spectrum near the kinematic endpoint @xcite . the results presented here supersede those of the preliminary analysis reported by the collaboration @xcite . the extraction of @xmath14 relies on two different theoretical calculations of the differential decay rates for @xmath39 and @xmath5 : the original work by defazio and neubert ( dn ) @xcite , and kagan and neubert @xcite , and the more comprehensive recent calculations by bosch , lange , neubert , and paz ( blnp ) @xcite . the dn calculations allow for the extrapolation of the observed partial @xmath0 decay rate above a certain electron momentum to the total inclusive @xmath0 decay rate using the measured sf parameters and a subsequent translation of the total decay rate to @xmath14 . the theoretical uncertainties on the rate predictions are estimated to be of order 1020% . the blnp authors have presented a systematic treatment of the sf effects , incorporated all known corrections to the differential decay rates , and provided an interpolation between the hqe and the sf regions . they have also performed a detailed analysis of the theoretical uncertainties . the calculations directly relate the partial decay rate to @xmath14 . while the calculations by blnp are to supersede the earlier work by dn , we use both approaches to allow for a direct comparison of the two calculations , and also a comparison with previous measurements based on the dn calculations . we adopt the sf parameters extracted by the collaboration : for the dn method we rely on the photon spectrum in @xmath5 decays @xcite ; for the more recent blnp method , we also use sf parameters derived from the photon spectrum , its moments , the hadron - mass and lepton - energy moments in inclusive @xmath6 decays @xcite , and the combination of all moments measured by the collaboration @xcite . the data used in this analysis were recorded with the detector at the 2 energy - asymmetric @xmath40 collider . the data sample of 88 million events , corresponding to an integrated luminosity of 80.4 @xmath41 , was collected at the resonance . an additional sample of 9.5 @xmath41 was recorded at a center - of - mass ( c.m . ) energy 40 @xmath42below the resonance , _ i.e. _ just below the threshold for production . this off - resonance data sample is used to subtract the non- contributions from the data collected on the resonance . the relative normalization of the two data samples has been derived from luminosity measurements , which are based on the number of detected @xmath43 pairs and the qed cross section for @xmath44 production , adjusted for the small difference in center - of - mass energy . the detector has been described in detail elsewhere @xcite . the most important components for this study are the charged - particle tracking system , consisting of a five - layer silicon detector and a 40-layer drift chamber , and the electromagnetic calorimeter assembled from 6580 csi(tl ) crystals . these detector components operate in a @xmath45-@xmath46 solenoidal magnetic field . electron candidates are selected on the basis of the ratio of the energy detected in the calorimeter to the track momentum , the calorimeter shower shape , the energy loss in the drift chamber , and the angle of the photons reconstructed in a ring - imaging cherenkov detector . the electron identification efficiency and the probabilities to misidentify a pion , kaon , or proton as an electron have been measured @xcite as a function of the laboratory momentum and angles with clean samples of tracks selected from data . within the acceptance of the calorimeter , defined by the polar angle in the laboratory frame , @xmath47 , the average electron identification efficiency is @xmath48 . the average hadron misidentification rate is about 0.1% . we use monte carlo ( mc ) techniques to simulate the production and decay of @xmath12 mesons , and the detector response @xcite , to estimate signal and background efficiencies , and to extract the observed signal and background distributions . the simulated sample of generic events exceeds the data sample by about a factor of three . information from studies of selected control data samples on efficiencies and resolutions is used to improve the accuracy of the simulation . comparisons of data with the mc simulations have revealed small differences in the tracking efficiencies , which have been corrected for . no significant impact of non - gaussian resolution tails has been found for high momentum tracks in the endpoint region . the mc simulations include radiative effects such as bremsstrahlung in the detector material and qed initial and final state radiation @xcite . adjustments for small variations of the beam energy over time have also been included . in the mc simulations the branching fractions for hadronic @xmath12 and @xmath49 decays are based on values reported in the review of particle physics @xcite . the simulation of charmless semileptonic decays , @xmath50 , is based on a heavy quark expansion to @xmath51 @xcite . this calculation produces a continuous spectrum of hadronic states . the hadronization of @xmath52 with masses above @xmath53 is performed by jetset @xcite . the motion of the @xmath9 quark inside the @xmath12 meson is implemented with the sf parameterization given in @xcite . three - body decays to low - mass hadrons , @xmath54 , are simulated separately using the isgw2 model @xcite and mixed with decays to non - resonant and higher mass resonant states @xmath55 , so that the cumulative distributions of the hadron mass , the momentum transfer squared , and the electron momentum reproduce the hqe calculation as closely as possible . the generated electron spectrum is reweighted to accommodate variations due to specific choices of the sf parameters . the mc - generated electron - momentum distributions for @xmath20 decays are shown in fig . [ f : bu_pe ] , for individual decay modes and for their sum . here and throughout the paper , the electron momentum and all other kinematic variables are measured in the rest frame , unless stated otherwise . above 2@xmath21 , the principal signal contributions are from decays involving the light mesons @xmath56 , and @xmath57 , and also some higher mass resonant and non - resonant states @xmath55 . decays : @xmath58 , @xmath59 , @xmath60 , @xmath61 , @xmath62 , the sum of @xmath12-meson decay modes to non - resonant and higher - mass resonance states ( @xmath63 ) , and the sum of all decay modes ( all ) . the spectra are normalized to a total rate of 1.0 . , height=245 ] for the simulation of the dominant @xmath64 decays , we have chosen a variety of models . for @xmath65 and @xmath66 decays we use parameterizations @xcite of the form factors , based on heavy quark effective theory ( hqet ) . decays to pseudoscalar mesons are described by a single form factor @xmath67 , where the variable @xmath68 is the scalar product of the @xmath12 and @xmath49 meson four - vector velocities and is equal to the relativistic boost of the @xmath49 meson in the @xmath12 meson rest frame . the linear slope @xmath69 has been measured by the cleo @xcite and belle @xcite collaborations . we use the average value , @xmath70 . the differential decay rate for @xmath71 can be described by three amplitudes , which depend on three parameters : @xmath72 , @xmath73 , and @xmath74 . we adopt values recently measured by @xcite : @xmath75 , @xmath76 , and @xmath77 . here the parameter @xmath72 is the slope assuming a linear dependence of the form factor on the variable @xmath68 . the quoted errors reflect the statistical and systematic uncertainties . we use the isgw2 @xcite model for various decays to higher - mass @xmath78 resonances . we have adopted a prescription by goity and roberts @xcite for the non - resonant @xmath79 decays . the shapes of the mc - generated electron spectra for individual @xmath64 decays are shown in fig . [ f : sp1 ] . above 2 @xmath21the principal background contributions are from decays involving the lower - mass charm mesons , @xmath80 and @xmath49 . higher - mass and non - resonant charm states are expected to contribute at lower electron momenta . the relative contributions of the individual @xmath64 decay modes are adjusted to match the data by a fit to the observed spectrum ( see below ) . decay modes : @xmath81 , @xmath82 , @xmath83 , @xmath84 , and @xmath20 , and the sum of all decay modes ( all ) . the signal @xmath50 spectrum is shown for comparison . the spectra are normalized to a total rate of 1.0 . , height=245 ] we select events with a semileptonic @xmath12 decay by requiring an electron with momentum @xmath85 . to reject electrons from the decay @xmath86 , we combine the electron candidate with any second electron of opposite charge and reject the combination , if the invariant mass of the pair falls in the interval @xmath87 . to suppress background from non- events , primarily low - multiplicity qed ( including @xmath88 pairs ) and @xmath89 processes ( here @xmath32 represents any of the @xmath90 or @xmath91 quarks ) , we veto events with fewer than four charged tracks . we also require that the ratio of the second to the zeroth fox - wolfram moment @xcite , @xmath92 , not exceed @xmath93 . @xmath92 is calculated including all detected charged particles and photons . for events with an electron in the momentum interval of 2.0 to 2.6 @xmath21 , these two criteria reduce the non- background by a factor of about 6 , while the loss of signal events is less than 20% . in semileptonic @xmath12 decays , the neutrino carries sizable energy . in events in which the only undetected particle is this neutrino , the neutrino four - momentum can be inferred from the missing momentum , @xmath94 , the difference between the four - momentum of the two colliding - beam particles , and the sum of the four - momenta of all detected particles , charged and neutral . to improve the reconstruction of the missing momentum , we impose a number of requirements on the charged and neutral particles . charged tracks are required to have a minimum transverse momentum of 0.2 @xmath21and a maximum momentum of 10 @xmath21 in the laboratory frame . charged tracks are also restricted in polar angle to @xmath95 and they are required to originate close to the beam - beam interaction point . the individual photon energy in the laboratory frame is required to exceed 30 @xmath42 . the selection of semileptonic @xmath12 decays is enhanced by requiring @xmath96 , and that @xmath97 points into the detector fiducial volume , @xmath98 , thereby effectively reducing the impact of particle losses close to the beams . furthermore , since in semileptonic @xmath12 decays with a high - momentum electron , the neutrino and the electron are emitted preferentially in opposite directions , we require that the angle @xmath99 between these two particles fulfill the condition @xmath100 . these requirements for the missing momentum reduce the continuum background from qed processes and @xmath89 production by an additional factor of 3 , while the signal loss is less than 20% . the stated selection criteria result in an efficiency ( including effects of bremsstrahlung ) of @xmath101% for selecting @xmath50 decays ; its dependence on the electron momentum is shown in fig . [ fig : p0 ] . decays as a function of the electron momentum . the error bars represent the statistical errors . , height=264 ] the spectrum of the highest momentum electron in events selected by the criteria described above is shown in fig . [ fig : p1]a , separately for data recorded on and below the resonance . the data collected on the @xmath23 resonance include contributions from events and non- background . the latter is measured using off - resonance data , collected below production threshold , and using on - resonance data above 2.8 @xmath21 , _ i.e. , _ above the endpoint for electrons from @xmath12 decays . the background to the @xmath102 spectrum is estimated from mc simulation , with the normalization of the individual contributions determined by a fit to the total observed spectrum . rest frame : ( a ) on - resonance data ( open circles blue ) , scaled off - resonance data ( solid circles green ) ; the solid line shows the result of the fit to the non- events using both on- and off - resonance data ; ( b ) on - resonance data after subtraction of the fitted non- background ( triangles blue ) compared to simulated background that is adjusted by the combined fit to the on- and off - resonance data ( histogram ) ; ( c ) on - resonance data after subtraction of all backgrounds ( linear vertical scale , data points red ) , compared to the simulated @xmath50 signal spectrum ( histogram ) ; the error bars indicate errors from the fit , which include the uncertainties in the fitted scale factors for non- and @xmath103 backgrounds . the shaded area indicates the momentum interval for which the on - resonance data are combined into a single bin for the purpose of reducing the sensitivity of the fit to the shape of the signal spectrum in this region . , height=321 ] to determine the non- background we perform a @xmath104 fit to the off - resonance data in the momentum interval of 1.1 to 3.5 @xmath21and to on - resonance data in the momentum interval of 2.8 to 3.5 @xmath21 . since the c.m . energy for the off - resonance data is 0.4% lower than for the on - resonance data , we scale the electron momenta for the off - resonance data by the ratio of the c.m . energies . the relative normalization for the two data sets is @xmath105 where @xmath106 and @xmath107 refer to the c.m . energy squared and integrated luminosity of the two data sets . the statistical uncertainty of @xmath108 is determined by the number of detected @xmath43 pairs used for the measurement of the integrated luminosity ; the systematic error of the ratio is estimated to be @xmath109 . the @xmath104 for the fit to the non- events is defined as follows , @xmath110 here @xmath111 and @xmath112 refer to the number of selected events in the off- and on - resonance samples , for the @xmath113-th or @xmath114-th momentum bin ( @xmath115 ) , and @xmath116 is the set of free parameters of the fit . for the function approximating the momentum spectrum , we have chosen an exponential expression of the form @xmath117 the fit describes the data well : @xmath118 for 58 degrees of freedom . above 2.8 @xmath21 , we observe @xmath119 events in the on - resonance data , compared to the fitted number of @xmath120 events . the electron spectrum from @xmath12-meson decays is composed of several contributions , dominated by the various semileptonic decays . hadronic @xmath12 decays contribute mostly via hadron misidentification and secondary electrons from decays of @xmath49 , @xmath121 , and @xmath122 mesons . we estimate the total background by fitting the observed inclusive electron spectrum to the sum of the signal and individual background contributions . for the individual signal and background contributions , we use the mc simulated spectra , and treat their relative normalization factors as free parameters in the fit . the non- background is parameterized by the exponential functions @xmath123 , as described above . we expand the @xmath104 definition as follows , @xmath124 where the first sum is for the off - resonance data and the second sum for the on - resonance data . the electron spectrum is approximated as @xmath125 , where the free parameters @xmath126 are the correction factors to the mc default branching fractions for the six individual contributions @xmath127 representing the signal @xmath128 decays , the background @xmath129 , @xmath130 , @xmath131 , @xmath132 decays , and the sum of other background events with electrons from secondary decays or misidentified hadrons . @xmath133 is the statistical error of the number of simulated events in the @xmath114-th bin . the momentum spectra @xmath127 are histograms taken from mc simulations . lrrrrr @xmath134 ( @xmath21 ) & 2.0 2.6 & 2.1 2.6 & 2.2 2.6 & 2.3 2.6 & 2.4 2.6 + total sample & 609.81 @xmath135 0.78 & 295.76 @xmath135 0.54 & 133.59 @xmath135 0.37 & 65.48 @xmath135 0.26 & 35.38 @xmath135 0.19 + non- background & 142.38 @xmath135 0.63 & 105.20 @xmath135 0.48 & 74.86 @xmath135 0.36 & 50.13 @xmath135 0.25 & 29.96 @xmath135 0.16 + @xmath103 background & 416.22 @xmath135 2.52 & 157.17 @xmath135 1.29 & 38.82 @xmath135 0.47 & 4.00 @xmath135 0.10 & 0.09 @xmath135 0.01 + @xmath121 and @xmath122 & 6.17 @xmath135 0.14 & 4.00 @xmath135 0.10 & 2.33 @xmath135 0.06 & 1.17 @xmath135 0.04 & 0.47 @xmath135 0.02 + other @xmath136 background & 1.61 @xmath135 0.05 & 0.62 @xmath135 0.02 & 0.24 @xmath135 0.01 & 0.08 @xmath135 0.01 & 0.03 @xmath135 0.00 + @xmath137 mis - identification & 1.34 @xmath135 0.04 & 0.98 @xmath135 0.03 & 0.64 @xmath135 0.02 & 0.34 @xmath135 0.02 & 0.10 @xmath135 0.01 + @xmath138 mis - identification & 0.47 @xmath135 0.02 & 0.26 @xmath135 0.01 & 0.13 @xmath135 0.01 & 0.05 @xmath135 0.01 & 0.01 @xmath135 0.00 + other mis - identification & 0.27 @xmath135 0.01 & 0.15 @xmath135 0.01 & 0.08 @xmath135 0.01 & 0.04 @xmath135 0.01 & 0.02 @xmath135 0.00 + @xmath139 background & 1.62 @xmath135 0.10 & 0.66 @xmath135 0.05 & 0.20 @xmath135 0.02 & 0.03 @xmath135 0.01 & 0.01 @xmath135 0.00 + @xmath140 signal & 39.72 @xmath135 2.70 & 26.72 @xmath135 1.49 & 16.31 @xmath135 0.71 & 9.64 @xmath135 0.38 & 4.70 @xmath135 0.25 + @xmath139 efficiency ( % ) & 42.1 @xmath135 0.3 & 41.2 @xmath135 0.4 & 40.2 @xmath135 0.5 & 39.5 @xmath135 0.7 & 37.9 @xmath135 1.0 + the fit is performed simultaneously to the on- and off - resonance electron momentum spectra in the range from 1.1 to 3.5 @xmath21 , in bins of 50@xmath22 . the lower part of the spectrum determines the relative normalization of the various background contributions , allowing for an extrapolation into the endpoint region above 2.0 @xmath21 . to reduce a potential systematic bias from the assumed shape of the signal spectrum , we combine the on - resonance data for the interval from 2.1 to 2.8 @xmath21into a single bin . the lower limit of this bin is chosen so as to retain the sensitivity to the steeply falling background distributions , while containing a large fraction of the signal events in a region where the background is low . the fit results are insensitive to changes in this lower limit in the range of 2.0 to 2.2 @xmath21 . the number of signal events in a given momentum interval is taken as the excess of events above the fitted background . the observed spectra , the fitted non- and backgrounds and the signal are shown and compared to mc simulations in fig . [ fig : p1 ] . the fit has a @xmath104 of @xmath141 for 73 degrees of freedom . above 2.3 @xmath21 , the non- background is dominant , while at low momenta the semileptonic background dominates . contributions from hadron misidentification are small , varying from 6% to 4% as the electron momentum increases . the theoretical prediction for the signal @xmath50 spectrum based on the blnp calculations uses sf parameters extracted from the combined fit @xcite to the moments measured by the collaboration . the fitting procedure was chosen in recognition of the fact that currently the branching fractions for the individual @xmath6 decays are not well enough measured to perform an adequate background subtraction . the mc simulation takes into account the form factor and angular distributions for the @xmath142 and @xmath143 decays . for decays to higher - mass mesons , this information is not available . as a result , we do not consider this fit as a viable method of measuring these individual branching fractions . nevertheless , the fitted branching fractions agree reasonably well with the measured branching fractions @xcite . for the decays to higher - mass states , the ability of the fit to distinguish between decays to @xmath144 and @xmath145 is limited . the sum of the two contributions , however , agrees with current measurements @xcite . table [ table : r1 ] shows a summary of the data , principal backgrounds and the resulting signal . the errors are statistical , but for the non- and @xmath103 background they include the uncertainties of the fitted parameters . the data are shown for five overlapping signal regions , ranging in width from 600 to 200 @xmath22 . we choose 2.6 @xmath21 as the common upper limit of the signal regions because at higher momenta the signal contributions are very small compared to the non- background . as the lower limit is extended to 2.0 @xmath21 , the error on the background subtraction increases . a summary of the systematic errors is given in table [ table : t2 ] for five intervals in the electron momentum . the principal systematic errors originate from the event selection and the background subtraction . the uncertainty in the event simulation and its impact on the momentum dependence of the efficiencies for signal and background are the experimental limitations of the current analysis . the second largest source of uncertainties is the estimate of the background derived from the fit to the observed electron spectrum , primarily due to the uncertainties in the simulated momentum spectra of the various contributions . in addition , there are relatively small corrections to the momentum spectra due to variations in the beam energies , and radiative effects . . summary of the relative systematic errors ( % ) on the partial branching fraction measurements for @xmath50 decays , as a function of @xmath146 , the lower limit of the signal momentum range . the common upper limit is 2.6 @xmath21 . the sensitivity of the signal extraction to the uncertainties in the sf parameters is listed as an additional systematic error , separately for the four sets of sf parameters . [ cols="<,^,^,^,^,^",options="header " , ] the errors listed for @xmath147 and @xmath14 are specified as follows . the first error reflects the error on the measurement of @xmath148 , which includes statistical and experimental systematic uncertainties , except for the uncertainty in the sf parameters . the second error is due to experimental uncertainty of sf parameters affecting both @xmath149 and @xmath148 . the third error is the theoretical uncertainty of @xmath149 . the fourth error on @xmath14 accounts for the theoretical uncertainty in the translation from @xmath147 to @xmath14 , as specified in eq . 5 . this error also depends on the @xmath9-quark mass and thus is correlated with the theoretical uncertainty on the sf . the results for the total branching fraction @xmath150 and @xmath14 obtained from the different momentum intervals are consistent within the experimental and theoretical uncertainties . for intervals extending below 2.3@xmath21 , the total errors on @xmath147 and @xmath14 do not depend very strongly on the chosen momentum interval . while the errors on @xmath148 are smallest above the kinematic endpoint for @xmath36 decays , the dominant uncertainty arises from the determination of the fraction @xmath151 and increases substantially with higher momentum cut - offs . the stated theoretical errors on @xmath151 , acknowledged as being underestimated @xcite , do not include uncertainties from weak annihilation and other power - suppressed corrections . assuming that one can combine the experimental and theoretical errors in quadrature , the best measurement of the total branching fraction is obtained for the momentum interval @xmath2@xmath152 @xmath21 . though the measurement of the photon spectrum @xcite results in the best estimate for the sf parameters , we have also considered sets of sf parameters obtained from photon spectra measured by the cleo @xcite and belle @xcite collaborations . these parameters are listed in table [ table : sf_all ] . in table [ table : comparison ] the results obtained for these different sf parameters based on the semileptonic data and on the dn calculations are listed for the momentum interval @xmath153 . the differences between the sf parameters obtained by the cleo and belle collaborations and the results are comparable to the experimental errors on these parameters . these differences affect the signal spectrum , and thereby the fitted background yield . the effect is small for high momentum region and increases for the signal intervals extending to lower momenta . the impact of the sf parameters on the partial branching fractions is included in the total error ( see table [ table : t2 ] ) . lrll & @xmath154 ( @xmath155 ) & @xmath156 ( @xmath157 ) + ' '' '' & ( spectrum ) @xmath158 @xcite & @xmath159 & @xmath160 + ' '' '' cleo & ( spectrum ) @xmath158 @xcite & @xmath161 & @xmath162 + ' '' '' belle & ( spectrum ) @xmath158 @xcite & @xmath163 & @xmath164 + & @xmath154 ( @xmath155 ) & @xmath165 ( @xmath157 ) + ' '' '' & ( spectrum ) @xmath158 @xcite & @xmath166 & @xmath167 + ' '' '' & ( moments ) @xmath158 @xcite & @xmath168 & @xmath169 + ' '' '' & ( moments ) @xmath170 @xcite & @xmath171 & @xmath172 + ' '' '' & ( comb1 . moments ) @xcite & @xmath173 & @xmath174 + llccc experiment & sf input & @xmath175 & @xmath176 & @xmath177 + ' '' '' & @xmath158 ( spectrum ) & @xmath178 & @xmath179 & @xmath180 + ' '' '' cleo & @xmath158 ( spectrum ) & @xmath181 & @xmath182 & @xmath183 + ' '' '' belle & @xmath158 ( spectrum ) & @xmath184 & @xmath185 & @xmath186 + ' '' '' & @xmath158 ( spectrum ) & @xmath187 & @xmath188 & @xmath189 + ' '' '' & @xmath158 ( moments ) & @xmath190 & @xmath191 & @xmath192 + ' '' '' & @xmath170 ( moments ) & @xmath193 & @xmath194 & @xmath195 + ' '' '' & combined fit to moments & @xmath196 & @xmath197 & @xmath198 + for the four sets of sf parameters ( see table iv ) based on the calculations of blnp , extracted from the photon energy spectrum ( short dash - red ) and from the photon energy moments ( dot - dash - green ) in @xmath5 , from the lepton energy and hadron mass moments in @xmath36 decays ( long dash - black ) , as well as from the combined fit to moments ( solid - blue ) measured by the collaboration . also shown are two straight lines indicting values of the sf parameters , for which the partial branching fraction ( dotted - magenta ) and @xmath14 ( solid - light blue ) are constant . , height=283 ] the second method for extracting @xmath14 is based on recent blnp calculations @xcite . in this framework the partial branching fraction @xmath148 is related directly to @xmath14 : @xmath199 where @xmath200 is the prediction for the partial rate for @xmath39 decays ( in units of @xmath201 ) . in these calculations the leading order sf is constrained by the hqe parameters , obtained either from the @xmath5 or @xmath36 decays , or both . the values of the sf parameters extracted from the analyses of inclusive @xmath202 @xcite , @xmath203 @xcite decays , and the combined fit @xcite to all moments measured by the collaboration are listed in table [ table : sf_all ] . note that the definitions of shape functions and the sf parameters are different for the dn and blnp calculations . the different sf parameters and their measurement errors are also shown in fig . [ fig : p4 ] . the sf parameters based on @xmath5 data only are extracted from either a fit to the photon spectrum or to the first and second moments of this spectrum in the `` shape function '' scheme . the hqe parameters extracted from fits to measured moments in the kinetic mass scheme have been translated into the `` shape function '' scheme at the appropriate scale . specifically , the hqe parameters extracted from the moments in @xmath36 decays have been translated based on two - loop calculations @xcite . the hqe parameters resulting from the combined fit to moments of the photon , lepton , and hadron mass spectra in the kinetic scheme are used to predict the first and second moments of the photon spectrum down to photon energies of 1.6@xmath204 , based on calculations by benson , bigi , and uraltsev @xcite . the lower limit on the photon energy is chosen such that the estimated cut - induced perturbative and non - perturbative corrections to the hqe are negligible . from these predicted moments , the sf parameters are extracted using the next - to - leading order calculations in a framework that is consistent with the one used for the determination of @xmath14 @xcite . the smallest errors on the sf parameters are obtained from the fit to the photon spectrum and the combined fit to all moments . the fit to the photon spectrum is most sensitive to the high end of the photon energy spectrum , and relies on the theoretical prediction for the shape of the spectrum down to low photon energies . since this shape is not directly calculable , several forms of the sf are used to assess the uncertainty of this approach . the use of two sets of the first and second moments of the photon spectrum , above 1.90 and above 2.09 @xmath204 , is less powerful , due to much larger statistical and systematic errors , but insensitive to the theoretical knowledge of the detailed shape of the spectrum . the sf parameters obtained from moments of the photon spectrum above 1.90 @xmath21agree with those obtained from the global fit to the moments , but also have larger errors . nevertheless , the inclusion of the photon energy moments significantly improves the sensitivity of the global fit to more than 30 measured moments . the results for the partial branching fractions @xmath148 and @xmath14 based on the blnp calculations are listed in tables [ table : br2f_bbrxsg_direct ] , [ table : br2f_bbrxsgm_direct ] , [ table : br2f_bbrxcenu_direct ] , and [ table : br2f_cf_direct ] for the four sets of sf parameters . ccrc @xmath205 ( @xmath21 ) & @xmath175 & @xmath206 & @xmath177 + ' '' '' @xmath207 & @xmath208 & @xmath209 & @xmath210 + ' '' '' @xmath211 & @xmath212 & @xmath213 & @xmath214 + ' '' '' @xmath215 & @xmath216 & @xmath217 & @xmath218 + ' '' '' @xmath219 & @xmath220 & @xmath221 & @xmath222 + ' '' '' @xmath223 & @xmath224 & @xmath225 & @xmath226 + ccrc @xmath205 ( @xmath21 ) & @xmath175 & @xmath206 & @xmath177 + ' '' '' @xmath207 & @xmath227 & @xmath228 & @xmath229 + ' '' '' @xmath211 & @xmath230 & @xmath231 & @xmath232 + ' '' '' @xmath215 & @xmath233 & @xmath234 & @xmath235 + ' '' '' @xmath219 & @xmath236 & @xmath237 & @xmath238 + ' '' '' @xmath223 & @xmath224 & @xmath239 & @xmath240 + ccrc @xmath205 ( @xmath21 ) & @xmath175 & @xmath206 & @xmath241 + ' '' '' @xmath207 & @xmath242 & @xmath243 & @xmath244 + ' '' '' @xmath211 & @xmath245 & @xmath246 & @xmath247 + ' '' '' @xmath215 & @xmath248 & @xmath249 & @xmath250 + ' '' '' @xmath219 & @xmath236 & @xmath251 & @xmath252 + ' '' '' @xmath223 & @xmath224 & @xmath253 & @xmath254 + ccrc @xmath205 ( @xmath21 ) & @xmath175 & @xmath255 & @xmath177 + ' '' '' @xmath207 & @xmath256 & @xmath257 & @xmath258 + ' '' '' @xmath211 & @xmath259 & @xmath260 & @xmath261 + ' '' '' @xmath215 & @xmath262 & @xmath263 & @xmath264 + ' '' '' @xmath219 & @xmath236 & @xmath265 & @xmath266 + ' '' '' @xmath223 & @xmath224 & @xmath267 & @xmath268 + ccc @xmath205 ( @xmath21 ) & @xmath149 & @xmath176 + ' '' '' @xmath207 & @xmath269 & @xmath270 + ' '' '' @xmath211 & @xmath271 & @xmath272 + ' '' '' @xmath215 & @xmath273 & @xmath274 + ' '' '' @xmath219 & @xmath275 & @xmath276 + ' '' '' @xmath223 & @xmath277 & @xmath278 + the errors cited in these tables are defined and determined in analogy to those in table [ table : br2f_bbrxsgdfn ] . the first error on the predicted rate @xmath279 accounts for the uncertainty due to the errors in measured parameters of the leading sf , the second error refers to the theoretical uncertainties in the subleading sfs , and variations of scale matching , as well as weak annihilation effects . for @xmath14 , the first error is the experimental error on the partial branching fraction , which includes the statistical and the experimental systematic uncertainty , the second error includes systematic uncertainties on the partial branching fraction and @xmath279 due to the uncertainty of the sf parameters , and the third error is the theoretical uncertainty on @xmath279 , estimated using the prescription suggested by blnp . in table [ table : comparison ] the results obtained for these different sf parameters based on the semileptonic data and on the blnp ( and dn ) calculations are listed for the momentum interval @xmath153 . the observed differences are consistent with the total error stated ; they are largest for the sf parameters extracted from the fit to the photon spectrum as compared to the moments of the photon spectrum . for all four sets of sfs we observe a tendency for the total branching fraction , and therefore also @xmath14 , to be slightly larger at the higher momentum intervals , but the uncertainties in the predicted rates @xmath279 are very large for the highest momentum interval . based on the blnp calculations @xcite of the inclusive lepton spectra , we have also determined the total @xmath280 branching fraction . the results are presented in table [ table : br2f_cf ] . the results for @xmath14 extracted for the blnp calculations are close to those obtained for the dn calculations ( see table [ table : comparison ] ) . in fact , the results based on the fit to the photon spectrum measured by the collaboration are identical for all electron momentum ranges , even though the partial branching fractions differ by one standard deviation of the experimental error ( see tables iii and vi ) . changing the ansatz for the sf from the exponential to a hyperbolic function @xcite has no impact on the results . in summary , we have measured the inclusive electron spectrum in charmless semileptonic @xmath12 decays and derived partial branching fractions in five overlapping electron momentum intervals close to the kinematic endpoint . we have extracted the partial and total branching fractions and the magnitude of the ckm element @xmath14 based on two sets of calculations : the earlier ones by defazio and neubert @xcite and kagan and neubert @xcite , and the more comprehensive calculations by lange , neubert and paz @xcite , as summarized in table v. within the stated errors , the measurements in the different momentum intervals are consistent for both sets of calculations . we adopt the results based on the more recent calculations ( blnp ) @xcite , since they represent a more complete theoretical analysis of the full electron spectrum and relate the sf to the hqe parameters extracted from inclusive @xmath281 and @xmath282 decays . we choose the sf parameters obtained from the combined fit to moments of inclusive distributions measured by the collaboration rather than the single most precise measurement of the sf parameters obtained from the recent measurement @xcite of the semi - inclusive photon spectrum in @xmath281 decays . assuming it is valid to combine the experimental and the estimated theoretical errors in quadrature , and taking into account the fraction of the signal contained in this interval , we conclude that the best measurement can be extracted from the largest momentum interval , @xmath2 to @xmath3 . for this momentum interval the partial branching fraction is @xmath283 here the first error is statistical and the second is the total systematic error , as listed in table [ table : t2 ] . in addition to the systematic uncertainty due to the signal extraction , the normalization , and various small corrections , this error also includes the observed dependence of the extracted signal on the choice of the sf parameters . based on the blnp method , we obtain a total branching fraction of @xmath284 and @xmath285 here the first error represents the total experimental uncertainty , the second refers to the uncertainty in the sf parameters from the combined fit to moments , and the third combines the stated theoretical uncertainties in the extraction of @xmath14 , including uncertainties from the subleading sfs , weak annihilation effects , and various scale - matching uncertainties . no additional uncertainty due to the theoretical assumption of quark - hadron duality has been assigned . the improvement in precision compared to earlier analyses of the lepton spectrum near the kinematic endpoint can be attributed to improvements in experimental techniques , to higher statistics , and in particular , to improved background estimates , as well as significant advances in the theoretical understanding of the sfs and extraction of the sf parameters from inclusive spectra and moments . while earlier measurements were restricted to lepton energies close to the kinematic endpoint for @xmath6 decays at 2.3 @xmath21and covered only 10% of the @xmath286 spectrum , these and other more recent measurements have been extended to lower momenta , including about 25% of the spectrum , and thus have resulted in a significant reduction in the theoretical uncertainties on @xmath14 . the determination of @xmath14 is currently limited primarily by our knowledge of sf parameters . an approximate linear dependence of @xmath14 on these parameters is @xmath287 for @xmath288 and @xmath289 . thus the uncertainty on the @xmath9-quark mass dominates . it should be noted that this dependence on @xmath33 is a factor of two smaller for measurements based on the dn calculations . these results are in excellent agreement with earlier measurements of the inclusive lepton spectrum at the resonance , but their overall precision surpasses them @xcite . the earlier results were based on the dn calculations . we observe that for the same experimental input , i.e. the same measured lepton and photon spectra , the extracted values of @xmath14 based on dn calculations agree very well with those based on bnlp calculations for the various momentum ranges under study , even though the corresponding partial branching fractions may differ by one standard deviation . the results presented here are also comparable in precision to , and fully compatible with , inclusive measurements recently published by the @xcite and belle @xcite collaborations , based on two - dimensional distributions of lepton energy , the momentum transfer squared and the hadronic mass , with sf parameters extracted from @xmath5 and @xmath6 decays . we would like to thank the cleo and belle collaborations for providing detailed information on the extraction of the shape function parameters from the photon spectrum in @xmath290 transitions . we are also indebted to m. neubert and his co - authors b. lange , g. paz , and s. bosch for providing us with detailed information on their calculations . we are grateful for the extraordinary contributions of our 2 colleagues in achieving the excellent luminosity and machine conditions that have made this work possible . the success of this project also relies critically on the expertise and dedication of the computing organizations that support . the collaborating institutions wish to thank slac for its support and the kind hospitality extended to them . this work is supported by the us department of energy and national science foundation , the natural sciences and engineering research council ( canada ) , institute of high energy physics ( china ) , the commissariat lenergie atomique and institut national de physique nuclaire et de physique des particules ( france ) , the bundesministerium fr bildung und forschung and deutsche forschungsgemeinschaft ( germany ) , the istituto nazionale di fisica nucleare ( italy ) , the foundation for fundamental research on matter ( the netherlands ) , the research council of norway , the ministry of science and technology of the russian federation , and the particle physics and astronomy research council ( united kingdom ) . individuals have received support from conacyt ( mexico ) , the a. p. sloan foundation , the research corporation , and the alexander von humboldt foundation . m. shifman and m. voloshin , sov . j. nucl . phys . * 41 * , 120 ( 1985 ) ; j. chay , h. georgi , and b. grinstein , phys . b * 247 * , 399 ( 1990 ) ; i. i. bigi and n. uraltsev , phys . b * 280 * , 271 ( 1992 ) ; a. v. manohar and m. b. wise , phys . d * 49 * , 1310 ( 1994 ) ; b. blok , l. koyrakh , m. shifman and a. i. vainshtein , phys . rev . d * 49 * , 3356 ( 1994 ) . collaboration , b. aubert _ et al . _ , _ measurement of the inclusive electron spectrum in charmless semileptonic b decays , _ contributions to ichep02 , amsterdam ( 2002 ) , hep - ex/0207081 ; collaboration , b. aubert _ et al . _ , _ determination of the partial branching fraction for @xmath291 and of @xmath14 from the inclusive electron spectrum near the kinematic endpoint , _ contribution to ichep04 , beijing ( 2004 ) , hep - ex/0408075 . collaboration , b. aubert _ et al . _ , _ measurement of the partial branching fraction for inclusive charmless semileptonic @xmath12 decays and the extraction of @xmath14 _ , hep - ex/0507017 , contribution to the int . symposium of lepton - photon interactions , uppsala ( 2005 ) .

we present a measurement of the inclusive electron spectrum in @xmath0 decays near the kinematic limit for @xmath1 transitions , using a sample of 88 million pairs recorded by the detector at the resonance . partial branching fraction measurements are performed in five overlapping intervals of the electron momentum ; for the interval of @xmath2@xmath3 we obtain @xmath4 . combining this result with shape function parameters extracted from measurements of moments of the inclusive photon spectrum in @xmath5 decays and moments of the hadron mass and lepton energy spectra in @xmath6 decays we determine @xmath7 . here the first error represents the combined statistical and systematic experimental uncertainties of the partial branching fraction measurement , the second error refers to the uncertainty of the determination of the shape function parameters , and the third error is due to theoretical uncertainties in the qcd calculations . -pub-05/45 , + slac - pub-11499 +

because of its closeness , the sun is the best - observed and the only spatially - resolved x - ray star . it is therefore natural to consider it as a template and a guide to analyze and interpret what we observe of the other stars . in this perspective two previous works ( orlando et al . 2000 , peres et al . 2000a , hereafter paper i and paper ii , respectively ) illustrate a method to put solar x - ray data collected by the _ soft x - ray telescope _ on board yohkoh into the same format and framework as stellar x - ray data . the method allows us to simulate accurately the observation of the sun at stellar distances with a stellar instrument and to apply to the relevant data the same analysis as if they were real stellar data : we can compare homogeneously stellar to solar data and use the latter as a template for stellar observations . paper i focussed on the details of the treatment of yohkoh data and paper ii outlined the method in its generality and showed some representative applications to observations in different phases of the solar cycle plus one flare case . the present work applies this analysis to several solar flares in the perspective to interpret stellar flares and some features of very active stellar coronae . coronal flares are transient , x - ray bright and localized events : they last from few minutes to several hours , they easily overcome the luminosity of the whole solar corona and they occur in relatively small regions , often in single coronal magnetic structures ( loops ) . as such their occurrence and evolution is mostly independent of the structure and evolution of the rest of the corona . since the phenomenology , duration and intensity can be very different from one flare to another it is difficult to take a single flare as representative . in this work we analyze the emission measure distribution vs. temperature and its evolution during some selected solar flares , representative of the wide range of possible events ; as we did in paper ii we then use these em(t ) s to synthesize relevant stellar - like spectra , which are then analyzed with standard analysis tools of stellar coronal physics . together with the general problem of the structure and heating of the solar corona , flares represent an unsolved puzzle of solar physics . their early impulsive phase is so fast that the trigger mechanism remains elusive , in spite of the high quality of the data collected with many instruments , and in particular with the yohkoh / sxt , optimized to observe flares . apart from experimental limitations , intrinsic physical reasons make the diagnostics of the flare heating very difficult : the flare starts when a low density plasma is heated at more than @xmath5 k ; due to the low emission measure , initially the plasma is not easily observable while thermal conduction is so efficient that it smoothes out in a few seconds any trace of local thermal perturbation . the rise of the brightness in the soft x - ray band , where flares are best - observed , does not follow the evolution of the heating ( e.g. reale & peres 1995 ) . while analysing the rise phase is useful to investigate the mechanisms which originate the flares , the decay phase has been shown to be useful in other respects . it has long been known that the duration of the decay is linked to the size of the flaring region , because the thermal conduction cooling time depends on the length of the flaring loops . this has direct implications on stellar flares : from the observed light curve it has been possible to estimate the size of the unresolved stellar flaring loops . on the other hand it has also been shown that the estimates based on simple scalings from the conduction or radiation cooling times only can be largely incorrect : indeed , significant heating can be present ( jakimiec et al . 1992 ) and sustain the solar and stellar flare decays ( reale et al . 1997 , reale & micela 1998 , schmitt and favata 1999 , favata et al . 2000 , maggio et al . 2000 ) longer than expected , therefore leading to very coarse loop lengths overestimates . a useful tool to identify the presence of such a prolonged heating is the flare density - temperature diagram : the longer the heating decay ( compared to the free cooling loop decay time , serio et al . 1991 ) , the smaller is the slope of the decay in this diagram ( sylwester et al . 1993 ) . knowing the decaying light curve and the path in the density - temperature diagram leads us to obtain reliable estimates of the flaring loop length and of the heating in the decay phase ( reale et al . 1997 ) . the analysis of stellar flares have been often based on loop models ( e.g. reale et al . 1988 , van den oord et al . 1988 , van den oord & mewe 1989 , reale & micela 1998 ) . the present study , instead , uses solar flares really observed and spatially resolved by the yohkoh / sxt as templates : we will apply here the method illustrated in papers i and ii to obtain the evolving em(t ) and stellar - like flare spectra as observed by non - solar instruments , in particular the _ solid - state imaging spectrometer _ ( sis ) on board the _ advanced satellite for cosmology and astrophysics _ ( asca ) and the _ position sensitive proportional counter _ ( pspc ) on board the _ rntgen satellit _ ( rosat ) , which have been among the most used x - ray observatories for the observation of stellar coronae . it is interesting to apply the standard stellar spectra analysis to the resulting flare spectra and see analogies and differences from really observed stellar flares . the advantages of this approach are that : i ) it is relatively model - independent ( only spectral models have to be included ) ; ii ) it analyzes the whole flaring region including contributions from structures adjacent but outside the dominant ( if any ) loop , or , in the case of arcade flares , from many loops . spectra of several intense ( and therefore yielding good count statistics ) stellar flares collected by rosat ( e.g. preibisch et al . 1993 , ottman et al . 1996 ) and especially by asca ( e.g. gotthelf et al . 1994 , gdel et al . 1999 , osten et al . 2000 , hamaguchi et al . 2000 , tsuboi et al . 2000 ) have been analyzed in the recent past . in order to yield a count statistics appropriate for a sound analysis , each spectrum is typically integrated on time intervals spanning significant fractions of the flare , therefore averaging on rapidly evolving plasma and emission conditions . in spite of this , spectrum models of isothermal plasma ( 1-t models ) are generally able to describe each spectrum ( subtracted by the quiescent `` background '' spectrum ) with temperatures and emission measures following expected solar - like trends . the peak temperatures of analyzed stellar flares are quite higher , and their emission measures orders of magnitude higher ( @xmath6 and @xmath7 @xmath8 , respectively ) , than typical solar ones . deviations from the single temperature description have been also detected ( ottmann & schmitt 1996 , favata et al . 2000 ) . it is interesting in this context to investigate if there is correspondence with analogous spectra synthesized from solar flares which may then be used as templates to interpret stellar flare spectra . a debated question about stellar flare spectra is the fact that many of them are better fit by allowing the metal abundance to vary in the fitting model . the spectra are acceptably fit with metal abundance increasing in the rising phase and then decreasing gradually in the decay ( ottman et al . 1996 , favata et al . the physical meaning of this result is still far from being settled . wherever possible , it has also been shown that the abundance variations seem to change element by element ( osten et al . 2000 , gdel et al . 1999 ) . although the problem of abundance variations in stellar coronae is intriguing , far from being settled and addressed by several investigations with xmm - newton ( _ x - ray multi - mirror mission _ ) and chandra satellites ( e.g. evidence for an inverse first ionization potential effect has been found on the star hr1099 , brinkman et al . 2001 ) , the approach presented here does not allow to investigate it properly . what we can do is to explore the effect of allowing abundances to vary in fitting spectra originating from parent multi - thermal emission measure distributions . in this study beyond the extrapolation of the solar flares to the stellar environment as isolated , self - standing and evolving events , we will also focus on their possible contribution to make up the emission of very active coronae . in this respect , evidence has been collected from multi - line xuv observations that the emission measure of some active stellar coronae has two peaks , one at a few @xmath9 k and the other at @xmath10 k and it has been proposed that the higher temperature peak is due to a continuous flaring activity ( e.g. gdel 1997 , drake et al . 2000 ) . paper ii addressed , although limitedly , the evidence from extensive rosat observations that solar - like stars cover an extended region in the plane flux / hardness ratio ( schmitt 1997 ) , while the sun , as a whole , spans the low hr - low flux part of the same region . in this context it is worth investigating the region occupied by flares . this work is structured as follows : in section [ sec : data ] , we describe the solar data and our analysis for the derivation of the flare em(t ) s and of the stellar - like spectra ; section [ sec : results ] shows the results obtained for our sample of yohkoh / sxt flares , the related stellar - like spectra , collected with asca / sis and rosat / pspc , and their analysis with standard stellar methods ; in section [ sec : discuss ] we discuss our results and draw our conclusions . in order to study the em(t ) distribution of solar flares , we have selected a sample of flares well - observed for most of their duration by yohkoh / sxt and covering a wide range of flare intensities and physical conditions . in particular we have selected eight flares ranging from relatively weak ( class c5.8 ) to very intense flares ( x9.0 ) . the flares and their sxt observations are listed in table [ tab : fl_list ] , along with the date and time of the flare beginning as measured with the _ geosynchronous operational environmental satellite _ ( goes ) , the goes class , the duration of the flare as in the goes log file , and the start , maximum emission measure and end times of the sxt observations . the observations monitor large fractions of the rise , peak and decay of the thermal phase of the flares . the analyzed data include the sxt data taken in flare mode , and , in particular , in the two filter passbands specific for flare mode observations , i.e. be 119 @xmath11 m and al 11.4 @xmath11 m . as discussed in section [ sec : emt ] , additional data taken in two other softer filters , al.1 and almg , are included to analyze flare 4 . the sxt data have been processed according to the standard yohkoh analysis system . the datasets consist of sequences of frames @xmath12 pixels of 2.5 arcsec side , taken alternatively in the two filter bands , with sampling cadence usually ranging between @xmath13 sec ( typically in the rise phase and around the flare peak ) and @xmath14 sec in late decay phase of long - lasting flares . the analysis then includes the derivation of temperature and em maps , as in papers i and ii , during the flare evolution . since the plasma conditions may change significantly in the time taken to switch between the two filters , , the data in the al 11.4 @xmath11 m filter band have been interpolated to the exact times of the be 119 @xmath11 m data ( as done routinely in the standard yohkoh data analysis ) , in order to improve the accuracy of the temperature and emission measure estimates . the distributions of em vs t ( em(t ) ) are then derived from the t and em maps with the procedure illustrated in paper i and ii . in any of the two filter bands we have screened out saturated pixels , typically in the brightest regions , and pixels which collected photons below a threshold of 10 photons . these pixels would introduce large errors and are therefore critical in the analysis of em vs t for localized events like flares . we have carefully selected for our analysis only those frames with no ( or just very few ) saturated pixels . the temperature bins for the em(t ) are those defined in papers i and ii , i.e. 29 bins uniform in @xmath15 between log t = 5.5 and log t = 8 . from each of the several em@xmath16 distributions of a flare obtained during its evolution we synthesize the relevant x - ray spectrum with the mekal spectral code ( kaastra 1992 ; mewe et al . 1995 ) , and filter it through the instrumental response of non - solar telescopes of interest . the process to generate the stellar - like spectra from an em@xmath16 distribution derived from the solar data is described in paper ii . metal abundances are assumed as in anders & grevesse ( 1989 ) . in this paper , we consider the stellar - like focal plane spectra that _ asca_/sis and _ rosat_/pspc would collect , observing the solar flares selected here at stellar distances . it would be very interesting to perform finely time - resolved spectroscopy , but in real stellar observations this is prevented by the limited counting statistics of the source . analogously to stellar observations , therefore , the flare spectra have to be integrated on time intervals of hundreds of seconds to increase the overall statistics of the fitting process . this means that the flares are binned into few time intervals . since flares evolve on time scales smaller or of the same order , the data collected in a time bin do not represent steady plasma conditions but integrate on significant variations of the plasma temperature and density in the same dataset . therefore when time - resolved flare data are analyzed with steady - state models , one should keep anyhow in mind that this is an _ a priori _ limited representation and description of the data . in order to approach the typical conditions of non - solar flare observations we make the following exercise : 1 . we sample the em(t ) s of a flare at constant time intervals of 60 sec 2 . from each em(t ) we synthesize the corresponding spectrum filtered through the spectral response and effective area of two instruments , namely asca / sis and rosat / pspc 3 . we bin the flare into three ( or more ) long time segments , ( at least ) one including the rise phase , one the flare maximum and one the decay phase . the duration of the time segments varies within each flare and from flare to flare , depending on the flare duration and on the data structure ( the presence of gaps , for instance ) , and span from a minimum of 180 s to a maximum of @xmath17 s. 4 . each spectrum is normalized so as to yield a total number of counts in each time interval typical of good stellar flare observations ( e.g. between 1000 and 10000 counts for asca ) ; to this end we have to assume the distance at which the solar flare would yield the appropriate statistics . we randomize the photon counts according to poisson statistics 5 . we sum all folded and randomized spectra within each time bin and therefore obtain a single spectrum for each time bin . we analyze each spectrum with the standard thermal models used for stellar data analysis . we apply the standard stellar analysis to the binned spectra and use the tools commonly used by the stellar community . in particular , we fit the stellar - like spectra with multiple - isothermal components models . in the rosat / pspc case , we exclude channels with less than three counts from the analysis to grant an appropriate evaluation of @xmath18 , and the first two channels because they are typically affected by systematic errors . in the asca / sis case , the energy channels are grouped so to have at least 20 photon counts per channel ; furthermore the channels with bad quality or empty are discarded . we use the x - ray spectral fitting package xspec v10.0 and the manipulation task ftools v4.0 . most fittings are performed with single temperature components , assuming negligible column density @xmath19 , and standard metal abundances kept fixed . some checks have been done by allowing metal abundance to vary all by the same amount . all flares listed in table [ tab : fl_list ] last between 9 and 90 min , except the huge flare 8 , of class x9 , whose duration is more than 6 hours ( despite the goes log reports less than 1 hour ) . the sxt observation of flare 8 is divided into 3 segments separated by two gaps lasting more than one and two hours , respectively . the first segment begins just around the flare maximum , and therefore most of the flare rise phase is not covered by sxt . we have selected anyhow this flare because it is a rather extreme case of intense flare and the information from the rise phase is not crucial for our findings , as described below . in all the other cases the related sxt observation covers quite well both the rise and the decay phase and the flare maximum is well within the observation . only for flare 7 ( class x1.5 ) the decay phase is monitored for a relatively short time . as part of our analysis we investigate the role of the morphology of the flaring region in determining the em(t ) , and the effect of the heating release , and in particular of its intensity , duration and decay time , on the evolution of the em(t ) . in this perspective we tag each flare with the light curve in both filters of sxt flare - mode , its path in the density - temperature ( hereafter _ n - t _ ) diagram ( the square root of the emission measure is used as proxy for the density ) , and the main morphological features observable in the sxt images . [ fig : lcnt ] and fig . [ fig : lcnt2 ] show the light curves in the al 11.4 @xmath11 m and be 119 @xmath11 m filter bands obtained by summing all the counts in each 64x64 pixels sxt frame , and the corresponding n - t diagrams , obtained from the ratio of the light curves data points . notice that the resulting temperature is a weighted average temperature of the whole region in each frame . flare 2 is particularly well - covered since the total luminosity at the end has decayed to the values at the beginning ; this corresponds to a closed cycle in the n - t diagram . the n - t diagram also shows that this is the only flare in which the temperature changes significantly during the rise phase ( from @xmath20 to @xmath21 ) . this may indicate that the heating which triggers the flare is released more gradually in the rise phase of this flare than in the others ( e.g. sylwester et al . 1993 ) . during the other six flares for which the rise phase has been observed , the temperature is in fact much more constant and stays above @xmath5 k during the rise phase . as for the density , yohkoh / sxt has detected an increase of more than half a decade of em@xmath22 for five flares ( flare 2 , 4 , 5 , 6 and 7 ) ; during the decay of flare 8 the decrease of em@xmath22 is particularly significant , i.e. almost one order of magnitude . table [ tab : fl_par ] shows relevant physical and morphological characteristics of the selected flares obtained with the flare mode filters , i.e. the slope of the flare decay path in the @xmath23 diagram , the decay time of the light curve in the al 11.4 @xmath11 m filter band ( the latter two quantities are used to estimate the flaring loop length according to reale et al . 1997 ) , conservative loop half - length ranges obtained from measuring the loop projections on the images and from applying the method of reale et al . ( 1997 ) , the morphology of the flaring region ( fig . [ fig : frames ] shows one grey - scale image sampled during each flare ) , the maximum temperature and emission measure obtained with the filter ratio method from the ratio of the data points of the light curves in the two flare - mode filter bands . the slope @xmath24 is an indicator of sustained heating during the decay , whenever significantly smaller than @xmath25 ( sylwester et al . 1993 ) . from table [ tab : fl_par ] we see that : * heating is negligible during the decay of three flares ( 1 , 3 and 6 ) , significant during the decay of the other five and in particular of the most intense ones ( 7 and 8) . the slope @xmath24 of the whole decay of flare 8 is below the minimum possible value ( @xmath26 ) predicted by single loop hydrodynamic modeling ( reale et al . 1997 ) , which means that this model is not applicable in this case , and that , therefore , significant magnetic rearrangements and complicated and continued heating release probably occur . * the light curve e - folding decay time is below 30 min for all flares except flare 4 , an arcade flare , and flare 8 . * the loop half - length is in the range 10 mm and 100 mm , typical of active region loops is to be taken as an indicative scale size . the morphology of the flaring regions ranges from simple single loops to multiple loops , an arcade and even more complicated structures . * the maximum values of the average temperatures , as measured with yohkoh filter - band ratio , are in the range @xmath27 , weakly increasing with the goes class . * the maximum total emission measure increases with the flare goes class and spans two orders of magnitudes from @xmath28 to @xmath29 @xmath8 . the above considerations make us confident that this sample of flares is enough representative of flare conditions typically met on the sun and can be used to derive general properties to be compared to those observed in stellar flares . [ fig : emt ] shows the evolution of the em(t ) obtained with the two hardest sxt filters , during the eight selected flares . for each flare we show the emission measure distributions sampled at a constant rate of one every 2 min since the beginning of the data selected as in section [ sec : data ] . the longer the flare , the more are the em(t ) s shown : those shown for the short flares 1 and 7 are much fewer than those shown for flare 8 , by far the longest one . for reference , [ fig : emt ] shows also the em(t ) s obtained in paper ii for the sun near the maximum and the sun near the minimum of the cycle . [ fig : emt ] shows that at any flare phase the em(t ) is typically quite narrow , practically independent of the flaring region morphology , and covers a temperature decade around @xmath10 k. it is much narrower than any em(t ) of the whole sun . exceptions are the em(t ) s obtained for flare 8 , the most intense one , which shows significant amounts of hotter plasma , at temperatures up to @xmath30 k. the em(t ) reaches for this flare maximum values @xmath31 @xmath8 . hot plasma at temperatures above 30 mk is present in flares more intense than m1.1 ( flare 3 ) . in fig . [ fig : emt ] the em(t ) of all flares clearly follows a common evolution path : it starts low but already at a relatively high temperature , centered at @xmath10 k ; it grows toward higher em values , maintaining a more or less constant shape and sometimes shifting slightly rightwards to higher temperatures ( e.g. flares 2 , 4 and 6 ) ; then it decays by gradually cooling ( leftwards ) and decreasing ( downwards ) . we can clearly identify envelopes of the evolving em(t ) s and notice that the slope of the envelopes in the decay phase ( determined by the relative rate of the emission measure decrease and the temperature decrease ) is linked to the slope in the n - t diagram ( see fig . [ fig : lcnt ] and fig . [ fig : lcnt2 ] ) . several em(t ) distributions late during the decay are partially hidden by the higher preceding ones ( e.g. flares 3 , 4 and 5 ) . this clearly shows that the em(t ) mostly evolves inside a common envelope " in the rise and the decay phase , with important implications on the interpretation of some stellar coronal em(t ) s ( see section [ sec : discuss ] ) . the em(t ) s shown in fig . [ fig : emt ] are derived from observations in the hardest sxt filters bands . since these filters are most sensitive to plasma around and above @xmath5 k , one may wonder whether contributions from plasma at lower temperatures , not detected by the two filters , may be important to the flare em(t ) , or not . the observation of flare 4 includes several frames taken in the al.1 and almg filter bands and allows us to investigate this item . [ fig : emt_4 ] shows the em(t ) s of flare 4 at the same times as those shown in fig.[fig : emt ] , but including contributions to the em(t ) s derived from images in the softer filters bands mentioned above . cooler contributions to the em(t ) s are clearly present from comparison with fig.[fig : emt ] . these contributions make each em(t ) flatter at temperatures below the em(t ) maxima ( @xmath10 k ) ; the slope in this rising region approaches the @xmath32 trend expected from loop structures ( see peres et al . 2001 ) . although the presence of such contributions modify the shape of the em(t ) s in the low temperature part , they are anyhow significantly smaller than the dominant components around @xmath5 k , and we have checked that the analysis of stellar spectra synthesized from em(t ) s with and without such contributions does not change significantly , as further discussed in section [ sec : spectra ] . since very few flares yielded good data in softer filters bands ( saturated pixels are much more frequent ) , and contributions from such bands are anyhow affected by uncertainties and contaminations from the coexistence of cool and hot plasma within the same pixel , we prefer to perform the analysis considering expositions only in the two hardest filters and keeping anyhow in mind the limitations that this choice implies . all em(t ) s obtained for flares are well separated and distinct from the em(t ) of the sun at minimum . they are also mostly `` higher '' than the em(t ) of the sun at minimum . the em(t ) s of the first three flares are instead all lower than the em(t ) of the sun at maximum , while the em(t ) s of the remaining five flares are comparable or higher . if we did the exercise , similar to that in paper ii , to build a single em(t ) of the flaring sun at maximum by combining that of the sun at maximum and any one of the latter flares , we would invariably obtain an em(t ) with two distinct peaks , the hotter one being associated with the flare . flare 8 involves emission measures quite higher than the emission measure of the full non - flaring sun at maximum ( see paper ii ) . [ fig : maxemt ] shows the maxima of the em(t ) s of all of the eight flares . this figure indicates that an increasing intensity in the goes class mostly corresponds to an increase in the em(t ) height and much less to an increase in temperature and/or in the em(t ) width . an exception is flare 8 which is significantly hotter and wider . from each em(t ) obtained during the flares we can synthesize the corresponding x - ray parent spectra for an optically thin plasma in thermal equilibrium , as described in paper ii . [ fig : spectra ] shows examples of x - ray spectra in the 0.2 - 20 kev band , synthesized from em(t ) at the beginning , peak and end of flare 2 and at the peak and data end of flare 8 . the maximum luminosities in the whole band result to be @xmath33 erg / s and @xmath34 erg / s , respectively . from fig . [ fig : spectra ] we see concentrations of emission lines around 1 kev , typical of plasma mostly at temperature around 10@xmath35 k , and mostly due to the fe - l complex . the spectrum at the peak of flare 8 shows prominent lines of the fe group at energies around 6 kev , which are sensibly reduced at the end of the data ( lower panel , dashed line ) . notice that the continuum of the spectrum at the peak of flare 8 is considerably flatter than the others shown and indicates that there are contributions of plasma at significantly hotter temperatures than in all other synthesized spectra shown . [ fig : spectra ] shows asca / sis spectra of flare 4 . these spectra have been analyzed by fitting them with spectral models consisting of one isothermal plasma component ( _ 1-t fit _ ) . table [ tab : fl_asca ] shows the results of the spectral analysis applied to four of the eight selected flares ( 2 , 4 , 6 , and 8) . for each flare , the table includes the distance at which we put the flaring sun , and , for each time bin , the phase of the flare it covers , the time range , the number of counts , the best - fit temperature and emission measure , the number of degrees of freedom ( the number of energy channels yielding a significant number of counts minus the parameters of the fitting ) and the reduced @xmath18 of the fitting . table [ tab : fl_asca ] tells us , first of all , that the single thermal component model already provides an acceptable description of the various phases of the flares , in agreement to the rather peaked em(t ) distributions that we obtain from the two hard sxt filters , even averaging over each time bin . in spite of the fast flare evolution , temperature and emission measure variations _ within _ each time bin do not affect the 1-t fitting in the rise phase , mainly for two reasons : i ) the emission measure at the end of the bin in the rise phase is much higher than , and dominates over , that at the beginning ; ii ) generally the temperature does not vary much during the rise phase . conditions are quite stable during the flare maximum and a single temperature therefore describes reasonably well this phase . on the contrary , in spite of the relatively slow evolution , during the decay plasma temperature may vary significantly from the beginning to the end of the same time bin , while the emission measure remains relatively constant ( or slowly decreasing ) . the single temperature component may therefore fail to describe time bins of the decay phase , especially in flares with a steep slope in the n - t diagram . indeed in bins e ) and f ) in the decay phase of flare 6 , the fitting is not as good ( @xmath36 ) as in the other ones or in the other flare decays : the slope of the decay of this flare in the n - t diagram is very high ( @xmath37 ) , higher than in all the other flares . 1-t fitting of spectra taken with asca during stellar flare decays , and subtracted of the spectrum of the background non - flaring corona , have sometimes failed ( e.g. favata et al . 2000 ) , and our results suggest a possible explanation . the best - fit temperature values approximately correspond to the maxima of the em(t ) distribution averaged in the respective time bin and the corresponding emission measure values are approximately proportional to the total emission measure in that bin . the maximum emission measure values are slightly smaller than the values listed in table [ tab : fl_par ] , both because some contributions are excluded by the instrument limited spectral band and because in the respective time bins of table [ tab : fl_asca ] the emission measure is not constantly at maximum . notice that due to scaling with respect to the maximum , the last bin of flare 8 yields only 400 photon counts , since the emission measure is two orders of magnitude smaller than at maximum . the last section of table [ tab : fl_asca ] reports again results for flare 4 but including the components obtained from the softer filters ( see fig.[fig : emt_4 ] ) . comparing to the results without the softer filters , we notice the slightly higher count statistics , the slightly lower best - fit temperature , the slightly larger emission measure , the generally higher @xmath18 . the higher @xmath18 indicates that the single thermal model is not as good as previously , but not to such a point to discriminate the presence of the cooler components , also considering the higher count statistics . finally , we have checked the effect of fitting the simulated asca data with 1-t models in which the global metal abundance is left free to vary , from one spectrum to the other . the resulting best - fit abundance values are generally different from the expected solar value and vary within a factor two , either in excess or in defect , from the central unity value . in the course of a flare we obtain either abundances all smaller than one , or all larger than one , or fluctuating around one . although in a few cases the abundance appears to decrease during the decay , we were not able to identify clear systematic trends of abundance variations during the flare evolution . the best - fit @xmath18 values are improved by 0 - 10% , and in no case by more than 20% , while the temperature values are practically unchanged ( while the emission measure values vary in inverse proportion to the abundance values ) . the detection of such abundance variations are clearly an artifact of the model fitting process , since the synthesized spectra are consistently built by assuming fixed solar metal abundances at any step and can be explained in the following terms : the em(t ) flare distributions are neither sharp enough to be described at best by a simple @xmath38 function ( an isothermal model ) , nor broad enough to require a multi - thermal ( e.g. 2-t ) model . moderate abundance variations are sufficient to account , and adjust the fit , for the presence of minor emission measure components around the dominating maximum component . this exercise tells us that although the limited broadness of the flare em(t ) may favor an improvement of spectral fitting by letting abundance vary , the resulting abundance variations seem to be random ; therefore we can not exclude that systematic abundance variations are to be explained by effects other than the one illustrated here . rosat / pspc has a lower spectral resolution than asca / sis , and is , therefore , less able to discriminate multi - thermal components . in fact , we find that a single thermal component model describes even better the pspc spectra obtained from the flare em(t ) s averaged over the same time bins as those used for asca / sis . we have analysed spectra yielding a maximum between 3000 and 5000 total counts : all fittings were acceptable ( reduced @xmath39 with less than 30 degrees of freedom ) . the temperature and emission measure values are very similar ( with a mean deviation within 10% and 15% , respectively ) to those obtained with asca / sis spectral fitting . on the average , the rosat temperature is slightly systematically lower ( @xmath40% ) than the corresponding asca one , and the emission measure slightly higher ( @xmath41% ) . analogously to paper ii , as an additional piece of analysis of the synthesized rosat / pspc data , we compute the surface x - ray flux @xmath42 , defined as the x - ray luminosity in the rosat / pspc spectral band divided by the pixel area over which the yohkoh / sxt photon count is larger than 10 cts / s , and the hardness ratio defined as in schmitt ( 1997 ) : @xmath43 where @xmath44 are the total counts in the ( soft ) pspc sub - band 0.13 - 0.4 kev and @xmath45 are the total counts in the ( hard ) sub - band 0.55 - 1.95 kev . fig . [ fig : rosat_flux ] shows @xmath42 vs hr for each time bin in which a spectrum has been collected , and for all flares . all data points appear to lie in a relatively thin vertical strip around a hardness ratio value of 0.3 and with @xmath42 ranging from @xmath9 to @xmath46 erg @xmath3 s@xmath4 . the smallest flares evolve mostly in the range @xmath5 to @xmath47 erg @xmath3 s@xmath4 , while the most intense ones seem to move in a wider flux range . the flare evolution has no well defined trend in this plane , mostly because of the very limited variation of the hardness ratio . the almost constant hr is mostly due to two factors : i ) the flare average temperature , of which the hardness ratio is a proxy , does not change much from @xmath48 k , during the flare evolution ; ii ) the hardness ratio , as defined above , is weakly sensitive to temperature variations ( and even multi - valued ) around @xmath49 k. for comparison , fig . [ fig : rosat_flux ] shows the plane region occupied by late - type stars observed by rosat / pspc ( schmitt 1997 ) : solar flares ( isolated from the remaining corona ) , practically in any phase of their evolution , are harder than typical stellar coronal emission and in a region of high surface flux . papers i and ii illustrated a method to use the sun as a template of x - ray stars , converting yohkoh / sxt solar data into corresponding focal plane data collected by non - solar telescopes , such as rosat and asca , from a sun located at stellar distances . this work presents the application of this method to solar flares . in order to sample the various flare conditions , we have selected a set of flares spanning from weak to extremely intense and occurring either in simple loop structures , or in complex regions or arcades . the method is , in general , the same as that used to analyse the full - disk non - flaring observations . however some important differences are in order : * yohkoh sxt flare observations are performed with different characteristics , the so - called flare - mode , which involves mostly the use of two harder filters , double ( full ) spatial resolution and a much higher sampling cadence . * flares evolve on short time scales and the evolution of the emission measure distribution is a mostly important item of the present study . * flares are highly localized in areas of the order of 1/1000 of the solar hemisphere ; although their luminosity is often comparable to that of the whole corona , their evolution is mostly independent of what happens in the rest of the corona , and we study them as self - standing phenomena also from the stellar point of view . the two flare - mode filters are mostly sensitive to plasma at temperatures above @xmath5 k , appropriate for flares ; however their sensitivity to plasma below @xmath5 k is low , and temperature and emission measure diagnostics are not at best . since the plasma is typically stratified inside coronal structures , even during flares , we may then miss contributions from relatively cool plasma components which may be significant , or even dominant , or coexisting with hotter plasma , in some pixels during the flare evolution . the presence of such a cooler plasma has indeed been detected in a flare observed both with flaring mode filters and with standard mode filters ( flare 4 ) : the emission measure distribution during the evolution is not modified for @xmath50 k but additional ( lower ) contributions appear for @xmath51 k making the em(t ) less steep on the cool side of the maximum . this should be kept in mind when discussing the shape of the em(t ) distributions . the slope of the cool side approaches 3/2 , the typical value expected for standard loop structures in equilibrium ( see also peres et al . 2001 ) , coherently to flare evolution being dominated by plasma confined in closed magnetic structures and suggesting that , after the initial ( not bright ) impulsive phases , the dynamics is significantly less important and the flaring plasma is very close to equilibrium conditions . we notice that the shape and evolution of the em(t ) during flares little depend on the detailed geometry of the flare region , as well as on the flare intensity ( except , of course , for the relative em(t ) height ) : in fig.3 one can hardly distinguish flare 4 , an arcade flare , and flare 8 , very intense and complex , from the other flares with simpler geometry . flare 4 and flare 8 last longer than the others , and their temperature decreases very slowly , but these features are not evident in fig.3 . the temperature of the em(t ) maximum is also weakly dependent on the flare intensity : this may depend in part on the flare - mode filters used , which are more sensitive to plasma around that temperature , but also on the strong efficiency of thermal conduction ( @xmath52 ) at higher and higher temperature , which implies a very large energy input even for a small temperature increase , to balance conductive losses to chromosphere . significant amount of plasma at very high temperature ( @xmath53 k ) , comparable to the peak temperatures of intense stellar flares , is detected , in particular during flare 8 . such hot plasma , however , has an overall low emission measure with respect to the dominant relatively cooler plasma at @xmath48 k. our results show that minor components of emission measure hotter than the dominant component described by the single temperature fitting are practically not detectable in low resolution spectra collected by rosat and asca during stellar flares . high resolution spectra collected by chandra and newton may show them ( fig.6 ) . rosat and asca spectra synthesized from our em(t ) s , even integrated over long time segments and at relatively high count statistics , are well - fitted by single isothermal components , at or around the temperature of the em(t ) peak . even when we include additional cooler contributions obtained from the softer sxt filters , 1-t fit is successful , albeit at a slightly lower best - fitting temperature . deviations from isothermal behavior in rosat and asca synthesized spectra are sometimes obtained in the decay phase because of variations of plasma temperature within the same time segment . these results can be used to interpret observations of stellar flares : successful isothermal fittings of time - resolved spectra ( such as those mentioned in section [ sec : intro ] ) detect the dominant component of a multi - component but single - peaked emission measure distribution , and the distributions of the solar flares presented here may be taken for reference . high @xmath18 values during the decay may not be indicative of a multi - temperature distribution of the flaring region , rather of the evolving temperature of the dominant emission measure component . we also notice that the single component fittings , both of rosat and asca spectra , generally well evaluate the total emission measure involved , missing only few percent of the parent emission measure . therefore the emission measure values obtained from the fittings are rather reliable . our approach does not allow us to address exhaustively the problem of metal abundance variations obtained to best - fit the spectra of several stellar flares ; however we made the exercise to fit the stellar - like solar spectra letting the global metal abundance free to vary . we have found that non - solar abundances help to improve somewhat the fitting quality , probably because they better account for the limited broadness of the flare em(t ) s , but they do not seem to vary systematically ( e.g. increase first and then decrease in the decay ) during each flare . therefore , we can not exclude that the evidence of systematic abundance variations requires explanations different from the one suggested by our results . the analysis of the rosat / pspc flare simulated spectra tells us that the solar flares group in the x - ray flux vs hardness ratio diagram , and in particular in a narrow strip spanning two orders of magnitude at relatively high flux values and hardness ratio value between 0.2 and 0.5 . this strip is rightwards ( harder ) , in practice completely outside , of the region occupied by the g stars sampled by schmitt ( 1997 ) , which however pertain to whole coronae , outside flares . this is consistent with both flux and hardness ratio of non - flaring solar - like stellar coronae being well below conditions of typical solar flaring regions , which also agrees with the range of temperatures seen in non - flaring stars vs. stellar or solar flares . we now discuss some major implications of our results on the analysis of stellar coronae . there is evidence that the emission measure distribution of very active stellar coronae , obtained from spectrally resolved xuv observations , is double - peaked ; the first peak is at a few @xmath9 k , and the second peak above @xmath5 k ( griffiths and jordan 1998 ) . this aspect is much debated and still open , but it has been suggested that this hot component may be due to continuous flaring activity ( gdel 1997 , drake et al . 2000 ) : the stellar surface is covered by active regions , flares are very frequent and their light curves overlap , cancelling out any variability due to the single events . in this framework , the present work shows that a double - peaked em(t ) distribution is indeed obtained if one combines the em(t ) of the whole corona to the envelope of the em(t ) s during the flares ( see fig . the two peaks are clearly evident also when summing the em(t ) s of flares from 4 ( m2.8 ) to 8 ( x9 ) to the em(t ) of the sun at maximum of its cycle of activity . this seems to suggest that uninterrupted sequences of overlapping proper flares , whichever their evolution , would not be able to fill the gap between the two em(t ) peaks , which , therefore , would be a permanent feature of the em(t ) of an active star . indeed this second distinct maximum associated with flares is not surprising also on the basis of theoretical modeling . from hydrodynamic modeling of flaring plasma confined in magnetic loops , it has been generally found that the impulsive heating originating the flare first causes a very fast local temperature enhancement above @xmath5 k which propagates along the whole loop in few seconds due to the highly efficient ( even in saturated regimes ) thermal conduction . the loop plasma density ( and therefore emission measure ) increases more slowly and gradually , because determined by evaporation of plasma up from the chromosphere , on typical dynamic time scales ( minutes ) . later , when heating decreases , temperature and density both decay , their relative decay rate dictated by the rate of the heating decrease . _ models never predict that temperature does decrease and density does not _ , which would result into an emission measure distribution maintaining the bell - shape , with little or no change of shape and shifting from high to lower temperature , thus filling the gap between the two em(t ) peaks . this scenario would be at variance with both hydrodynamic models and observations as represented in the density- temperature diagrams shown in fig.1 : for all flares except flare 2 the temperature is high and the emission measure low ( upper left extremes of the paths ) already at the beginning of the observation . in the decay the path is never vertical ( i.e. maintaining a constant em and decreasing t ) , but has a maximum slope ( theoretically found to be less than 2 , jakimiec et al . all this means that the em(t ) during flares , whatever its exact shape , will : 1 . increase starting already from @xmath10 k 2 . cool but also invariably decrease , in the flare decay . the cooling may be much slower than the decrease , if the heating is sustained during the decay , as it seems to occur in many flares ( sylwester et al . 1993 , reale et al . 1997 ) . these considerations and results , in our opinion , show that a major continuous flaring activity on a stellar corona would produce an em(t ) with two distinct peaks , the higher temperature one at @xmath54 k ; the peak will be sharper if flares are heated during their decay . this work provides a key to interpret stellar flare x - ray data in terms of solar ones . in particular : it provides templates of stellar flare spectra and it tells us that the single thermal components which typically fit stellar flares low - resolution spectra are in agreement with single peaked , relatively sharp , emission measure distributions vs temperature . our simulations indicate that the best - fit temperature of flare spectra collected with rosat / pspc and asca / sis corresponds to the maximum of the flare em(t ) , and that the emission measure values obtained from the fitting well reproduce the total parent emission measures . this work also allows us to put solar flares in relationship with the surface emission of stellar coronae and explains why a continuous , moderate , flaring activity may produce a second peak in the emission measure distribution vs temperature . we expect more detailed information , such as the detection of minor emission measure components at very high temperatures , from high resolution spectra of stellar flares collected by chandra and xmm / newton . anders , e. , grevesse , n. , 1989 , geochimica et cosmochimica acta , 53 , 197 . brinkman , a.c . , et al . , 2001 , a&a , 365 , l324 . drake , j.j . , peres , g. , orlando , s. , laming , j.m . , maggio , a. , 2000 , apj in press favata , f. , reale , f. , micela , g. , sciortino , s. , maggio , a. , matsumoto , h. , 2000 , a&a , 353 , 987 . gotthelf , e.v . , jalota , l. , mukai , k. , white , n.e . , 1994 , apjl , 436 , l91 griffiths , n.w . , jordan , c. , 1998 , apj 497 , 883 gdel , m. , 1997 , apj , 480 , l121 . gdel , m. , linsky , j. l. , brown , a. , nagase , f. , 1999 , apj , 511 , 405 hamaguchi , k. , terada , h. , bamba , a. , koyama , k. , 2000 , apj , 532 , 1111 jakimiec , j. , sylwester , b. , sylwester , j. , et al . 1992 , a&a , 253 , 269 . kaastra , j. s. , an x - ray spectral code for optically thin plasmas , internal report , updated version 2.0 ( sron - leiden ) . maggio , a. , pallavicini , r. , reale , f. , tagliaferri , g. , 2000 , a&a , 356 , 627 . mewe , r. , kaastra , j. s. , liedahl , d.a . , 1995 , legacy , 6 , 16 . orlando , s. , peres , g. , reale , f. , 2000 , apj , 528 , 524 ( paper i ) osten , r. a. , brown , a. , ayres , t. r. , linsky , j. l. , drake , s. a. , gagn , m. , stern , r. a. , 2000 , apj , 544 , 953 ottmann , r. , schmitt , j.h.m.m . , 1996 , a&a , 307 , 813 peres , g. , orlando , s. , reale , f. , rosner , r. , hudson , h. , 2000a , apj , 528 , 537 ( paper ii ) peres , g. , orlando , s. , reale , f. , rosner , r. , 2001 , apj , submitted preibisch , t. , zinnecker , h. , schmitt , j.h.m.m . , 1993 , a&a , 279 , l33 reale , f. , betta , r. , peres , g. , serio , s. , mctiernan , j. , 1997 , a&a 325 , 782 reale , f. , micela , g. , 1998 , a&a , 334 , 1028 reale , f. , peres , g. , 1995 , a&a , 299 , 225 schmitt , j. h. m. m. , 1997 , a&a , 318 , 215 . schmitt , j. h. m. m. , favata , f. , 1999 , nat . , 401 , 44 serio , s. , reale , f. , jakimiec , j. , sylwester , b. , sylwester , j. , 1991 , a&a , 241 , 197 . sylwester , b. , sylwester , j. , serio , s. , et al . , 1993 , a&a , 267 , 586 . tsuboi , y. , imanishi , k. , koyama , k. , grosso , n. , montmerle , t. , 2000 , apj , 532 , 1089 van den oord , g.h.j . , mewe , r. , brinkman , a.c . , 1988 , a&a , 205 , 181 . van den oord , g.h.j . , mewe , r. , 1989 , a&a , 213 , 245 .

in previous works we have developed a method to convert solar x - ray data , collected with the yohkoh / sxt , into templates of stellar coronal observations . here we apply the method to several solar flares , for comparison with stellar x - ray flares . eight flares , from weak ( goes class c5.8 ) to very intense ones ( x9 ) are selected as representative of the flaring sun . the emission measure distribution vs. temperature , em(t ) , of the flaring regions is derived from yohkoh / sxt observations in the rise , peak and decay of the flares . the em(t ) is rather peaked and centered around @xmath0 k for most of the time . typically , it grows during the rise phase of the flare , and then it decreases and shifts toward lower temperatures during the decay , more slowly if there is sustained heating . the most intense flare we studied shows emission measure even at very high temperature ( @xmath1 k ) . time - resolved x - ray spectra both unfiltered and filtered through the instrumental responses of the non - solar instruments asca / sis and rosat / pspc are then derived . synthesized asca / sis and rosat / pspc spectra are generally well fitted with single thermal components at temperatures close to that of the em(t ) maximum , albeit two thermal components are needed to fit some flare decays . rosat / pspc spectra show that solar flares are in a two - orders of magnitude flux range ( @xmath2 erg @xmath3 s@xmath4 ) and a narrow pspc hardness ratio range , however higher than that of typical non - flaring solar - like stars .